EPT<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>
