<style type="text/css">#TLKHwrap { background: #f4f1ec; color: #555; width: 736px; font-family: Tahoma, Verdana, Segoe, sans-serif; } #TLKHwrap a { color: #0894dd; text-decoration: none; } #TLKHwrap a:hover { color: #4792de; } #TLKHbanner { background: url('/staff/HawaiianPig/knowhow/banner.jpg') no-repeat; height: 68px; padding: 24px 20px 25px 260px; text-align: right; overflow: hidden; } #TLKHbanner h1 { margin: 0px; padding: 0px; color: #000; font-size: 20px; font-family: "Franklin Gothic Medium", "Franklin Gothic", "ITC Franklin Gothic", Arial, sans-serif; } #TLKHbanner h2 { margin: 0px; padding: 0px; font-size: 11px; font-family: "Gill Sans", "Gill Sans MT", Calibri, sans-serif; text-transform: uppercase; letter-spacing: 2px; } #TLKHbody { padding: 0px 40px 40px 40px; } #TLKHbody h2 { font-size: 18pt; margin:24px 24px 10px 0px; display:inline-block; } #TLKHbody h3 { margin: 10px 0px 10px 0px; font-size: 18px; font-weight: bold; font-family: inherit; } #TLKHfooter { border-top: 1px solid #999; padding: 30px; font-size: 10px; } .border { border: 10px solid white; margin-bottom: 3px; } .resize { width: 300px; } #TLKHbody > .blockquote { background: #5f5a53; color: white; margin: 10px; border-radius: 10px; -webkit-border-radius: 10px; -moz-border-radius: 10px; padding: 20px; } #TLKHbody p { margin: -2px 0px 0px 0px; color: #555; } #TLKHbody li { font-size: 10pt; }</style><style type="text/css">#TLKHwrap table { font-size:0.9em; color: #555; } #TLKHwrap table td { color: #555; } #TLKHwrap table.ref { text-align:left; color: #555; } #TLKHwrap span { font-size:0.9em;} #TLKHwrap span.cl { float:left; width:580px; } #TLKHwrap span.cr { display:block; float: right; width:63px; text-align: right; color:#555; } #TLKHwrap span.cl2 { float:left; width:560px; padding-left:20px; } #TLKHwrap span.cl3 { float:left; width:580px; text-indent:40px; } #TLKHwrap span.section {font-size:0.8em;} #TLKHwrap .fig { display:block; margin-left:auto; margin-right:auto; padding: 0px 0px 0px 0px;} #TLKHwrap span.fig { text-align:center; font-style:italic; margin:25px 0px 25px 0px; font-size:10pt; } #TLKHwrap a.ext { font-style:italic; } #TLKHwrap th { text-align:left; } #msr { width:163px; float:left; vertical-align:top; } #TLKHwrap p {margin:0px 0px 0px 0px;} #TLKHwrap p.foot { font-size: 100%; font-style:italic;} #TLKHwrap h4 { margin: 0px; padding: 0px; font-size:14px;} #TLKHwrap .majorminortable { font-size:0.9em; color: #555; width: 630px; } </style><a name="top"></a><div id="TLKHwrap"><div id="TLKHbanner"><h1>A Music Theory Primer</h1><h2>Part I: Introductory Materials</h2><h2>author: wo1fwood</h2></div><div id="TLKHbody"> <a name="ii"></a><table><tr><td><span class="cl"><a href=#i>Introduction</a></span><span class="cr">i</span></td></tr><tr><td><span class="cl"><a href=#ii>Table of Contents</a></span><span class="cr">ii</span></td></tr><tr><td><span class="cl"><a href=#1.0>A Preliminary Discussion: Reading the Staff</a></span><span class="cr">1.0</span></td></tr><tr><td><span class="cl"><a href=#2.0>Western Fundamentals: An Introduction</a></span><span class="cr">2.0</span></td></tr><tr><td><span class="cl2"><a href=#2.1>Pitch</a></span><span class="cr">2.1</span></td></tr><tr><td><span class="cl2"><a href=#2.2>Scale</a></span><span class="cr">2.2</span></td></tr><tr><td><span class="cl2"><a href=#2.3>Key</a></span><span class="cr">2.3</span></td></tr><tr><td><span class="cl2"><a href=#2.4>Intervals</a></span><span class="cr">2.4</span></td></tr><tr><td><span class="cl2"><a href=#2.5>Rhythm and Meter</a></span><span class="cr">2.5</span></td></tr><tr><td><span class="cl3"><a href=#2.5.1>A Return to the Staff</a></span><span class="cr">2.5.1</span></td></tr><tr><td><span class="cl3"><a href=#2.5.2>Meter Signatures</a></span><span class="cr">2.5.2</span></td></tr><tr><td><span class="cl3"><a href=#2.5.3>Crashing the Party: Asymmetrical Meter</a></span><span class="cr">2.5.3</span></td></tr> <tr><td><span class="cl3"><a href=#2.5.4>Other Rhythmic Techniques</a></span><span class="cr">2.5.4</span></td></tr> <tr><td><span class="cl3"><a href=#2.5.5>Staff Notation: Beaming and Beat Divisions</a></span><span class="cr">2.5.5</span></td></tr> <tr><td><span class="cl"><a href=#3.0>References</a></span><span class="cr">3.0</span></td></tr> <tr><td><span class="cl"><a href=#4.0>Terms to Know</a></span><span class="cr">4.0</span></td></tr></table> <a name="i"></a><h2>i Introduction</h2> <span class="section"><p><em>"He is an artist. Before him is a scale of colors, and in his mind he approaches the reds. For his brush's immediate use he sees a carmine, a vermilion, a scarlet, a crimson, a cerise, a garnet, a ruby, and verging off into other color values are an orchid and a magenta, a nasturtium and an orange, and a sienna, a rust, and an ochre.</p> <p>He ponders leisurely. No, the exact shade he envisions - despite the great variety at hand - isn't here. With the assurance born of a life spent in being able to get what he wants, he then mixes - in just the right proportions - a bit of white, rust, and cerise to his vermilion, and - there! He has it!"</em></p> <p>I've started this article with a quote from the American composer Harry Partch as it is a imaginative description of the way in which composers approach writing music. From this description you can get a small sense of some of his feelings towards writing, and going about the process of creating a piece of music from the ground up. Today I would like to tackle at least part of this enormous topic of "how" music is constructed, through an introductory piece on music theory.</p> <p>This series of articles will be geared towards a basic understanding of theoretical topics, beginning with the most basic elements and then touching upon some of the advanced items to be aware of in order to gain a better understanding of how music functions and why (including some important historical considerations). This information should also be useful in understanding how to read music, whether it be the first time, or in gaining a broader perspective from what you might know already.</p> <p>The article itself will be pared down into <em>three major sections</em>, where section one will cover most of the necessary information you will need in gaining a basic understanding of musical considerations. The other two sections will consist of a discussion of the fundamentals and some of the advanced topics surrounding tonal harmony, and a look into how music outside the common practice period or popular music genres function, or how to approach them (in the hopes of a better understanding and appreciative perspective). Additionally, because the study of music theory is an extraordinarily complex topic and one that continually reveals layers upon layers within pieces of music, there will be a lot of advanced topics that I simply cannot cover in detail for parts 2 and 3, else this write-up could easily be over 800 pages long. These advanced topics and source materials will then be listed via external texts in the Further Materials section for each part, and will require much personal study in order to gain a full understanding of them.</p> <p>I'd also like to note that the topics covered here will mostly be skewed towards tonal music (also known as western music, or the western classical tradition) due to both the concentration of my studies, but for a number of other reasons as well, including an important trajectory (seen in the commercialization of music today) that has been globally shaping the landscape since around the time the phonograph was introduced.</p></span> <a name="1.0"></a><h2>1.0 A Preliminary Discussion: Reading the Staff</h2> <span class="section"><p>Before we get any further into any of these terms or items, we should discuss the staff and how musical ideas are communicated from composer to performer. It should be noted that this section will discuss terms and areas that will be covered in more detail later, so if a term is not defined here, it will be in subsequent sections.</p><h3>Staff</h3><p>The modern <a class="ext" href="http://en.wikipedia.org/wiki/Staff_(music)" target="_blank">staff</a> (pl. staves) is the main container where all the necessary information needed for a performance is normally contained. It defines the relationship between pitches horizontally and vertically, and of the rhythmic patterns and that occur in linear time. The staff itself, which has gone through many iterations over the centuries, today contains five horizontal lines and four spaces which, in conjunction with a <a class="ext" href="http://en.wikipedia.org/wiki/Clef" target="_blank">clef</a>, allows a musician to identify what pitches mean what visually, and the relationship these pitches have to one another.</p> <span class="fig"><img class="fig" src="/staff/wo1fwood/TLKH/1_1/1_1.png" alt="blank staff"/></span> <p>Additionally, certain instruments due to their wide ranges or how they are played sometimes require the use of more than one staff in order to be able to communicate what can be written effectively to the performer. A common usage can be found in piano notation where two staves are grouped together with the help of a visual symbol called a brace. This visual representation is commonly known as the grand staff (as it is also perfectly sequential in how notes are named vertically, see <em>Scale</em>). It is interesting to note here that the grand staff has roots in very early iterations of multi-lined staves before use of 5 line staves became the standard practice (more or less by the mid 16th century). Finally, in larger orchestral scores when writing where many instruments are playing together, it is common for another symbol called a bracket to group together instrument families, such as strings so that the score can be more easily read. These are advancements that both slowly developed over hundreds of years.</p> <span class="fig"><img class="fig" src="/staff/wo1fwood/TLKH/1_1/1_2.png" alt="grand staff and string family"/><br/>A grand staff, and a group of similar instruments.</span> <h3>Clef</h3><p>There are four to five common clefs. The most common clef is the treble clef, also referred to as the G clef, as it's identifying symbol centers around the note G (crosshair-like center). The F clef or the Bass clef as you would expect defines where F is on the staff. The C clef like the others, indicates where C is located on the staff. It should be noted that in modern day notation that the treble and bass clefs have been standardized so that they will always appear the same (fixed on a specific line on the staff). In music a couple hundred years ago this was not always the case as G could have been on the first line, or even the third. These kinds of odd clef uses are much more commonly seen in the <a class="ext" href="http://en.wikipedia.org/wiki/Baroque_music" target="_blank">Baroque</a> period and further back, as the notational system had not been entirely unified or standardized. The C clef is the only clef that has retained some of this behavior as a movable clef, and has essentially two variants as the alto and tenor clef, centered on either the third or fourth line of the staff (lines and spaces are counted from bottom to top as seen above). The final two clefs that you will most commonly see are the neutral clef which is most often used in percussion notation, and the tablature clef, which is used in guitar notation (though guitar has another tablature notation that is entirely different from this notation).</p> <span class="fig"><img class="fig" src="/staff/wo1fwood/TLKH/1_1/2.png" alt="clefs"/><br/>G clef (treble), C clef (alto/tenor), F clef (bass), neutral, and tablature notations.</span> <p>With the combination of the staff and a clef, you can now figure out which notes are which on each staff (see <em>Scale</em> for ordering of notes), though there are a few other additional symbols that help to identify what pitches mean.</p> <h3>Other Symbols</h3><p>Ledger lines are additional lines above and below the staff that help to track the pitches (readability) once they go beyond the confines of the staff itself. These lines extend the staff in both directions as if there were many more than just five lines to a staff, however these lines are much easier to read than if one simply added more staff lines (which used to actually be the case long ago). Note that for the grand staff that many ledger lines can be used in between the two staves, however if using the traditional treble and bass clefs, the first ledger line above the lower staff, and below the upper staff represent the same note (in this case C, shown below).</p> <span class="fig"><img class="fig" src="/staff/wo1fwood/TLKH/1_1/31.png" alt="ledger lines"/><br/>Ledger lines on a single staff, and grand staff.</span> <p>There are generally two ways to show octave designations, as a separate symbol (It. ottava or quindicesima sign), or as part of the clef. Separate octave symbols help to alleviate the problem of too many ledger lines (again, for readability) and always appear as 8va or 15ma (quindicesima, or at the 15th); they either indicate the pitches being one or two octaves above or below (bassa for below) what is written. Occasionally you'll find the symbols va, ma, vab, or mab in different contexts; these variations are either incorrect, or are only used when the ottava bassa sign cannot be placed below the staff visually (a practice unnecessary today). Similarly, clefs that have octave designations are essentially the same clef but indicate that the part is one octave higher or lower through an additional symbol. The most common instance of this is the Tenor clef in choral writing, though there are a number of others.</p> <span class="fig"><img class="fig" src="/staff/wo1fwood/TLKH/1_1//32.png" alt="octave symbols"/><br/>Octave symbols, and a tenor clef (vocal).</span> <p>The two other non-temporal items that help to identify pitches are the use of accidentals and key signatures. Accidentals are chromatic alterations of pitches (normally alterations of a half step or semitone) that can either be used for voice leading purposes, or are part of scales or key signatures. The most commonly used accidentals are sharps (♯), flats (♭), and naturals (♮), though double sharps (x) are often encountered as well. There are a number of other accidentals that have become more and more important as music had progressed, but we'll leave those for part three. The second most common use of accidentals are when they are grouped together at the beginning of each staff line into key signatures which will also be thoroughly covered a little later.</p> <span class="fig"><img class="fig" src="/staff/wo1fwood/TLKH/1_1/4.png" alt="accidentals"/><br/>Common accidentals, and a sample key signature.</span> <p>One other non-temporal and very important set of markings should be discussed before we continue into our western fundamentals, dynamics. Dynamic markings are Italian terms, often only used in shorthand, that describe the relative loudness of sounds being played or heard. They indicate to a performer, how loudly (forte) or softly (piano) specific notes or groups of notes should sound. Traditional markings range from pianississimo (ppp) to fortississimo (fff) though later composers such as Puccinni and Verdi are notorious for using dynamic markings ranging from pppppp-ffff in their scores. Along with these descriptions we also have another symbol called a crescendo or decrescendo that indicates a gradual increase or decrease in the loudness of the music being played. Traditionally they are represented below:</p> <span class="fig"><img class="fig" src="/staff/wo1fwood/TLKH/1_1/5.png" alt="dynamics"/></span> <p>The final piece to being able to read the staff begins the discussion into rhythm, and its constituent elements. This will be discussed after we get through our look into pitch-related materials.</p> <p class="foot">Note: all standard designations for <a class="ext" href="http://en.wikipedia.org/wiki/Glossary_of_musical_terminology" target="_blank">symbols, expressions, or terminology</a> are written in Italian (e.g. mezzo forte, ritardando, dal segno al fine, ottava, etc...) which is the standard convention, though regional languages are often used in imparting expression or tempo indications. This is done as a global standard so that a Chinese musician can still read and understand what a Finnish composer has written, though this practice it not strictly adhered to where expression or tempo markings are concerned.</p></span> <a name="2.0"></a><h2>2.0 Western Fundamentals: An Introduction</h2> <span class="section"><p><em>"Western music written during the Baroque, Classical, and Romantic periods (ca. 1650-1900) is called tonal music, or music of the common practice period. Compositions written during these three centuries have a point of gravitation, and explicit or implicit center around which all its pitches orbit. This phenomenon is called tonality, and the gravitational center-a single pitch (labeled using letters of the alphabet from A -G, plus various possible modifiers, including "flat", "sharp", "major", and "minor")-is called the tonic."
- Steven G. Laitz, The Complete Musician</em></p> <p>Laitz' description of tonality and the hierarchy of pitches draws out an important note about tonal music, or tonality in general. Tonality is defined by a single pitch that exhibits a centered quality where all other pitches are continually referring to this pitch, either implicitly or explicitly. The tonic is what gives a piece a sense of movement and goal oriented direction, as it is both the start and endpoint to musical structure (form, harmonic, and melodic processes), as well as a marked point of rest or resolution. Let us now look into the fundamentals surrounding pitch so that when we get to part two that we have a firm understanding of the basic terminology and a sense of how things are organized musically.</p> <a name="2.1"></a><h3>2.1 Pitch, Frequency, and Sound Waves</h3><p>Generally speaking, a pitch is a periodic <a class="ext" href="http://en.wikipedia.org/wiki/Waveform" target="_blank">waveform</a> that is generated by a physically vibrating mechanism. This could be anything from making a sound with your voice, to drawing a bow across a string of a violin, to striking a surface of some kind, or from pulsations of a loudspeaker. The speed or periodicity in which these vibrations/waves occur is what determines the pitch, and is referred to in acoustics as its frequency (most commonly measured in hertz). The shorter the periodicity (faster vibrations), the higher the frequency, and the higher the pitch; so to in the reverse. If for example one waveforms frequency is vibrating at twice the speed of another, they are what we call an octave apart, or vibrating at a 2:1 ratio. Note that pitch is the psychoacoustic definition of how frequencies are perceived by an individual, and is not necessarily reflective of what is happening with sound acoustically (an important consideration for understanding electronic music later). Also note that noise is characterized by its aperiodic nature, in that the waveform representing noise does not have a periodicity or regularly occuring pattern.</p> <p>When written down on a staff, pitches are referred to as <a class="ext" href="http://en.wikipedia.org/wiki/Note" target="_blank">notes</a> and are given the most commonly used (and standard) letter designations A, B, C, D, E, F, and G (in that sequential order). Other traditions have had variants such as H which for the austro-germanic tradition was B-natural, but because of standardized practices is no longer used in modern notation. Combined with these names, notes also have an octave designation that uses C as a base and changes at every subsequent C. As an example, the note C that oscillates at ~16Hz is identified as C0, while C8 is ~4100Hz. You will also see how each other note around C is organized in the image below. These differences in pitch as either low, medium, or high is known generally speaking as (the notes or pitches) register, or the relative height or range of these frequencies. </p> <span class="fig"><img class="fig" src="/staff/wo1fwood/TLKH/1_1/6.png" alt="piano range"/><br/>Notes between C4 and C5 (white keys only) are ordered thusly: D4, E4, F4, G4, A4, B4, C5, D5, E5, etc...</span> <p>Additionally, all pitches also belong to a pitch class. A pitch class refers to all pitches that share the same letter name, regardless of octave. As an example D3 and D6 are the same pitch class, even though they are separated by three octaves. One big difference between pitch and pitch class is that enharmonically equivalent pitches, or notes that are spelled differently but sound the same, are considered to be in the same pitch class (e.g. E♯, F and G♭♭ (double flat) are all one pitch class). This is an important distinction to keep in mind when we touch upon atonality and serialism in part three.</p> <a name="2.2"></a><h3>2.2 Scale</h3><p>So now that we know what a pitch/note is, how did we get from knowing this to having one of these frequencies more important than all the rest? This result occurs mainly because of natural acoustic properties and the analysis of those properties in the western tradition (discussed in part two, or in a different light <a class="ext" href="http://www.teamliquid.net/blogs/viewblog.php?id=397937" target="_blank">here</a>), but also because in tonality pitches are sequentially ordered (because of the above considerations) into what we call scales that create a hierarchy. In tonal music, the most basic scale consists of a collection of seven pitches that spans one octave, and is known as the diatonic scale (a scale that extends the tonic). Each pitch in the diatonic scale has a corresponding scale degree (1-7) and a designation which is as follows, beginning with the tonic: <ol><li>Tonic</li><li>Supertonic</li><li>Mediant</li><li>Subdominant</li><li>Dominant</li><li>Submediant</li><li>Leading Tone, or Subtonic (used situationally)</li></ol><p>Of course the names of these scale degrees are more than just generic descriptions, they also describe the function of each of these notes in the context of the scale, but that is something that we'll cover later when talking about goal oriented voice leading (part two).</p> <p>It should be noted that there are a number of scales beyond the most basic diatonic scales, which includes seven traditional modes, whole tone, pentatonic, octatonic scales, and many, many others. Each of these scales are unique because they exhibit a specific ordering of notes into whole and half step sequences (tones and semitones). We will however start with the two most important and widely used scales, major and minor.</p> <p>The major scale will always exhibit the same pattern of whole and half steps, in the following succession of W-W-H-W-W-W-H. Below you can see the C-major scale spanning one octave, beginning on C4 and ending on C5. The A-major scale is also shown here; note that in order to preserve the whole/half step ordering for A-major that accidentals (chromatic alterations of notes) need to be used. This would also be the case if the major scale began on B♭, except that flats would be needed.</p> <span class="fig"><img class="fig" src="/staff/wo1fwood/TLKH/1_1/7.png" alt="c/a scales"/></span> <p>The minor scale is somewhat different than the major scale in that it has a lowered third scale degree (mediant) and three variations to its pattern. The three patterns are natural minor, harmonic minor, and melodic minor. The natural minor pattern is the base from which harmonic and melodic minor scales are then altered. It should be noted that in the natural minor (and descending melodic) because the 7th scale degree is lowered it is not labeled the leading tone but the subtonic (because of its functional difference).</p> <ul><li>Natural minor has a lowered 6th and 7th scale degree and its overall pattern is W-H-W-W-H-W-W.<br/><span class="fig"><img class="fig" src="/staff/wo1fwood/TLKH/1_1/81.png" alt="natural minor"/></span></li><li>Harmonic minor has lowered 6th, but a raised 7th scale degree which retains the leading tone and creates the space of a tone and a half between the 6th and 7th scale degrees (also more commonly heard as a minor 3rd).<br/><span class="fig"><img class="fig" src="/staff/wo1fwood/TLKH/1_1/82.png" alt="natural minor"/></span></li><li>Melodic minor is different in that both scale degrees 6 and 7 are raised when in an ascending motion, but when in a descending motion revert to the natural minor.<br/><span class="fig"><img class="fig" src="/staff/wo1fwood/TLKH/1_1/83.png" alt="harmonic minor"/></span></li></ul> <p>Finally, in returning to our understanding of reading the staff, an important piece was initially left underdeveloped. Where are notes located on the staff? The answer is relative based upon the clef sign, but now that we've discussed the basic ordering of pitches in a scale and have a reference point from the clef, you can now find any note on any staff (including the more obscure ones). The two most common configurations for the treble and bass clefs are however:</p> <span class="fig"><img class="fig" src="/staff/wo1fwood/TLKH/1_1/9.png" alt="melodic minor"/><br/>Treble • Lines: EGBDF, Spaces: FACE Bass • Lines: GBDFA, Spaces: ACEG</span> <a name="2.3"></a><h3>2.3 Key</h3><p>In the notation of music, a key signature is used to convey the pitch classes of the major or natural minor scales to the performer. Key signatures are primarily used as a shorthand so that accidentals that would normally need to accompany notes to convey certain scales (such as A-major, shown above) do not have to be. The use of key signatures also serves to indicate the tonal center of a work, though some additional analysis is often needed to identify exactly what key (major/minor, modalities). Below you can see the convenience of using key signatures when certain scales use a large number of accidentals.</p> <span class="fig"><img class="fig" src="/staff/wo1fwood/TLKH/1_1/101.png" alt="g-flat no key"/><br/>G♭-Major without a key signature<br/><br/> <img class="fig" src="/staff/wo1fwood/TLKH/1_1/102.png" alt="g-flat with key"/><br/>G♭-Major with a key signature</span> <p>It should be noted that in modern music notation the use of key signatures in scores is rarely seen nowadays, largely due to the breakdown of tonality at the beginning of the last century, thought it is still commonly used in popular genres, movies, song, and video game music where tonality is still widely used or is the standard practice.</p> <p>Below is the collection of the major and minor keys, the number of accidentals accompanying each, and their relationship to one another. Note that in this chart each key that is next to one another vertically is separated by a perfect fifth (intervals discussed below). This is because adjacent keys (or closely related keys) follow a circle of fifths relationship, which is also expressed by the number of accidentals that accompany each key (this is due in part again, to our natural acoustic properties). Additionally in staff notation, you will notice that accidentals in key signatures also follow this circle of fifths relationship in how they are ordered visually (seen in the accidentals in key column). You can also see another relationship in that each major key has a corresponding minor key that shares the same key signature and vice versa. This relationship is known as relative keys. Parallel major/minor keys in contrast share the same tonic pitch, but do not share the same key signature. As an example, C-major, and c-minor are parallel keys of one another, though c-minor contains three flats in its key signature, while C-major has no accidentals.</p> <table class="majorminortable"><th>Major Key</th><th>Minor Key</th><th>Number of Accidentals</th><th>Accidentals in Key</th><tr><td>C♯</td> <td>a♯</td><td>7♯</td><td id="c5r">F♯ C♯ G♯ D♯ A♯ E♯ B♯</td></tr><tr><td>F♯</td> <td>d♯</td><td>6♯</td><td>F♯ C♯ G♯ D♯ A♯ E♯</td></tr><tr><td>B</td><td>g♯</td><td>5♯</td><td>F♯ C♯ G♯ D♯ A♯</td></tr><tr><td>E</td><td>c♯</td><td>4♯</td><td>F♯ C♯ G♯ D♯</td></tr><tr><td>A</td><td>f♯</td><td>3♯</td><td>F♯ C♯ G♯</td></tr><tr><td>D</td><td>b</td><td>2♯</td><td>F♯ C♯</td></tr><tr><td>G</td><td>e</td><td>1♯</td><td>F♯</td></tr><tr><td>C</td><td>a</td><td>-</td><td>-</td></tr><tr><td>F</td><td>d</td><td>1♭</td><td>B♭</td></tr><tr><td>B♭</td><td>g</td><td>2♭</td><td>B♭ E♭</td></tr><tr><td>E♭</td><td>c</td><td>3♭</td><td>B♭ E♭ A♭</td></tr><tr><td>A♭</td><td>f</td><td>4♭</td><td>B♭ E♭ Ab D♭</td></tr><tr><td>D♭</td><td>b♭</td><td>5♭</td><td>B♭ E♭ Ab D♭ G♭</td></tr><tr><td>G♭</td><td>e♭</td><td>6♭</td><td>B♭ E♭ Ab D♭ G♭ C♭</td></tr><tr><td>C♭</td><td>a♭</td><td>7♭</td><td>Bb E♭ Ab D♭ G♭ C♭ F♭</td></tr></table> <a name="2.4"></a><h3>2.4 Intervals</h3><p>Intervals are one of the last items that we need to define before we can begin to understand more complicated topics such as voice leading, counterpoint, and chordal analysis. An interval is simply the distance between two pitches. This distance uses ordinal numbers that represent the vertical distance between notes. For example the distance between C and G is a fifth (C, D, E, F, G). Intervals are also either simple or compound. Simple intervals are intervals that stay within a single octave while compound intervals are larger than this. A tenth for example (D3 to F4) is a compound interval. Furthermore, intervals while having a generic label (second, fifth, seventh, etc...), may also have a specific quality to them. These qualities are major, minor, perfect, augmented, and diminished.</p> <span class="fig"><img class="fig" src="/staff/wo1fwood/TLKH/1_1/11.png" alt="intervals"/></span> <p>Perfect intervals are comprised of unisons, fourths, fifths, and octaves. Major and minor intervals consist of seconds, thirds, sixths, and sevenths. All intervals can be augmented or diminished (except for the unison, which can only be augmented), though it should be noted that perfect intervals that when raised or lowered do not become major or minor, but augmented or diminished intervals. As an example, a Perfect 5th (P5) raised a semitone is an augmented 5th (a5), or diminished 5th (d5) if lowered by semitone. A Major 3rd (M3) however, if raised by one semitone becomes an a3, but must be lowered by two semitones (or a tone) to become diminished. The following is the hierarchy of perfect and imperfect intervals as each are raised or lowered by semitone:</p> <ul><li>Perfect Intervals (diminished > Perfect > augmented)</li><li>Imperfect Intervals (diminished > minor > Major > augmented )</li></ul> <p>Finally, intervals can be either consonant or dissonant (or, stable or unstable intervals). As you might suspect, perfect intervals are consonant, along with thirds and sixths, though these two are referred to as imperfect consonances (as they are not perfect intervals). Dissonant intervals are made up of the remaining intervals, seconds, and sevenths, as well as any augmented or diminished interval. I did not mention fourths because the fourth while having the designation 'perfect' is generally considered to be a dissonant interval, though whether this interval is classified as consonant or dissonant has much to do with the context of the interval itself (out of context and naturally speaking the 4th is an unstable interval).</p> <p class="foot">Throughout this write-up you may have begun to notice that when using these various musical symbols in designating key, or pitch, or intervallic content, that capitalization is important, as it reflects whether or not this content is Major, minor, Perfect, or otherwise. This is important later when we begin to talk about the quality of chords.</p> <a name="2.5"></a><h3>2.5 Rhythm and Meter</h3><p>Now at last we can finally start discussing linear time and how rhythm and meter play a role in the germination of music temporally in the western tradition.</p> <p>The very first thing we need to talk about is pulse. Pulses are undifferentiated clicks, taps, or punctuations in time that have a certain temporal duration to them (length), and can be accented (a process where stress is applied to the pulse to give it more emphasis) or unaccented. When pulses of the same duration are placed in succession they in essence punctuate the passage of time with a marked regularity. When an accented pulse is added into a group of pulses a periodicity forms, in that for every x pulses, one is accented or seems to draw more emphasis to it (there exists a centrality or weighted-ness to it). Because of this addition, we no longer call them pulses but accented or unaccented beats (or strong/weak beats, seen as | (accented/strong) and • (unaccented/weak)). Additionally, these beats begin to exhibit patterns, the most common and basic fundamental groupings being two and three, but we'll get to that in just a bit.</p> <a name="2.5.1"></a><h4>2.5.1 A Return to the Staff: Notes, Beat Divisions, Accents, and Tempo</h4> <p>Before we continue to talk about meter and meter signatures, we need to discuss how beats are represented on the staff, and how they are named.</p> <p>The standard unit of measure for rhythmic duration that arose early on was known as the Longa, and as music started to mature more and more this beat was subdivided in half into smaller and smaller parts, resulting in the Breve, Semibreve, Minima, Semiminima, Fusa, and Semifusa (the semibreve is 1/2 the rhythmic duration of the breve, the minima is 1/2 the duration of the semibreve, and so on and so forth as they rhythmic durations get smaller and smaller). The naming of these divisions of beats are derived from early <a class="ext" href="http://en.wikipedia.org/wiki/Mensural_notation" target="_blank">mensural notation</a> (the link also is a great visual to see how notation matured over the centuries). More recently however, these naming schemes were dropped for more modern systems. The two systems that developed over a number of hundreds of years are the European tradition, and the American tradition. The older European tradition as you would expect uses the older nomenclature that arose in the late <a class="ext" href="http://en.wikipedia.org/wiki/Renaissance_music" target="_blank">Renaissance</a> (later half of the 16th century), while the American system favors a different and more modern nomenclature (even more modern than the European tradition). As I was trained in the US I will be using the latter but I have included the older tradition as it is important to note some things historically, or so that those familiar with a different system can still orient themselves (note that many European countries also no longer use this nomenclature and have adopted this modern nomenclature).</p> <p>When the transition from mensural notation to more standard notational conventions began, the naming system was changed, and even some notes stopped being used over time (the Maxima and Longa for example), while other smaller subdivisions arose. Below is how each system relates to one another:</p>
<table><th>Mensural Notation</th><th>Older European Tradition</th><th>American System</th><th>Modern Notation</th><tr><td>Maxima</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Longa</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Breve</td><td>Breve</td><td>Double Whole Note</td><td><img src="/staff/wo1fwood/TLKH/1_1/dwn1.png" alt="double whole note"/> <img src="/staff/wo1fwood/TLKH/1_1/dwn2.png" alt="double whole note"/> <img src="/staff/wo1fwood/TLKH/1_1/dwr.png" alt="double whole rest"/></td></tr><tr><td>Semibreve</td><td>Semibreve</td><td>Whole Note</td><td><img src="/staff/wo1fwood/TLKH/1_1/wn.png" alt="whole note"/> <img src="/staff/wo1fwood/TLKH/1_1/wr.png" alt="whole rest"/></td></tr><tr><td>Minima</td><td>Minim</td><td>Half Note</td><td><img src="/staff/wo1fwood/TLKH/1_1/hn.png" alt="half note"/> <img src="/staff/wo1fwood/TLKH/1_1/hr.png" alt="half rest"/></td></tr><tr><td>Semiminima</td><td>Crotchet</td><td>Quarter Note</td><td><img src="/staff/wo1fwood/TLKH/1_1/qn.png" alt="quarter note"/> <img src="/staff/wo1fwood/TLKH/1_1/qr.png" alt="quarter rest"/></td></tr><tr><td>Fusa</td><td>Quaver</td><td>Eighth Note</td><td><img src="/staff/wo1fwood/TLKH/1_1/8n.png" alt="8th note"/> <img src="/staff/wo1fwood/TLKH/1_1/8r.png" alt="8th rest"/></td></tr><tr><td>Semifusa</td><td>Semiquaver</td><td>Sixteenth Note</td><td><img src="/staff/wo1fwood/TLKH/1_1/16n.png" alt="16th note"/> <img src="/staff/wo1fwood/TLKH/1_1/16r.png" alt="16th rest"/></td></tr><tr><td>-</td><td>Demisemiquaver</td><td>Thirty-second Note</td><td><img src="/staff/wo1fwood/TLKH/1_1/32n.png" alt="32nd note"/> <img src="/staff/wo1fwood/TLKH/1_1/32r.png" alt="32nd rest"/></td></tr><tr><td>-</td><td>Hemidemisemiquaver</td><td>Sixty-fourth Note</td><td><img src="/staff/wo1fwood/TLKH/1_1/64n.png" alt="64th note"/> <img src="/staff/wo1fwood/TLKH/1_1/64r.png" alt="64th rest"/></td></tr><tr><td>-</td><td>Semihemidemisemiquaver</td><td>One Hundred Twenty-eighth Note</td><td><img src="/staff/wo1fwood/TLKH/1_1/128n.png" alt="128th note"/> <img src="/staff/wo1fwood/TLKH/1_1/128r.png" alt="128th rest"/></td></tr></table> <p>In today's notation it is uncommon to see music that uses divisions past a 64th note, unless you are talking about contemporary or more complex pieces in which these divisions occur more regularly.</p> <p>So how do these rhythmic durations look? Both the notehead used, and the stem (and flag if it is present) are all used to indicate the approximate temporal value of each division. As an example an 8th note has a filled in notehead and a single flag attached to its stem; as the divisions get smaller and smaller, more and more flags are added. So a 16th note for instance, which is 1/2 the duration of an 8th note, has two flags. In the reverse, flags are removed, and past the quarter note, the stem and even noteheads change to indicate these durations. Note that flags can be drawn in two ways, with their traditional curved look, or in the shape of an angled rhomboid (a practice that became more common in the 1950's, but due to spacially notated music also is somewhat problematic).</p> <span class="fig"><img class="fig" src="/staff/wo1fwood/TLKH/1_1/12.png" alt="flags and stems"/></span> <p>Similarly to indicating when beats happen and when, it is important to identify periods of rest, where beats are not present. This is indicated by symbols known as rests and they follow the same subdivision properties of beats, but instead indicate periods of silence. Their modern notation can also be seen in the notation chart above, although the use of double whole notes/rests is somewhat rare today.</p> <p>One pertinent question might have surfaced in your mind at this point, how are groups of three beats or rests visually represented? Much like the use of flags to delineate rhythmic subdivisions, dots are used in conjunction with notes or rests to indicate half durations. As an example, a dotted quarter note is 1.5 times the value of a un-dotted quarter note, but if we subdivide the quarter note into two eighth notes, we can see that a dotted quarter also represents three eighth notes. The same applies to dotted rests. It should be noted that double dots and triple dots are used occasionally in the classical tradition, though not as often. In their operation, each subsequent dot is half the value from the previous dot, so expanding on our previous example, a double dotted quarter equals the value of 1.75 quarter notes, 3.5 8th notes, or 7 16th notes if we subdivide our rhythmic values again. In our talk about meter signature however this practice is unimportant.</p> <p>Another visual practice that is important to note is how musical stress is represented on the staff. While natural accents are present through a variety of musical elements (discussed briefly below), stressing certain notes or phrases can also be done via visual symbols that indicate the performer play that note louder, or with more stress applied to it than other notes. The most common accent symbols are tenuto, staccato, sforzando, and marcato though there are a number of variations to these accents.</p> <ul><li>Tenuto markings are represented by a line above or below the notehead and indicate a small amount of stress should be applied to the note.</li><li>Staccato markings in contrast are small dots above of below the notehead that indicates that these notes be played short. As music performance is wildly different from person to person, the length of these notes can be somewhat different, but in a general sense, a staccato note cuts the duration of that note in half (ex. a staccato 8th note will have the rhythmic duration of one 16th note).</li><li>The sforzando accent is between the stress of the tenuto and marcato markings and is the most predominantly used accent in music.</li><li>Marcato markings are the strongest type of accents and indicate notes be played with a very strong attack. It is also known as a forzando.</li></ul> <p>These accents or stress markings can also be combined to create other markings as well that have specific qualities to them. As an example, a combination of a staccato marking and marcato marking can indicate that the note be played with a short but very strong attack. Another example would be the tenuto-staccato that can be found in the music of Johannes Brahms (this symbol indicates a small stress is applied, but that each note be slightly separated). Additionally, these markings have a direct relationship to certain dynamic markings (sforzando and forzato) that we discussed in our initial look at the staff. They are indicated below.</p> <span class="fig"><img class="fig" src="/staff/wo1fwood/TLKH/1_1/13.png" alt="accents"/></span> <p>One other question may have also crossed your mind as well, how does one know how long the duration of these notes are? Long ago these indications were passed much in the folk manner, in a wrote fashion. As music matured however composers began to use expressive markings and text to indicate the speed or tempo, and eventually once metronomes and a reliable way to track periodicities over time emerged, numbers began to enter the fray. So what is a tempo marking? Today you will primarily find tempo markings indicating what the beat represents over the span of one minute (BPM or beats per minute) and are primarily seen as a number along with an indication written in Italian. If I notated for example that a piece was to be played where the eighth note equals 100bpm (♪=100), this would indicate that there are 100 eighth notes in the span of one minute, and would allow the performer to play at the appropriate tempo.</p> <span class="fig"><img class="fig" src="/staff/wo1fwood/TLKH/1_1/14.png" alt="tempo marking"/></span> <p>In the opposite manner, there is a specific symbol that temporarily suspends the tempo of a piece, and these are called fermatas. Simply, a note or rest that has a fermata is held for a undisclosed amount of time, that is to say, often it's up to the performer to determine just how long these fermatas are. However, in modern notation often an indication in seconds accompanies these symbols.</p> <span class="fig"><img class="fig" src="/staff/wo1fwood/TLKH/1_1/15.png" alt="fermatas"/></span> <p>Let us now return to our discussion of pulse, beats, and meter.</p> <a name="2.5.2"></a><h4>2.5.2 Meter Signatures</h4> <p>In returning to our organization of pulse, we remember that once accents are applied to pulses they become beats. The next step is to organize simple beat groupings into regularly occurring patterns of two and three. We call this organization Meter. Meter can either be duple, triple, or quadruple, each having their own ordering of strong and weak beats. Duple and triple are divided thusly:</p> <ul><li>Duple: strong-weak | • </li><li>Triple: strong-weak-weak | • •</li></ul> <p>Quadruple is slightly different from the other two in that there are two layers of accented beats. The first group of two beats consists of a very-strong beat, and then a weak beat, followed by a strong beat, and another weak beat. By adding this very-strong beat into the mix, there is a second (and more global) level of accent that makes quadruple have both a four beat and two beat emphasis contained in its overall pattern:</p><ul><li>Quadruple: very strong-weak, strong-weak <strong>|</strong> • | •</li></ul> <p>The last organizational pattern that we need to introduce is how the beat unit itself is divided (its subdivision) in a meter. This as you might expect is either in groups of two or three (ex. two 8th notes = 1 beat vs three 8th notes = 1 beat) and determine whether the meter is simple or compound. In simple meter, beat units are divided into two parts, while in compound meter they are divided into three.</p> <p>Confused yet? Let's review how we started with a pulse, and ended with a meter:</p> <ol><li>We begin with undifferentiated pulses of the same duration: • •</li><li>An accent is applied to the first pulse, now we have an accented beat and an unaccented beat: | •</li><li>Grouping this same pattern in two, and making the fist accent very strong, we now have constructed what is known as a <em>simple quadruple meter</em>: <strong>|</strong> • | •</li></ol> <p>What may not be immediately apparent about the very strong accent is that it exhibits its pull not entirely by it being more accentuated (or stressed), but by various other qualities that center the temporality of the meter towards this point. These nontemporal factors could be the contour of the musical line, the articulation of the note or phrase, the dynamic marking, the harmonic emphasis, pitch and that pitch's register, or from textural choices made by the composer. All of these other elements help to center the rhythmic pattern around this central point.</p> <p>So now that we know how a beat can be represented visually, have an understanding of the basic groupings, and know how a beat can be subdivided into twos or threes, how is this information then imparted to the performer? They are written down as a meter signature, more colloquially known as a time signature. These meter signatures as briefly mentioned before can either be simple or compound in their nature (where the beat unit divides into two or three parts). They have appeared in various forms over the years, and especially with the advent of the 20th century became highly varied in both how they were constructed and what they looked like, but generally speaking a meter signature is divided into two numbers aligned vertically. Each of these numbers impart an important piece to understanding what the meter signature means.</p> <span class="fig"><img class="fig" src="/staff/wo1fwood/TLKH/1_1/16.png" alt="3/8 meter signature"/><br/>An example time signature.</span> <p class="foot">I'm going to approach this somewhat differently than most texts or descriptions do, because I feel that most textbooks are very limiting in their view of meter signatures, or go about them in a very convoluted manner, including the Laitz that I have been referencing.</p> <p>When looking at a meter signature, the top number indicates how many of the bottom number (and its corresponding rhythmic value) are contained within a single measure (described below) of that meter signature. Using the above meter signature as an example, in 3/8 there are a total of three 8th note beats contained in one measure. Simple. Easy. Another example and far more complex would be to look at a meter signature of 12/16, which is not often used. With our current knowledge we can easily figure out that there will be 12 16th note beats in this meter signature. This is where things begin to become more complicated however. How does one figure out when a meter is simple or compound so that we can figure out what the rhythmic value of one beat unit is?</p> <ul><li>For simple meters, remember that the beat unit is always divided into two equal parts. Because of this, we can see that our time signature of 3/8 is in fact a simple meter (in this case simple triple as there are three beats in it).</li><li>For compound meters, remember that beats are made up of three equal parts. If we were to look at our 12/16 example you could group by twos and come up with 6 beat groupings in that time signature. If you remember however, traditional meters are always duple, triple, or quadruple. Having 6 beat groupings then is very out of place, so if we go back and group by threes, we can see that in fact there are four groupings of three 16th notes in our 12/16 bar. This allows us then to identify this meter as compound quadruple. Another more common example would be a time signature of 6/8. By grouping eighth notes into groupings of three we can see two groups form, and can thusly identify this as a compound duple meter.</li></ul> <p>Unfortunately two questions from these examples arise around this. For the 12/16 bar, couldn't you group those six groupings into two 3s or three 2s? If that were done, then our bottom number would have to be 8 and not 16 (we would have created a double grouping, in essence we would have subdivided our subdivided beat, or simply subdivided our beat twice). The 6/8 bar however presents a more obvious problem and if you were paying attention, something might have jumped out at you...Couldn't a composer divide the measure however they wanted, like in 3 groups of two? The answer is a rather complicated <em>yes</em>, but this is something that will be covered a little later (metrical disturbances below, and in part three) as a large portion of this flies in the face of hundreds of years of tradition, and would be somewhat confusing to talk about right now. In order to be concise however and return to the conventions of tonal harmony, the following chart shows many common simple and compound meters prevalently used, and their beat groupings:</p> <table><tr><th>Simple Meters</th><th></th></tr> <tr><td id="msr">Top Number:</td><td>2 = two beats in a measure 3 = three beats in a measure 4 = four beats in a measure</td></tr> <tr><td id="msr">Bottom Number:</td><td>2 = one half note makes up the beat unit 4 = one quarter note makes up the beat unit 8 = one eighth note makes up the beat unit 16 = one sixteenth note makes up the beat unit</td></tr></table> <table><tr><th>Compound Meters</th><th></th></tr> <tr><td id="msr">Top Number:</td><td>6 = two dotted beats in a measure 9 = three dotted beats in a measure 12 = four dotted beats in a measure</td></tr> <tr><td id="msr">Bottom Number:</td><td>2 = three half notes makes up the beat unit (dotted whole note) 4 = three quarter notes makes up the beat unit (dotted half note) 8 = three eighth notes makes up the beat unit (dotted quarter note) 16 = three sixteenth notes makes up the beat unit (dotted eighth note)</td></tr></table> <p>I should briefly note that there are two special symbols of meter signature that are quite commonly used. Because of the prevalent nature of the 4/4 meter signature, pieces began to colloquially be known as written in 'common time'. Similarly to this we also see a similar use or meters in 'cut-common time' or just 'cut time' (essentially 2/2 to be played twice as fast). While commonly used, these meter signatures should not be used as they are both redundant symbols to already existing meter signatures, but also in that 4/4 is increasingly not the commonly used meter signature, especially in contemporary classical music.</p> <p>The last piece of information surrounding meter signatures that I've mentioned quite a bit, but have yet to define is beaming and measures. In parsing out all this information and returning to our beat groupings, if we combined a very strong-weak, strong-weak grouping three times, we would see a strong-weak pattern which would look like this, <strong>|</strong> • | • <strong>|</strong> • | • <strong>|</strong> • | •, or if on the modern staff:</p> <span class="fig"><img class="fig" src="/staff/wo1fwood/TLKH/1_1/171.png" alt="4/4 no beams"/></span> <p><em>Note: Noteheads that are visually represented by an x are most commonly used to indicate rhythm, but with the absence of pitch.</em></p> <p>This is somewhat problematic however as there is no visual indication of our recurring pattern. To solve this, meter signatures are visually separated in to measures or bars, where one measure contains the rhythmic duration of a single meter signature grouping (example: one measure of 4/4 contains four quarter notes). Compared with the image above, the additions of beams (the vertical lines that separate measures, also called barlines) makes the periodicity and repetitive nature of time signatures visually apparent, and we gain a much better sense of how these patterns over time look to the performer.</p> <span class="fig"><img class="fig" src="/staff/wo1fwood/TLKH/1_1/172.png" alt="4/4 with beams"/></span> <p>Additionally to these beams, there are also a few <a class="ext" href="http://en.wikipedia.org/wiki/Bar_(music)" target="_blank">special types</a> that indicate certain important markings such as the end of a piece of music (known colloquially as the double bar or final bar), at tempo changes, or to indicate the beginning or ending of repeated sections (a process where a set of measures within these repeat signs are played one or more times).</p> <span class="fig"><img class="fig" src="/staff/wo1fwood/TLKH/1_1/18.png" alt="special beams"/><br/>Tempo change, beginning and end repeats, and final bar.</span> <p>There are also special types of measures that can be used in music notation, though most of these are used from the 20th century onward and will be discussed later. There is one however that you will often encounter, and that is an incomplete measure. An incomplete measure is a type of unaccented musical event (called an anacrusis) that leads to an accented musical event. Another way to look at this is looking at our beat. Knowing that the very beginning of a measure is a strong temporal accent, we actually have a name given to that first beat, called the downbeat, its name having its roots in conducting nomenclature (where a conductor moves his hand downward in a specific way to indicate the beginning of a measure). In these measures only part of the meter signatures duration is used. This type of measure can only be found at the beginning or ends of the piece or movement as they lead into full measures, or at the end of the piece complete that first incomplete measure (by using the remainder of beast that were not initially used).</p> <span class="fig"><img class="fig" src="/staff/wo1fwood/TLKH/1_1/19.png" alt="incomplete measure"/><br/>An incomplete measure at the beginning of a piece.</span> <p>One more important note should be made about measures. When no rhythms or notes are present in a measure and it is blank or empty, the whole rest is used to indicate this, but is instead labeled as a measure rest to convey this. In this instance this measure rest does not necessarily represent four quarter notes in duration, but the total duration of the measure. As an example, in 9/8 a measure rest has the total duration of one whole note and one 8th note, or if going by beat groups three dotted quarter notes.</p> <a name="2.5.3"></a><h4>2.5.3 Crashing the Party: Asymmetrical Meter</h4> <p>Repetitive and equally divided beat groupings in the western tradition were and at least in tonally-oriented music are still quite prevalent today, however there is one more configuration of meter signature that we have yet to talk about. Asymmetrical meters. On their own and in folk settings they have existed for centuries, though it wasn't until around the turn of the century that these asymmetrical meters started to be used with more regularity in western music.</p> <p>These meters can exhibit their asymmetricality in one of two ways, by the number of beats in the measure, and how beat groupings exist. In the latter example, if we began with the meter signature 4/8 (a simple quadruple meter), but say we were to add one 8th note to this meter, now we have a time signature of 5/8 and our nice grouping of 2+2 is disturbed. So how do we divide the measure? Remember how we mentioned that groups of two and three are the most basic units? Well the simple solution is to make one beat unit two 8th notes, and the other three. This then again provokes the question we previously asked about dividing meter however you want. In this case the beat grouping of 3 can be either at the beginning or at the end of a measure, however the composer chooses to do so. </p> <p>This asymmetrical beat grouping also highlights a special type of stressing in music, called an agogic accent. Simply put an agogic accent is just like any other accent (in that it stresses the material in some way), but this stress is due to the length of the note, or in this case beat unit. Next to our grouping of two 8th notes, our grouping of three has a naturally stronger accent applied to it as it occupies more linear time than the other. Knowing this you can see a typical pattern arise in asymmetrical meters, in that they have this natural tendency to stress the number of 2s and 3s that make up the meter. As an example, a 11/8 meter can be ordered in a number of different ways, but is most commonly grouped into divisions of 2+2+3+2+2. This grouping specifically is found quite commonly in the folk music of eastern European countries such as Bulgaria, or others around the <a href="http://en.wikipedia.org/wiki/Georgia_(country)" target="_blank"><em>Georgia Caucausus</em></a>.</p> <p>The other asymmetrical grouping is when meters have a single strong beat and all of the others are weak. For example, a measure of 7/4 could be broken up into groups of 2+2+3, but often times in more contemporary music, you will see that the beat group is actually all seven quarter notes so that there is only a single stress and not three.</p> <p>There are a few other considerations surrounding time signatures and how rhythm and form is approached, but these will be left out until we start discussing folk music traditions or advanced 20-21st century techniques.</p> <a name="2.5.4"></a><h4>2.5.4 Other Rhythmic Techniques</h4> <p>In closing this first part into music principles, I'd like to talk about two more rhythmic items that are often used in tonal music. These are the use of metrical disturbances, and borrowed divisions, starting with the former.</p> <p>There are two general ways in which we can disturb the natural rhythmic feel of a measure. In the time signature 4/4 for example, stronger accents occur on the 1st, and 3rd beats of the measure. If we look at the 8th note level, we can see that these accents correspond to the 1st and 5th eighth notes. If we look a little deeper however, we can also see that the 3rd and 7th eighth notes also have lesser accents to them (following our quarter note beat). What we can surmise from this is that stronger beats are naturally left-aligned in a measure, and that they occur at the beginning of each rhythmic subdivision (quarter to 8th to 16th, and so on). Coming back to the eighth note level, we can also see that the 2nd, 4th, 6th and 8th notes because of this are inherently weaker halves of the overall quarter note beat. In musical terms we call these halves off-beats (the second half of a beat, or parts of the beat that are not accented if we include compound meters). Essentially the hierarchy of strong accents are as follows, beginning with the most global accents and then working downward:</p> <span class="fig"><img class="fig" src="/staff/wo1fwood/TLKH/1_1/20.png" alt="accent hierarchy"/><br/>Hierarchy of important accents in the meter signature 4/4.</span> <p>One of the most common ways of disturbing this natural rhythmic tendency is to place longer durations of notes on weak beats, or weaker parts of beats. When this is done we call this phenomenon syncopation. This can happen at any rhythmic level, but is most commonly seen at either the half, quarter, or eighth note levels. If, using again our 4/4 example, these longer durations are found on the weak 8th notes (2, 4, and/or 6, I'll get to 8 in a second) our syncopation is also on the off-beat. This is the most common use of syncopation that you will find in music. One thing to note is that in order to perceive this syncopation, often times a composer will write other parts of their composition with the normal rhythmic pulsation of the beat, so the differences are aurally clear to the audience (in more advanced works or sections of works composers will however drop this and just use the syncopation as a way to disturb the listener, and create an additional level of conflict that can then be resolved later).</p> <span class="fig"><img class="fig" src="/staff/wo1fwood/TLKH/1_1/21.png" alt="syncopation"/><br/>Quarter notes on the off-beats in a 4/4 measure.</span> <p>Somewhat closely related, another type of metrical disturbance that is often used is called hemiola. In hemiola the established meter is displaced by another competing meter. Commonly you will see this when duple meter temporarily displaces an established triple meter. When this is done the accents of the duple meter are used within the established triple meter temporarily suspending our triple feel and inserting a duple feel.</p> <span class="fig"><img class="fig" src="/staff/wo1fwood/TLKH/1_1/221.png" alt="3/4 no hemiola"/><br/>3/4 with normal beat unit.<br/><br/> <img class="fig" src="/staff/wo1fwood/TLKH/1_1/222.png" alt="3/4 with hemiola"/><br/>3/4 with hemiola accents applied.</span> <p>This stressing of beats does not have to be at the beat unit but can be done supermetrically as well as within the above example (supermetrical refers to longer or larger than a beat, e.g. a half note in 4/4 time). When this is done we can use a musical symbol called ties to extend the duration of a note. Using a tie extends the duration of a note similar to dotting a note, but is more flexible and is the only way in traditional notation to extend a duration between measures, as you cannot write for instance a half note that is part of beat 3 of one measure and beat 1 of the next. In order to accomplish that rhythmic duration we use quarter notes on beats 3 and 1 and tie them together, which sounds exactly the same as a half note would, but satisfies our rules for maintaining traditional notation practices (I do mention traditional because the use of polymeter can easily get around this notational issue, but is something I'll leave for later). A composer who uses this technique quite commonly is Johannes Brahms, but it is a very popular technique amongst common practice composers nonetheless.</p> <span class="fig"><img class="fig" src="/staff/wo1fwood/TLKH/1_1/223.png" alt="3/4 half hemiola"/><br/>3/4 hemiola with half note duration.</span> <p>The last portion of this introductory part will talk about the last traditional rhythmic choices that composers make in writing, and something that is seen quite commonly in all common practice period music; this is the use of borrowed divisions.</p> <p>The most common and easiest to understand borrowed divisions are when beat groups from a simple meter are imported in to a compound meter, or when done in the reverse. In these cases, the subdivision of the beat unit (remembering simple is divided into 2, compound in 3) is changed from 2 to 3, or 3 to 2; in essence this means three beats are played in the duration or span of what was two beats, or the reverse. In the meter signature 12/8 for example if we took our normal beat group, which is normally divided into three 8ths (remembering that we are in a compound quadruple time signature) and instead subdivided the beat into two eighth notes, we have successfully borrowed a division from a simple meter and used it in a compound meter. There is one additional problem however. As our beat temporally speaking does not change its duration, it would appear that the rhythmic duration of our new 8th notes and old 8th notes do not match, and indeed this is the case. To illustrate this and to identify this clearly, the notation and nomenclature that composers gave to these borrowed divisions are what we call tuplets.</p> <span class="fig"><img class="fig" src="/staff/wo1fwood/TLKH/1_1/23.png" alt="duplets and triplets"/></span> <p>The word tuplet is an ambiguous term that describes in a general way that a borrowed division is being used somewhere, but many of these tuplets have specific names as well. In our 12/8 example specifically, as we re-divided our original beat from three into two, we used a duplet to accomplish this. If we were to do this in reverse and use three notes in the span of what was two notes, we would call this a triplet. Further tuplet subdivisions that can be often found are quadruplets, quintuples, sextuplets, septuplets, and so on. In fact, as many as 13-note tuplets can be seen with some regularity in classical music. Additionally, as music developed over time, these borrowed divisions evolved to include the use of irregular divisions and not just simple duplets or triplets. In the 20th and 21st centuries, composers refined this practice and even began to place tuplets within tuplets (e.g. a triplet within a quintuplet), creating exceptionally complex rhythms.</p> <p>Irregular divisions of tuplets are generally shown in one of two ways. The simpler notation indicates the tuplet (example: a septuplet) and then leave it up to the performer to see, through the rest of the score, what the ratio or durational value is. A more modern technique uses the ratio between the old division and new division to indicate exactly what they represent durationally. An 8th note septuplet that spanned five 8th notes would, if using this notation, be shown as a ratio of 7:5 (to indicate 7 8th notes within the span of 5). Notationally you can see below the difference between the two notations:</p> <span class="fig"><img class="fig" src="/staff/wo1fwood/TLKH/1_1/24.png" alt="7:5 tuplet"/></span> <p>Two more considerations regarding the notation of tuplets are as follows. In cases where simple tuplets (duplets and triplets) are often used, sometimes the composer will drop the number or bracket signifying the tuplet in their notation, indicating the tuplet only once and then subsequently show from the notation that the duplet or triplet continues. This saved ink and engraving time, though today is not an acceptable practice as it is somewhat ambiguous to a performer and may require more rehearsal time.</p> <span class="fig"><img class="fig" src="/staff/wo1fwood/TLKH/1_1/25.png" alt="disappeared triplet notation"/></span> <p>Finally, and historically speaking, dotted-note tuplets also have existed, though in modern day notation should be avoided as this practice has been known to cause a great deal of <a class="ext" href="https://www.facebook.com/photo.php?fbid=10150950314323865" target="_blank">confusion</a>, even amongst trained musicians.</p> <a name="2.5.5"></a><h4>Staff Notation: Beaming and Beat Divisions</h4> <p>The very last item that I wanted to briefly mention was in regards to some of the above notation you have seen in this first part, especially when talking about tuplets. Earlier we discussed flags and stems that indicate rhythmic durations. In an effort to clarify the beat unit in meter signatures, these flags were over time eventually beamed together in a way that visually indicates these divisions (and eventually subdivisions). They follow the same principles of flags in adding additional beams as note values get smaller, and especially for tuplets are a strong visual indicator for how long a set of notes are in linear time. Note that in the example below we have also preserved our subdivisions visually by how we beamed our subdivisions together. This is a more modern practice that clearly indicates the submetrical rhythmic structure (the more traditional notation is to next to it) of a piece and can be a great asset for when writing rhythmically complex works.</p> <span class="fig"><img class="fig" src="/staff/wo1fwood/TLKH/1_1/26.png" alt="beaming differences"/><br/>Traditional beaming and modern beaming.</span> <hr/><br/> <p>This concludes part I of III in this series of articles. Hopefully you have gained an understanding, or expanded upon what you currently know regarding these introductory materials, and will be ready to delve into more complex theoretical topics such as counterpoint, functional harmony, non-western traditions, and the myriad of stylistic and theoretical models that arose in the 20th century.</p> <span class="fig"><img class="fig" src="/staff/wo1fwood/TLKH/1_1/27.png" alt="mozart sonata"/><br/>Hopefully, you now have enough knowledge to be able to read and understand most of this excerpt.</span></span> <a name="3.0"></a><h2>References</h2> <span class="section"><ul><li><a class="ext" href="http://www.amazon.com/Complete-Musician-Integrated-Approach-Listening/dp/0199742782" target="_blank">The Complete Musician</a>, Steven G. Laitz Oxford University Press, New York; Third Edition (2011)</li><li><a class="ext" href="http://www.amazon.com/Music-Notation-Twentieth-Century-Practical/dp/0393950530" target="_blank">Music Notation in the Twentieth Century: A Practical Guidebook</a>, Kurt Stone W. W. Norton & Company; First Edition (November 17, 1980)</li><li><a class="ext" href="http://www.amazon.com/Music-Notation-Crescendo-Book-Gardner/dp/0800854535" target="_blank">Music Notation: A Manual of Modern Practice</a>, Gardner Read Taplinger Publishing Company; Second Edition (April 1, 1979)</li><li><a class="ext" href="http://www.amazon.com/History-Western-Music-Eighth/dp/0393931250" target="_blank">A History of Western Music</a>, Peter J. Burkholder, Donald J. Grout, and Claude V. Palisca W. W. Norton & Company; Seventh Edition (July 6, 2005)</li></ul></span> <a name="4.0"></a><h2>Terms to Know</h2> <span class="section"><table class="ref"><tr><td>staff</td><td>clefs</td><td>ledger lines</td><td>octave</td><td>accidentals</td></tr><tr><td>key signatures</td><td>dynamic markings</td><td>pitch</td><td>frequency</td><td>notes</td></tr><tr><td>register</td><td>pitch class</td><td>enharmonic</td><td>scales</td><td>scale degrees</td></tr><tr><td>modes</td><td>tones/semitones</td><td>major/minor</td><td>augmented/diminished</td><td>keys</td></tr><tr><td>circle of fifths</td><td>relative keys</td><td>parallel keys</td><td>intervals</td><td>consonant/dissonant</td></tr><tr><td>rhythm</td><td>pulse</td><td>accents</td><td>beats</td><td>beat groups</td></tr><tr><td>note durations (8th, 16th, etc..)</td><td>notehead</td><td>flag</td><td>stem</td><td>rest</td></tr><tr><td>subdivision</td><td>dotted notes/rests</td><td>BPM (beats per-minute)</td><td>fermatas</td><td>meter signature/time signature</td></tr><tr><td>simple meter</td><td>compound meter</td><td>nontemporal factors (contour, articulation, phrasing, register, harmonic emphasis, textural choices)</td><td>duple/triple/quadruple</td><td>beams</td></tr><tr><td>measures/bars</td><td>downbeat/upbeat</td><td>anacrusis</td><td>asymmetrical meter</td><td>agogic accents</td></tr><tr><td>syncopation</td><td>off-beat</td><td>submetrical/supermetrical</td><td>hemiola</td><td>ties</td></tr><tr><td>borrowed divisions</td><td>duplets</td><td>triplets</td><td>irregular tuplets</td><td>tempo</td></tr><tr><td>bracket</td><td>brace</td></tr></table></span> </span></div><div id="TLKHfooter"><em>Brought to you by the TL Knowhow Team</em><br/> <strong>Written by:</strong> wo1fwood</div>
"It should be noted that in modern music notation the use of key signatures in scores is rarely seen nowadays, largely due to the breakdown of tonality at the beginning of the last century, thought it is still commonly used in popular genres, movies, song, and video game music where tonality is still widely used or is the standard practice."
Can you elaborate on this? Because from personal experience they seem to be used everywhere (classical guitar). Does it only pertain to newer compositions/composers?
As a musician and songwriter, really nice guide - and so accurate!
However, I cannot get rid of the feeling that it still is not simple enough for real beginners. Although, then again your post would have like 800 pages... ;-)
On April 03 2013 18:06 IBringUFire wrote: As a musician and songwriter, really nice guide - and so accurate!
However, I cannot get rid of the feeling that it still is not simple enough for real beginners. Although, then again your post would have like 800 pages... ;-)
I think you are trying to go into too much detail though. As an introduction to music theory/notation you don't need to know about mensural notation and 7:5 rhythm immediately
You might want to explain about how groups of eighth and 16th etc notes are combined into larger symbols. Your example only includes a string of 16th notes. It would be more clear to include a combined example with eighth notes and dots i think. You could also add a figure/example of a dotted note and the same duration with a tied notation. Those type of figures are much easier to read then written text often (for me at least)
On accents within measures and how to achieve those; you can add the agogische(?) accents (i don't know the English term for this :S ) The ones where you make a beat slightly longer to accentuate it.
(p.s In the figure where you show how all the note durations are related you forgot to double the eighth notes )
On April 03 2013 16:08 LaLuSh wrote: "It should be noted that in modern music notation the use of key signatures in scores is rarely seen nowadays, largely due to the breakdown of tonality at the beginning of the last century, thought it is still commonly used in popular genres, movies, song, and video game music where tonality is still widely used or is the standard practice."
Can you elaborate on this? Because from personal experience they seem to be used everywhere (classical guitar). Does it only pertain to newer compositions/composers?
Yea this mostly pertains to what would be classified as 'contemporary classical' music (music from about 1908-ish and onward). This really does depend on the genre that its being written for, as well as the composer though. Eric Whitacre for example does use key signatures in his choral music, as does Morton Lauridsen (in choral music this is somewhat more common). But music from John Corigliano for example mostly will not have key signatures, unless he's writing for a certain audience.
As a general rule, contemporary composers will not use key signatures unless there's a good reason to, which is why it's more rare to see as composers do still use tonally-influenced harmonies, but the use of strict 'tonal harmony' amongst my ilk is more uncommon.
Of course as I did mention in more popular genres that key signatures are still widely used because tonality is still the dominant harmonic frame there (popular music, 'rock' bands, hollywood and video game scores, etc...).
I think you are trying to go into too much detail though. As an introduction to music theory/notation you don't need to know about mensural notation and 7:5 rhythm immediately
You might want to explain about how groups of eighth and 16th etc notes are combined into larger symbols. Your example only includes a string of 16th notes. It would be more clear to include a combined example with eighth notes and dots i think. You could also add a figure/example of a dotted note and the same duration with a tied notation. Those type of figures are much easier to read then written text often (for me at least)
On accents within measures and how to achieve those; you can add the agogische(?) accents (i don't know the English term for this :S ) The ones where you make a beat slightly longer to accentuate it.
(p.s In the figure where you show how all the note durations are related you forgot to double the eighth notes )
never ceases to amaze me that no matter how long I stare and proofread, I still miss things. Yea I 'll have to edit that image. Also I do talk about agogic accents briefly.
This was extremely helpful as noone did successfully explain the meter signatures to me before (regardless of if it was their or my fault :-)). Listening examples would however be even more amazing!
On April 03 2013 16:08 LaLuSh wrote: "It should be noted that in modern music notation the use of key signatures in scores is rarely seen nowadays, largely due to the breakdown of tonality at the beginning of the last century, thought it is still commonly used in popular genres, movies, song, and video game music where tonality is still widely used or is the standard practice."
Can you elaborate on this? Because from personal experience they seem to be used everywhere (classical guitar). Does it only pertain to newer compositions/composers?
Composers for the strictly classical concert music pieces tend to not use key signatures as often (I should know since I am one ). With the "breakdown" of tonality (I don't like the wording here since I'm a fairly tonal composer) it meant that pitches could be assigned freely without the need to keep to a strict key signature, therefore dumping key signatures altogether simply made sense from a logistical standpoint. If we kept using key signatures some pieces would look nonsensical since you would be adding accidentals anyway.
excellent Everything familiar so far Was great to get to know most of the terms in english though Without having to dig out theory books off the internets :D
On April 03 2013 16:08 LaLuSh wrote: "It should be noted that in modern music notation the use of key signatures in scores is rarely seen nowadays, largely due to the breakdown of tonality at the beginning of the last century, thought it is still commonly used in popular genres, movies, song, and video game music where tonality is still widely used or is the standard practice."
Can you elaborate on this? Because from personal experience they seem to be used everywhere (classical guitar). Does it only pertain to newer compositions/composers?
The lack of key signatures is definitely something that came to be in the 20th century. The "tonal" music that is referenced has some really essential components: establishment of key, a cadence in a key, and a bunch of stuff in between.
In some ways, it's also easier to read and think about the music if there is no key signature, as each note is presented as is, without some insinuation as to its relation to the key. If you are a beginner for classical guitar, then most of your pieces are going to be tonal in nature, and will use key signatures for pedagogical functions. The idea is that you start understanding the usual positions for a given key. With the lack of key signatures, the composer tries to avoid any confusion as to his/her intentions.
My worry/critique is that it might be too detailed that it would scare beginners, whom I assume is your main target audience, away with an article as concise as this, depends on your intention though
I also suggest putting up some examples or youtube links when you talk about this that are harder to grasp or imagine like the Georgian 2+2+3+2+2 rhythms to us hear some music :D
My worry/critique is that it might be too detailed that it would scare beginners, whom I assume is your main target audience, away with an article as concise as this, depends on your intention though
I also suggest putting up some examples or youtube links when you talk about this that are harder to grasp or imagine like the Georgian 2+2+3+2+2 rhythms to us hear some music :D
they're not pros like the other one but just look at that atmostphere!
It's fucking hard because they syncopate the hell out of it but it's fucking goooooooooood so just listen and have fun :D
It certainly walks a line so to speak. Hopefully I've been concise enough and not too obtuse that subsequent readings can reveal more for those who are pretty new to this stuff. That being said, I sort of had to at least touch upon some things now if I ever want to try to approach some of the more advanced topics (like microtonality and tuning systems, or serialism), which is something that I feel is incredibly important to talk about, and not often touched on.
I don't think you've been obtuse, but I always found teaching this stuff or learning any sort of theory by myself is much easier with an instrument on hand and lots singing/clapping.
This is a very nice writeup. However, I think this stuff is all moot, if you just read it and do not learn it practically while learning to play an instrument. It is pointless to know what minor-subdominant-cadenza is if you don't know what it sounds like and cannot hear it in your head.
My worry/critique is that it might be too detailed that it would scare beginners, whom I assume is your main target audience, away with an article as concise (edit:complete and detailed) as this, depends on your intention though
I also suggest putting up some examples or youtube links when you talk about this that are harder to grasp or imagine like the Georgian 2+2+3+2+2 rhythms to us hear some music :D
they're not pros like the other one but just look at that atmostphere!
It's fucking hard because they syncopate the hell out of it but it's fucking goooooooooood so just listen and have fun :D
It certainly walks a line so to speak. Hopefully I've been concise enough and not too obtuse that subsequent readings can reveal more for those who are pretty new to this stuff. That being said, I sort of had to at least touch upon some things now if I ever want to try to approach some of the more advanced topics (like microtonality and tuning systems, or serialism), which is something that I feel is incredibly important to talk about, and not often touched on.
Fair enough, I guess your intention is to provide people with a complete education, including the different and lesser known parts, and it definitely does.
(Btw I used the wrong word, it should be "complete and detailed" instead of concise)
When you've the time please do add video/audio examples. It'll definitely be a more interesting learning experience for me to hear it instead of just reading about it. That's the only thing that's lacking imo.
When I was in primary school my teacher lost me when she started telling us that there's a C here and another C a few lines below and another C a few lines above. I was like "wtf is this shit can't they pick a C and stick with it?" hahahaha. Of course I wasn't really paying attention back then :D
I must admit, I came here and expected a rather poor guide. Sorry for that, it seems well structured and filled with knowledge.
However, I can't imagine anyone being able to learn music theory like this, especially if they can't read a single note yet. If you first try to learn relationships between scales and the cycle of fifths, after you have just learned how to read notes and probably have forgotten all the intervals and what they are called and which intervals belong in a scale... I just think turning here as a novice won't help you. Or maybe I don't remember if learning music theory the slow way wasn't any better.
Great effort nonetheless, it is obvious you have put a lot of thought into it.
I agree with everyone's assessment regarding the excellence of this guide. As someone who studied music, it's superbly concise and filled with a breadth of information. That being said, I think it reaches a decent amount higher than what a beginner or even novice person in music might completely absorb.
I also have to second the notion someone brought up earlier regarding not having audible examples. Learning straight theory is one thing, but utilizing audible examples for certain techniques and other musical notation would prove incredibly valuable. Perhaps this is something that can be added to the piece over time to better illustrate the theory.
On April 07 2013 06:58 divito wrote: I agree with everyone's assessment regarding the excellence of this guide. As someone who studied music, it's superbly concise and filled with a breadth of information. That being said, I think it reaches a decent amount higher than what a beginner or even novice person in music might completely absorb.
I also have to second the notion someone brought up earlier regarding not having audible examples. Learning straight theory is one thing, but utilizing audible examples for certain techniques and other musical notation would prove incredibly valuable. Perhaps this is something that can be added to the piece over time to better illustrate the theory.
Audio examples do help, but you really need to perform the actions of creating music to internalize a lot of the basics. The so-called ABCs of music don't need much intellectualization, practice is the key. Perhaps tailoring the guide to someone with either a keyboard or guitar would help matters. As I said before though, this guide is very well written for beginners who simply want to understand the basic elements of music theory.
On April 07 2013 06:58 divito wrote: I agree with everyone's assessment regarding the excellence of this guide. As someone who studied music, it's superbly concise and filled with a breadth of information. That being said, I think it reaches a decent amount higher than what a beginner or even novice person in music might completely absorb.
I also have to second the notion someone brought up earlier regarding not having audible examples. Learning straight theory is one thing, but utilizing audible examples for certain techniques and other musical notation would prove incredibly valuable. Perhaps this is something that can be added to the piece over time to better illustrate the theory.
While you are right, the current article probably took already quite some effort. Making good audio examples would require even more labor. I once wrote a tool which can play modal scales through Windows Midi, which I planned to use in a Youtube series (however that would be in German) but I found it surprisingly hard to make clear what the listener should hear.
For my blog series (which wo1fwood thankfully commented on and corrected some mistakes) I chose a different approach. I am still not sure what is better: To explain staff and clef, or to explain acoustic phenomena?
I am very thankful for wo1fwood's effort because it took me almost two years to understand the concept of scales. With articles like his (that one here is only the primer) finally a good explanation is out there to show others. I noticed that an understanding of the theory behind music greatly amplifies the pleasure when I listen to music.
On April 05 2013 00:17 Chrono000 wrote: too bad i dont understand music
If string theory is right, many concepts of music apply to the core of the universe itself. There would be overtones and other things. The entire cosmos would be one giant concert. And you have a ticket!
Adding in audio examples to this write-up was something I definitely considered and even am considering adding after the fact for Part I as it is an important portion of learning what music is (not just theoretically but aurally as well). This does however, require a lot more time to set up and do effectively. I might try to at least demonstrate what I've talked about in this Part (pitch, rhythm, intervals) and add them in later, but I'm sort of split on this at the moment.
The main issue that I keep dancing around is that adding in audio examples would make this write-up much better, but its really an issue of how much time I have to spend at the moment on it (and also the complexity of the topics), as well as the trajectory of the overall piece. As [F_]aths said, if I wanted to create all the content myself, this becomes quite the undertaking (especially for the more advanced concepts).
Also, it's definitely something worth noting, but audio examples should never be used as a crutch for the explanation. If you need an audio example to explain the theoretical concept, you simply haven't explained it well enough. They should enhance, or clarify what you've seen and read (or put into practice) so that they further reinforce the concepts you just learned. This kind of dances around the differences between formalized learning and wrote learning, but as this is geared towards tonal harmony the wrote method is kind of sub-optimal and for a number of reasons (that and the medium is somewhat limiting for the latter).
I should also note that I had been originally playing around with the idea of beginning the audio examples in part II and sort of unraveling and explaining things in part I as we went along. But this is/was still in a conceptual phase right now. Additionally, I had also originally planned on this being a single article, but due to my real life commitments and my overall nature I felt that I needed to prioritize and break this up at least in getting out what I had already started, rather than let it sit for months on end until it was fully fleshed out.
This is pretty awesome... Great formatting, TONS of useful information! Definitely going to give this a second read-through when I start teaching myself guitar...
Also, lol at 128th notes.. (I'm a drummer who's a big fan of DCI and all dem fancy drum corps n what not) ... that's pretty funny.
On April 07 2013 10:34 wo1fwood wrote: Also, it's definitely something worth noting, but audio examples should never be used as a crutch for the explanation. If you need an audio example to explain the theoretical concept, you simply haven't explained it well enough. They should enhance, or clarify what you've seen and read (or put into practice) so that they further reinforce the concepts you just learned. This kind of dances around the differences between formalized learning and wrote learning, but as this is geared towards tonal harmony the wrote method is kind of sub-optimal and for a number of reasons (that and the medium is somewhat limiting for the latter).
I would disagree very strongly in this regard. When teaching the very simplest of concepts in music, connecting the learner to a tangible product can really help. It also takes in to account the many ways that people learn. Further down the line, say if you are discussing dominant substitutions or the like (simple harmonic ideas), then I agree you should not use examples as a crutch constantly. But to help the uninitiated really "get" the basics of rhythm and pitch, you need to have a level of physical engagement. It is far easier to move tangible ideas into abstract concepts than vice-versa.
On April 07 2013 10:34 wo1fwood wrote: Also, it's definitely something worth noting, but audio examples should never be used as a crutch for the explanation. If you need an audio example to explain the theoretical concept, you simply haven't explained it well enough. They should enhance, or clarify what you've seen and read (or put into practice) so that they further reinforce the concepts you just learned. This kind of dances around the differences between formalized learning and wrote learning, but as this is geared towards tonal harmony the wrote method is kind of sub-optimal and for a number of reasons (that and the medium is somewhat limiting for the latter).
I would disagree very strongly in this. When teaching the very simplest of concepts in music, connecting the learner to a tangible product can really help. It also takes in to account the many ways that people learn. Further down the line, say if you are discussing dominant substitutions or the like (simple harmonic ideas), then I agree you should not use examples as a crutch constantly. But to help the uninitiated really "get" the basics of rhythm and pitch, you need to have a level of physical engagement. It is far easier to move tangible ideas into abstract concepts than vice-versa.
I'm not sure we're actually in disagreement here. You are talking about the practical application of theory as a necessity to understanding music as it functions in the real world, and I wouldn't really disagree with this. Without hearing how music and sound in general behaves acoustically, theoretical concepts can remain somewhat nebulous and disconnected from the actual product. That being said however, it is not necessary to include this practical application in order to understand the underlying theory itself. It's a very subtle difference when dealing with instruction and understanding (a purely theoretical understanding, and theoretical understanding from the practical application standpoint).
You do touch on the possible learning implications of the current piece and it's definitely something worth noting, and perhaps changing (e.g. what is the point and end goal).
On April 07 2013 10:34 wo1fwood wrote: Also, it's definitely something worth noting, but audio examples should never be used as a crutch for the explanation. If you need an audio example to explain the theoretical concept, you simply haven't explained it well enough. They should enhance, or clarify what you've seen and read (or put into practice) so that they further reinforce the concepts you just learned. This kind of dances around the differences between formalized learning and wrote learning, but as this is geared towards tonal harmony the wrote method is kind of sub-optimal and for a number of reasons (that and the medium is somewhat limiting for the latter).
I would disagree very strongly in this. When teaching the very simplest of concepts in music, connecting the learner to a tangible product can really help. It also takes in to account the many ways that people learn. Further down the line, say if you are discussing dominant substitutions or the like (simple harmonic ideas), then I agree you should not use examples as a crutch constantly. But to help the uninitiated really "get" the basics of rhythm and pitch, you need to have a level of physical engagement. It is far easier to move tangible ideas into abstract concepts than vice-versa.
I'm not sure we're actually in disagreement here. You are talking about the practical application of theory as a necessity to understanding music as it functions in the real world, and I wouldn't really disagree with this. Without hearing how music and sound in general behaves acoustically, theoretical concepts can remain somewhat nebulous and disconnected from the actual product. That being said however, it is not necessary to include this practical application in order to understand the underlying theory itself. It's a very subtle difference when dealing with instruction and understanding (a purely theoretical understanding, and theoretical understanding from the practical application standpoint).
You do touch on the possible learning implications of the current piece and it's definitely something worth noting, and perhaps changing (e.g. what is the point and end goal).
Fair points, I would just encourage people to use this more as reference than text. Also, minor quibble, you've been using the word "wrote" when your intention is for the word "rote." Unless this is some spelling from a subset of English I am not aware of (which could certainly be the case, then my bad).
On April 06 2013 02:05 nimbim wrote: I must admit, I came here and expected a rather poor guide. Sorry for that, it seems well structured and filled with knowledge.
However, I can't imagine anyone being able to learn music theory like this, especially if they can't read a single note yet. If you first try to learn relationships between scales and the cycle of fifths, after you have just learned how to read notes and probably have forgotten all the intervals and what they are called and which intervals belong in a scale... I just think turning here as a novice won't help you. Or maybe I don't remember if learning music theory the slow way wasn't any better.
Great effort nonetheless, it is obvious you have put a lot of thought into it.
As someone who just started trying to learn piano I would agree with this. There is a lot of information in this guide that looks fantastic and I'm sure you put a lot of work into, but it's a little bit too advanced for someone like me with very limited music experience.
On April 06 2013 02:05 nimbim wrote: I must admit, I came here and expected a rather poor guide. Sorry for that, it seems well structured and filled with knowledge.
However, I can't imagine anyone being able to learn music theory like this, especially if they can't read a single note yet. If you first try to learn relationships between scales and the cycle of fifths, after you have just learned how to read notes and probably have forgotten all the intervals and what they are called and which intervals belong in a scale... I just think turning here as a novice won't help you. Or maybe I don't remember if learning music theory the slow way wasn't any better.
Great effort nonetheless, it is obvious you have put a lot of thought into it.
As someone who just started trying to learn piano I would agree with this. There is a lot of information in this guide that looks fantastic and I'm sure you put a lot of work into, but it's a little bit too advanced for someone like me with very limited music experience.
You have to accept to do some work to get an understanding of music. Playing notes from the sheet is (more or less) easy. To understand what one is playing, is much harder. But it helps me to memorize a piece or to accompany it with chords.
The piano has the advantage that the keyboard layout is close to the notation on the sheet. If you see example notes which explain a context (for example the concept of a cadence regarding tonality and modulation) and then play them, you get an understanding deeper than if you just read words or look at sheets.
Whhhaaaat! Amazing! This is exactly like something I've been looking for. I am bookmarking and will read over the summer, and hopefully put music into a year-long project. I will of course post any music I make here. Thanks for writing this.
It's not impressive until you show us some kind of trick that you can explain with the theory. Like I can be a PhD in mathematics but it won't be impressive until I can do some indepth insight for a business or something.
On April 08 2013 15:54 RoieTRS wrote: It's not impressive until you show us some kind of trick that you can explain with the theory. Like I can be a PhD in mathematics but it won't be impressive until I can do some indepth insight for a business or something.
I don't quite get to whom you are talking about what? Music theory has no tricks. You can only go more in-depth or decide which classical rules you don't like and ignore them.
On April 08 2013 15:54 RoieTRS wrote: It's not impressive until you show us some kind of trick that you can explain with the theory. Like I can be a PhD in mathematics but it won't be impressive until I can do some indepth insight for a business or something.
Without knowing the theory, a composition will look arbitrary to you. With knowledge, you see what the composer actually has written.
It is a bit like seeing letters versus understanding the language.
With knowledge of music, you know how to create tension and release, so you can create nice effects in your piece. You know how you can express yourself through music. It is not just writing down a melody you have in your head.
Music Theory, at least from a compositional standpoint, is best seen as a tool. There are many ways to write a song, and western music theory happens to be a highly evolved thing. Another practical application for learning theory is that it helps with playing, both in a memorized and more importantly an improvisatory fashion. It also aids in learning music very quickly. If you ever have a teacher who says that theory is not essential to becoming any sort of musician, then leave him/her.
On April 08 2013 15:54 RoieTRS wrote: It's not impressive until you show us some kind of trick that you can explain with the theory. Like I can be a PhD in mathematics but it won't be impressive until I can do some indepth insight for a business or something.
Without knowing the theory, a composition will look arbitrary to you. With knowledge, you see what the composer actually has written.
It is a bit like seeing letters versus understanding the language.
With knowledge of music, you know how to create tension and release, so you can create nice effects in your piece. You know how you can express yourself through music. It is not just writing down a melody you have in your head.
I major in music. I can write down the fundementals of garchenspoosh but if I don't show the purpose of studying garchenspoosh you get some longwinded useless piece of info like this OP.
As a full sail student its really nice to have so much MTH knowledge on one giant post. The full sail resources are nice but its individual sections. I struggle with intervals personally, this enlightened me a little bit. I will definetly try to spread the word to my classmates and Music Production staff as an external resource.
On April 13 2013 16:47 SchfiftyFive wrote: As a full sail student its really nice to have so much MTH knowledge on one giant post. The full sail resources are nice but its individual sections. I struggle with intervals personally, this enlightened me a little bit. I will definetly try to spread the word to my classmates and Music Production staff as an external resource.
In this regard I have a slight critique regarding the article. It does not tell much about how intervals sound. And why unison, octave, fifth and fourth perfect are intervals while the others come in two versions. And what the pitch relationships of those intervals are.
Of course the primer is already very large and one can guess that the next parts will be going deeper into intervals.
I like to praise the introduction into rhythm. Music is not only about which notes to play, but also when and how. Tempo and rhythm is very important.
On April 14 2013 05:18 MichaelDonovan wrote: Have you considered writing textbooks? You might be able to make some money.
Thanks for the kind words, but I would have to do so much more work in order to feel competent in doing so, at least in this specific arena. I have occasionally mused on the idea of writing or working in other areas though where I would be considered more of an expert by my peers.
I did want to mention that I decided that I would add audio examples to this initial piece, but I'm entirely too busy at this current juncture to really tackle this properly. Hopefully sooner rather than later, but I need to attend to other tasks first.
On April 13 2013 16:47 SchfiftyFive wrote: As a full sail student its really nice to have so much MTH knowledge on one giant post. The full sail resources are nice but its individual sections. I struggle with intervals personally, this enlightened me a little bit. I will definetly try to spread the word to my classmates and Music Production staff as an external resource.
Hey, is there a place I could message you about Full Sail? I'm considering going there for Music Production.
On April 13 2013 16:47 SchfiftyFive wrote: As a full sail student its really nice to have so much MTH knowledge on one giant post. The full sail resources are nice but its individual sections. I struggle with intervals personally, this enlightened me a little bit. I will definetly try to spread the word to my classmates and Music Production staff as an external resource.
If you are having trouble with intervals, I'd suggest practicing at musictheory [dot] net if you haven't already. Simple tools, just practice like 10-20 minutes a day.
On April 13 2013 16:47 SchfiftyFive wrote: As a full sail student its really nice to have so much MTH knowledge on one giant post. The full sail resources are nice but its individual sections. I struggle with intervals personally, this enlightened me a little bit. I will definetly try to spread the word to my classmates and Music Production staff as an external resource.
If you are having trouble with intervals, I'd suggest practicing at musictheory [dot] net if you haven't already. Simple tools, just practice like 10-20 minutes a day.
The aural skills tools are about the only thing on that website that I could honestly recommend to anyone. They are good for listening and drilling basics of interval and pitch identifications, but If you wanted to really learn theory properly they make really obvious mistakes or omit incredibly important materials all the time. Also somewhat related, I have yet to find a good online tool for drilling chord progressions as well, most are either really basic, or use bizarre progressions (substitutions, incorrect functions of chords) in tonal harmony which entirely negates any effectiveness of what it's trying to teach in the first place.
On April 13 2013 16:47 SchfiftyFive wrote: As a full sail student its really nice to have so much MTH knowledge on one giant post. The full sail resources are nice but its individual sections. I struggle with intervals personally, this enlightened me a little bit. I will definetly try to spread the word to my classmates and Music Production staff as an external resource.
If you are having trouble with intervals, I'd suggest practicing at musictheory [dot] net if you haven't already. Simple tools, just practice like 10-20 minutes a day.
The aural skills tools are about the only thing on that website that I could honestly recommend to anyone. They are good for listening and drilling basics of interval and pitch identifications, but If you wanted to really learn theory properly they make really obvious mistakes or omit incredibly important materials all the time. Also somewhat related, I have yet to find a good online tool for drilling chord progressions as well, most are either really basic, or use bizarre progressions (substitutions, incorrect functions of chords) in tonal harmony which entirely negates any effectiveness of what it's trying to teach in the first place.
Agreed, use mainly for intervals. It's not something one should use to learn the whole topic. As to chord progressions, I think I've only ever had a teacher play it live for quizzes and whatnot. Again, another thing that is learned better by doing and slowly figuring out. Maybe we should just start an Oxford debate, wo1fwood.
This is exactly what I have been looking for, in terms of covering the fundamental concepts and providing just the right amount of depth to maintain readability. I haven't found any on-line source of this information that is presented as well as your OP. Great work, thanks!
Huh, it's odd that the picture you use has both the Helmholtz notation and simultaneously the more modern traditions. In Helmholtz system he'd use commas and capitalization to distinguish octave designations, as well as naming each octave (reference here). With the advent of MIDI technology helmholtz octave designations are kind of archaic and imprecise now, though they do give some good foundations in describing registers for organ music and their manuals.
What does Hz exactly mean when i tune my instrument? i am reading about this Norwegian guy thats saying that tuning instruments at 440 is wrong and that they should be tuned at 432 Heartz instead. The reason why i got into this is i am seeying a lot of music posted with stuff like 432 re-equilibrize yourself bla bla bla. I listen a little and i find it highly disturbing.
So if someone could clear for me what exactly it means you know. If i understood correctly, music at 432 Hertz simply means tuning the instrument a bit lower? And the difference anyway is not understandable by the human ear?
You most certainly can hear 8Hz difference. At that register the difference between 440Hz and 432Hz is about 1/3rd of a semitone in Equal Temperament. As an example, I can reliably discern pitch changes of +/- 3 cents, which is a lot smaller than an 8Hz difference (there are 1200 cents in one octave, or 100 cents per semitone in 12-tet).
As for tuning your instrument it depends on your instrument and what you are playing. If you are playing standard repertoire that is either classical, or just mainstream pop you should probably be tuning your instrument to 440Hz. All this means is that you are tuning your instrument where the pitch A4 is the frequency 440Hz.
If you play a Baroque instrument (or older instrument/music) it depends on the ensemble you are playing with as well as the type of music because you can tune anywhere from 432Hz to 402Hz in some cases. As a note, A4 used to be ~415Hz during the Baroque period, so what we think of as A-flat today was actually A-natural 3-400 years ago... Sheds an important light on music from that period, especially vocal music.
Unfortunately without context I don' know what the Norwegian you're referencing is getting at, but just know that tuning systems are a lot more variable and flexible than you might initially think.
Okay, i understand a bit better now. Ye i play an acoustic guitar, was perfectly happy with the 440Hz; I just could not understand what this guy is talking about.
The Name of the guy that suggested this 8 Hz change is Ananda Bosman. In the article it says he is not actually a musician.
Fun fact: in Italian unlike French and English we don' t say i play an instrument, we say i sound an instrument. (unless its singing then its just singing)
On May 28 2013 00:07 wo1fwood wrote: Huh, it's odd that the picture you use has both the Helmholtz notation and simultaneously the more modern traditions. In Helmholtz system he'd use commas and capitalization to distinguish octave designations, as well as naming each octave (reference here). With the advent of MIDI technology helmholtz octave designations are kind of archaic and imprecise now, though they do give some good foundations in describing registers for organ music and their manuals.
Helmholtz pitch notation is what I meant, I'm pretty sure that is still most commonly used in Europe. Sorry, I couldn't find a better picture at that time. I meant to post this one:
On May 30 2013 10:51 wo1fwood wrote: You most certainly can hear 8Hz difference. At that register the difference between 440Hz and 432Hz is about 1/3rd of a tone in Equal Temperament.
Did you mean semitone? The difference of 8 Hertz for the concert pitch is about 1/7th of a tone (or roughly 1/3 of a semitone.)
On May 28 2013 00:07 wo1fwood wrote: Huh, it's odd that the picture you use has both the Helmholtz notation and simultaneously the more modern traditions. In Helmholtz system he'd use commas and capitalization to distinguish octave designations, as well as naming each octave (reference here). With the advent of MIDI technology helmholtz octave designations are kind of archaic and imprecise now, though they do give some good foundations in describing registers for organ music and their manuals.
Helmholtz pitch notation is what I meant, I'm pretty sure that is still most commonly used in Europe. Sorry, I couldn't find a better picture at that time. I meant to post this one:
Thanks for your reply.
I still use these names (German: Subkontraoktave, Kontraoktave, Große Oktave, kleine Oktave, eingestrichene Oktave, zweigestrichene Oktave, ...) but as German I also use the notenames C, D, E, F, G, A, H and call B-flat just a B
On May 30 2013 12:08 pebble444 wrote: Okay, i understand a bit better now. Ye i play an acoustic guitar, was perfectly happy with the 440Hz; I just could not understand what this guy is talking about.
The Name of the guy that suggested this 8 Hz change is Ananda Bosman. In the article it says he is not actually a musician.
Fun fact: in Italian unlike French and English we don' t say i play an instrument, we say i sound an instrument. (unless its singing then its just singing)
Fun fact about 440 Hertz: If you use just intonation for the C major scale, 440 Hertz for the concert pitch will result in integer frequency values for all white-key notes.
In just intonation, A is 5/3 of C, so C turns out to be 264 Hertz. Derived from C, we get:
This is excellent, and perfectly timed in my opinion. I just completed a piano grading and my teacher recommended do a theory grading as well. Greatly looking forward to the next article.
Gotta say, I've never seen any of the 'modern beaming' that you showed at the end of the article in any of the pieces that I've played.
As he says in the relating paragraph, the modern beaming is used to show rhythmic intent. It's not necessarily that more recent composers all use that style, but that it is sometimes used to do a specific thing. I find it unneeded and distracting for the standard repertoire, but some works really benefit from and are easier to read with the the "newer" beam style.
On May 31 2013 23:13 Horn wrote: wo1fwood do u play music professionally? this article looks really good
I am a professional composer, singer, and dancer.
On June 01 2013 00:57 cassydd wrote: Gotta say, I've never seen any of the 'modern beaming' that you showed at the end of the article in any of the pieces that I've played.
Beaming this way didn't really become a 'thing' until the later romantic period, so it's a pretty modern addition in the grand scheme of things. You can find early forays into this practice in the music of Strauss, Mahler, or others of this time period.
On June 01 2013 02:11 sharkeyanti wrote: As he says in the relating paragraph, the modern beaming is used to show rhythmic intent. It's not necessarily that more recent composers all use that style, but that it is sometimes used to do a specific thing. I find it unneeded and distracting for the standard repertoire, but some works really benefit from and are easier to read with the the "newer" beam style.
Yea, the main purpose for this type of beaming is for clarity, so there is can be no question as to what is intended by the composer (a major can of worms when talking about dead peoples music). The main question to be asked when looking at this type of beaming vs a more traditional one is whether you gain anything in terms of readability or intent from whichever you use.
Often times being more precise can be more tedious and time consuming for the performer, but as a general rule I would always tell students to follow modern practices here. As an example in 4/4 it is common to see four 8th notes beamed together as beats 1+2 or 3+4, but technically this is incorrect because 4/4 is a quadruple meter and therefore doesn't reflect visually the beat unit divisions accurately.
Similarly while my 6/8 example is more precise if the intent is subdividing in 3, commonly you'll just see 6 16ths tied together to show the beat unit. But how you beam here can also be dependent on whether the composer intends a certain subdivision (like 3), or if the division is indeed 6 16ths to a beat unit. So this can be a very specific and situational thing. In common practice music this really shouldn't be worried about, but for music past 1900 it becomes an increasingly important question.
On May 31 2013 23:13 Horn wrote: wo1fwood do u play music professionally? this article looks really good
I am a professional composer, singer, and dancer.
I recently used Melodyne to test if I can sing a major scale. Most times I manage to get the structure right. But when I sing a song, it takes only a few measures to deviate up to a whole tone step. Also my vocal range is quite limited, A♭2 to B♭3 with somewhat clarity, F2 to C3 if I push it. Singing a D3 sounds horrible. So I cannot earn any money as singer
If you do some basic breathing exercises, you should be able to widen your range at least a little bit. Lots of singers have a limited range, though none of them would be in a classical/jazz field for sure.
If you do some basic breathing exercises, you should be able to widen your range at least a little bit. Lots of singers have a limited range, though none of them would be in a classical/jazz field for sure.
I am too bad for real music anyway. But since my keyboard is not here, I cannot play anything so I have to sing. (Actually I extended on a delphi program on mine so I can play scale degree triads with my computer, so I can accompany Youtube songs to find out their key and the chord progression.)
Damn that there are very, very few guys interested in actual music theory. Today I developed a new understanding why the fifth is so dominant and why the fourth is so unstable. I also explored a new way to explain the harmony of the minor triad (with undertones, but this explanation is superficial because it is just an elaborate way to say a minor triad is a major one upside down.) I also looked into the notes one can get with the overtone/untertone series. To my surprise, the major sixths / minor third is completely missing.
I also looked deeper into combination tones to find out that a major third indeed seeks resolution downwards (to the lower note.)
I still try to understand if a sus4 chord can be viewed as a chord for its supposed root or if it actually represents its fourth degree.
@ wo1fwood. Love it. Its hard to find easy to understand documents on music theory. Traditional books go way into depth or take too much time. I like how you compiled it into one post
On May 30 2013 09:15 pebble444 wrote: Hi guys i have a question:
What does Hz exactly mean when i tune my instrument? i am reading about this Norwegian guy thats saying that tuning instruments at 440 is wrong and that they should be tuned at 432 Heartz instead. The reason why i got into this is i am seeying a lot of music posted with stuff like 432 re-equilibrize yourself bla bla bla. I listen a little and i find it highly disturbing.
So if someone could clear for me what exactly it means you know. If i understood correctly, music at 432 Hertz simply means tuning the instrument a bit lower? And the difference anyway is not understandable by the human ear?
Its interesting modern symphonies actually tune a bit above 440 Htz. Not by 8HZ, but it creates the illusion of a brighter and more magical sound.
In the overtones of any note, lets take C as example, appear some useful intervals. For note C they are represented by E, G, D and B. (D appears later than E and G, because it is less consonant.) We can create some chords with these notes, these are:
Overtone scale use is not an exact science at all. When you start getting to those less common chords, context is as important as anything in their use. We can define these things in a strict manner, but relation to music is the real issue. In the practice of music, these exactitudes generally have little bearing. They become interesting rather than relevant, beautiful, enjoyable, etc.
A m(add4) chord doesn't really have much meaning by itself.
I know that context is of great importance, for example to tell an inverted minor chord from a sixth chord. I also know that the harmonic series (which the overtone series follows) only provides us some intervals, while the chromatic scale consists of many notes which do not appear in overtones.
Currently I think that the harmonic series or overtone series can be used to establish basic harmony. For example it is possible to explain why a just major triad is extremely consonant and why we perceive the root note as root indeed. While a minor chord is also consonant, even though a bit less, it is not possible to explain it directly with overtones. So I am trying out if the reversing of the intervals of consonant chords (so consonant that they can be explained using overtones) also leads to a, maybe somewhat less, but still harmonic chord.
If this hypothesis is correct, an m(add4) should be a bit less consonant than an add2 chord.
edit: I just tried it. m(add4) is suprisingly harmonic – compared to add2. m(add4) is a bit less consonant, but still creates a colorful sound with no sharp dissonance.
edit2: Of course there are a lot of useful chords which use way more tones than appear in the overtones of the root note. I am not trying to creating a complete chord theory here.
Consonance and dissonance are relative to the listener as well. While there is some degree of science related to the topic, I would be very wary of giving much value to anything beyond the basic and common chord structures. I think it's good to go through that overtone system in an organized fashion if it helps you understand your "ear" better. Something you might be interested in is general psycho-acoustic effects. Specifically, check out writings on Steve Reich's "Music for 18 Musicians." That stuff is fascinating.
I listened through a piece available on Reich's webpage. http://www.stevereich.com/mp3.html Strange, yet interesting. I felt taken on a journey. It was summer, but the sun was low and the wind already a bit chilly. But I did feel great. It somewhat reminded me of earlier works of Mike Oldfield.
It takes of course a lot of work and creativity to write and to perform good music. I view notes and chords similar to colors. The artist still have to paint the picture and I will not be able to explain the picture when I just describe the wavelength of the color red or the chemical composition of the red paint he used.
I am not always theorizing with numbers. I have a keyboard, a glockenspiel, and a flute. Even though I play neither instrument good enough for an actual song, I like to have this experience of structured sound. I heard the effect of the dominant or the leading tone long before I read about such concepts. A great part of my theoretical effort is to find a satisfying explanation why I do feel tension, or resolution. There is surprisingly little literature about those topics. But I am now satisfied with the explanations I found why the bass note of a perfect fifth interval appears as root: Overtones and combination tones both show, that the bass note if amplified and contains the higher note, but the higher note does not contain and therefore does not amplify the bass note.
Most disquisitions about the minor triads I read however are just playing with numbers or geometric figure. The "explanations" are so elaborate that I feel they are not really an explanation but rather a mathematical coincident.
Modern music theory is not concerned with natural acoustics (except psychoacoustics), even for the "simplest" triads, but rather with theorizing musical behaviors and patterns. For example, the enharmonic respellings that crop up in Haydn and Mozart are understood as having functional structural connotations -- i.e., double meanings, modulatory significance -- rather than as pointing to some abstract desire for a "pure" natural triad. While there have always been critics of the acoustical basis for a music theory, the main reason for this almost universal change of approach is that harmonics does a poor job of explaining even the most "basic" products of Western musical culture (and a terrible job for many other cultures).
Recent experiments (e.g., Neil McLachlan's "Consonance and Pitch" from this year, which also gives a historical overview) show that harmonics can't even account for the basic experience of consonance and dissonance. It might be useful, then, when conceptualizing the various "sus" chords, to consider how they got their names and what sorts of expectations they carry in a conventional tonal context.
That being said, there's always a place for discussion of historical ideas about ratios and harmonics, the old musica theorica. Zarlino in the late sixteenth century devised a theoretical tuning based on the syntonic diatonic, using ratios going up to the number 6 (the senario). Rameau tried to base his theory on "natural" properties of sound and even ingeniously argued that the supposed acoustical basis for a chord root (the "fundamental") also explained the behavior of chords in a musical context. He ran into the same problem as [F_]aths with the minor third, which (along with its inverted "shadow," the major sixth) does not occur over the fundamental. As Joel Lester writes in Compositional Theory in the Eighteenth Century:
Rameau never abandoned the search for an origin of the minor third. He was not alone in this search -- theorists from antiquity to the present have sought the sources of their musical system in divisions of the string or [numerically identical] harmonic resonance...Ultimately, the reason a solution has never been found is the fallacious basis of the search: an ethnocentric belief that music, a cultural phenomenon, is a Natural phenomenon... (pp. 102-104)
Lester points out some assumptions this kind of theorizing requires. My two favorites: (1) octave equivalence, which is useful in early music pedagogy but deeply flawed in both practice and in mathematical reality. (2) the assumption that simple ratios (or string divisions) represent actual musical intervals that one hears in various tunings.
I'm really glad the OP has opened up basic Western musical concepts to these forums. You don't have to be a great performer to develop subtle listening skills; likewise, conceptual development helps performance and listening alike, regardless of genre.
Those are actually my current two big issues: Not being able to order the intervals of the chromatic scale by consonance, and not being able to explain the major sixth. Treating it as an octave complement of the minor third is the best I can do, and the minor third only appears as fifth complement of the major third in the first place.
I read the entire Harmonielehre vom Diether de la Motte. He does not go deep into the theory, he rather focuses on harmony concepts used in practice. I also read most of “13 Tonstufen in der Duodezime” from Heinz Bohlen. He got deep into the theory and proposes to use 13 notes spanning a twelfth. The “Unterweisung im Tonsatz”, The Craft of Musical Composition by Paul Hindemith does not seem to be available as book, nor online, so I depend on Wikipedia here. Hindemith tried to offer a new explanation for our 12 step chromatic scale, but while many of his ideas seem to be good, he does not argue in a consistent way.
Still I think, the consonance of the minor third is explainable:
A) The minor third appears not in overtones, but as an interval in major triads. We only slightly modify the major triad by a minute change of the inner interval. It gets shifted by just a chromatic semitone, to get its relation of the middle note from being 1/4 higher than the root note, to 1/5 higher than the root. The gestalt of a triad is still there. While the minor third does not support the root note with overtones or combination tones, the perfect fifth still does. We still use both types of thirds, which are not that different anyway. We still see that it is a triad.
B) In music practice, we use the concept of scales. If we construct a scale from a string of three major triads (lets say the triads for F, C and G) that scale will also include minor triads. This has to happen. Let us connect F-A-C, C-E-G and G-B-D.
F-(maj)-A-(min)-C-(maj)-E-(min)-G-(maj)-B-(min)-D
We get A-minor and E-minor. Many tuning systems also allow for an acceptable D-minor. Only the root note determines if we perceive the scale as major or minor, the white-key notes allow for as many major as minor triads.
On June 07 2013 04:31 gorbonic wrote:Lester points out some assumptions this kind of theorizing requires. My two favorites: (1) octave equivalence, which is useful in early music pedagogy but deeply flawed in both practice and in mathematical reality. (2) the assumption that simple ratios (or string divisions) represent actual musical intervals that one hears in various tunings.
I think the mathematical properties of the octave justify its use. Your second point is in my opinion the great advantage of the western chromatic scale: One can construct a pentatonic, or diatonic, or chromatic scale only using octaves and fifths. A diatonic scale can also be constructed with three triads. To get the five missing steps for a chromatic scale, we calculate the octave inversions of the other intervals. While the tuning is a bit different (when we begin with C, the notes A and E-flat have to be adjusted by the syntonic comma) we can use the context to underline if we use distances by fifths for the relationship, or simple ratios.
I consider equal temperament the height of western music. While it sacrifices the purity of thirds, it allows to use all notes in different contexts and thus provides a rich palette with only 12 steps. From any note, it includes the ratios 1:2, 2:3, 3:4 with extremely good, and 4:5 and 5:6 with acceptable precision. It also provides a circle of (slightly tempered) fifths.
"Natural minor has a lowered 6th and 7th scale degree and its overall pattern is W-H-W-W-H-W-W."
I have thought long and hard about this. I am now confident to say that this description is not correct even though the outcome is the same. Western scales are not constructed by tone steps. The scale is constructed by strings of fifths or by triads (leading to different tunings) and then arranged into a chromatic scale of half-tone steps.
No-one understands why the minor scale is "w-h-w-w-h-w-w". But one can understand it as a string of three minor triads. Or as one out of seven modal scales (this approach takes a bit longer to explain, but extends the major-minor-dualism to seven modes, which is useful to explain diatonic scale functions. I see some issues in common diatonic function descriptions as well.)
Why are the half-tone steps of a minor mode there and not somewhere else? One cannot explain it unless one explains how major works with major triads and that one have to flatten the third to get a minor triad. Since three triads are involved, three notes have to be flattened. Since the triad in the middle is the tonic one, one can also compute on which position the flattening occurs.
OK, I finally got around to updating this thread since the site move, so now all the images have returned to their normal place. And I did want to address something you mentioned previously Faths:
Why are they arranged WHWWHWW in that specific order? This is explained very simply by understanding that the natural minor scale is simply the aeolian mode. Period. While the definition from the text is being written from a post-analysis perspective, it's not wrong in the slightest.
Music has always been more practical than scientific, and that trying to ascribe purely scientific principles to post-describe what happened in practical terms is somewhat problematic. It's interesting to look at this way for sure, but it's not indicative of how music evolved.
Let me give you an example. In the text A History of Western Music, Peter Burkholder writes
"Aristoxenus (a pupil of Aristotle) distinguishes between continuous movement of the voice, gliding up and down as in speech, and diastematic movement, in which the voice moves between sustained pitches separated by discrete intervals. A melody consists of a series of notes, each on a single pitch (in ascending or descending order)."
Additionally, in ancient Babylonia/Mesopotamia and Greece, tetrachords which were the common unit or space for four-note combinations (to which there are three most common iterations), span a 4th and not a 5th. There's a lot more, but a lot of what we know of reinforces the idea that melodic content was formed around speech patterns and scalar melodic tendencies rather than the harmonic series (which ofc was only 1st proofed in the 14th century). Of course I don't know 100% because I'm not a musicologist with expertise in ancient theoretical models, but I'm going off a lot of what I've learned, intuit, and have in books.
On November 22 2014 01:10 kaleidoscope wrote: will Part II and Part III be posted anytime soon? i'm quite interested in this
As a "playing what feels right" guitarist trying to learn music theory, I'm also quite interested in this. I also have a mathematics degree so that makes me doubly interested in what the exact science is behind music.
On November 22 2014 01:10 kaleidoscope wrote: will Part II and Part III be posted anytime soon? i'm quite interested in this
As a "playing what feels right" guitarist trying to learn music theory, I'm also quite interested in this. I also have a mathematics degree so that makes me doubly interested in what the exact science is behind music.
It is not exact science in the sense of tuning. If you try to compute pitches with the different approaches I describe here, you get slightly different pitches. Guitars and piano use a tuning which does not get either approach exactly right, but which offers a good approximation.
This tuning leads to C# and Db (and other enharmonic notes) having the same pitch, while the perfectly tuned frequency would be slightly different. But I focus on white-key notes for the basics.
Older penatonic scales, like C-D-E-G-A offer quite nice ratios to note C:
C = 1:1 D = 1/8 higher than C E = 1/4 higher than C G = 1/2 higher than C A = 2/3 higher than C C' = 2x as high as C. 2x or 1/2x the pitch fits extremely well, so that we use the same note name.
Simple ratios are easy to discern and to understand, and easy to sing along with. C-G, ratio 3:2, is especially nice. Not only the ratios to note C, the ratio of any note to any note is a more or less harmonic, "consonant" interval.
There is another principle at work here, the already mentioned 3:2 ratio G-C, the "perfect fifth" (since it is the fifth degree in our today's 7-tone, diatonic scale.)
Lets view fifths.
C - G - D' - A' - E''
We have a chain of fifths. And getting the notes of the pentatonic scale. Some notes are in different octaves, but they don't matter for the note name.
Lets view the common diatonic scale C-D-E-F-G-A-B in fifths:
F-C-G-D-A-E-B
Again, we have a chain of fifths. There is another principle at work at the same time, important for tonality (for establishing a note as tonal center.) Lets look at a major triad, C-E-F. The pitch ratio of a just intoned major triad is 4:5:6.
4:5:6 is a part of the overtone spectrum. (Every sound has its base pitch, as well as its double, triple, quadruple and so on frequency at the same time.) That means, each sound contains its own major triad! Therefore the major triad is a very natural sound.
Lets us have a look at the C-major tonic triad: C-E-G. (Note a triad also spans the fifth, C-G.)
So if we use a chain of three triads, the subdominant F-A-C, a fifth below the tonic triad C-E-G, and the dominant triad a fifth above the tonic, G-B-D, we again get all seven diatonic notes.
These three triads are important to create a tonic center, in this case "C".
The diatonic scale includes some disharmonious, "dissonant" intervals like F-B or B-C. This expands not only the melodic possibilities, these intervals can be used to generate tension and to create the desire for resolution.
That was the best intro to music theory I've ever read. Was Part 2 ever published? Sorry to bump but it's so good and I do hope there's a sequel somewhere.