A Strange Exercise in Footnotes...or, Pretty Much Everything I Know About Music Theory
What follows is "experimental writing," and as such it's avant-garde (weird) -- in other words it might not be as much fun to actually read as it was to conceive and create; conceptual writing, if you will, but I ain't no Gertie Stein. The piece started off as a blog entry in ca. 2004 -- I was informally writing about my electric guitar effects setup, and realized that I was introducing terms a "normal" audience probably would not know. So, I added a few footnotes (hypertext is great for footnotes). Then, in writing these footnotes, I introduced more unfamiliar terms. So, footnotes for the footnotes. Etc. Eventually I found myself talking about the physical/numeric foundations of Western music theory, having begun several hundred words earlier with my Boss digital delay/reverb pedal. I close with a subsection (perhaps the most "readable" part) describing the place to which my writing seemed to be pointing: how is music like math? That also became the title, til I came to think there was too much culture-of-math stuff on my site painting an unflattering portrait of intellectual vanity, and changed it to something descriptive.
I use a Boss digital delay/reverb pedal[1]. Playing with a delay, in DADGAD[2] and strictly with my fingernails[3], it becomes impossible to make an unpleasant sound. Everything I plays sounds like the crystal fairy symphony of the snow-covered hills of mystery. My tube combo[4] amp has gain[5] and reverb knobs, and for some reason a built-in phaser[6] that's pretty good, but that gets really old really fast. The delay never gets old.
Effects[7] are cool, but I think ultimately that kind of approach can be self-defeating.
Guitar effects can be a slippery slope -- one day you're plugged into a delay pedal, and the next you're shelling out $1,500 for a rack-mounted[8] effects processor[9].
I have mixed feelings about electric and electronic instruments. On the plus side, you can craft your own sound. On the minus side, crafting your own sound can take up an inordinate amount of your time and creative energy, and you end up tweaking your crunchy distortion[10] as opposed to learning to play better; it really does get obsessive quickly. Take a look at those rock 'n' roll music stores that sell drums, basses, electric guitars, and a huge amount of effects and amps. I think for many electric players, obsessing over their sound and tweaking it electronically takes precedence over playing. Also on the minus side: the sound is coming out of an amp-speaker, and sounds sort of artificial, at east compared to a resonating acoustic or classical guitar.
But, of course, it's possible to do both (focus on playing and at the same time be interested in your sound). I've just seen it get out of hand so much that I'm leery of it.
[1] Digital delay simulates an echo. The returning echo strength and time until the next echo are adjustable. Reverb is more or less the same thing -- it simulates the sound of a noise in a gymnasium. The length and "resonance" of the reverb are similarly adjustable. Many electronic guitar effects come in the form of a pedal that you can easily turn on and off by stepping on it. They usually do only one thing, or, in the case of a delay/reverb pedal, two very closely related things (reverb is simply echo/delay that is so rapid it sounds like one continuous sound).
[2] DADGAD is a way of referring to an "alternate" guitar tuning[a]. The name is derived from the notes[b] to which the six strings are tuned, from lowest-pitched[c] to highest-pitched: D, A, D, G, A, D. The bottom "D" is pitched at about 73Hz[d]. The top "D" is pitched two octaves[e] higher, or at about 294Hz.
[3] Most electric players play with a pick. A pick is advantageous in playing rapid melodies with a clear "attack" (a sharp, un-slurred beginning of the sound). Picks are also nice for playing rhythmic[f] strum-patterns[g], useful for playing along with a rhythm section. Using the fingers, with or without fingernails to aid attack, is advantageous obviously in that two or more strings can be plucked precisely and simultaneously, regardless of whether they are adjacent or not. This process can be simulated with a pick, but it's comparatively difficult, and of course multiple notes will never sound at precisely the same time using one pick-stroke to pluck them all.
[4] Guitar amplifiers either use vacuum-tube[h] amplification or transistor[i] amplification. vacuum-tubes are generally regarded as sounding better (you'll hear the phrase "warm tone," which refers to a pleasant, uniform signal distortion and fuzziness, associated with tube amps a lot), but transistors are cheaper and more reliable. Many amps, like mine, use a combination of transistor and tube amplification -- hence, "combo amp."
[5] Gain strengthens a signal before it is amplified. A stronger signal will overload an amplifier, producing a distorted sound. Distortion used to be something to avoid, but Jimi Hendrix changed that (there may have been other significant early pioneers, or he may not have been the first, but whatever). Over the years, the sound quality of an amplifier driven into distortion has been tweaked and honed. Now-a-days, there are many different kinds and levels of distortion, from a Neil Young "bluesy" tone to a Pantera "crunch" sound.
[6] A "phase"' simulates the wobbly sound of rotating speakers by modulating the strength, or volume, of certain frequencies in one direction while modulating others in the opposite.
[7] "Effects" is an umbrella term for everything one can do to process the sound of an electric guitar. Common effects include distortion, delay/reverb, phaser, and chorus (several voices at slightly different pitches). As far as I know, guitar effects fall into a category of delay-based (delay/reverb), volume-based (phaser), distortion-based, or pitch-based (chorus) effects. However, my analysis might be a little off.
[8] If a player has a lot of guitar effects that aren't pedals, he might put them all in a case for convenience. Such a case full of effects is called a "rack." Rack effects are usually higher-quality than pedal effects.
[9] An effects processor, usually rack-mounted, is a battery of multiple effects. Sometimes there's even a tube in them to improve the quality of distortion-based effects. Processors are pretty expensive.
[10] "Crunchy" is a term that describes the sound quality of guitarists from bands like Pantera, Metallica, Korn, etc -- it's so distorted that it almost sounds like white noise. Fun, but possibly somewhat limited. With that much distortion, chords[j] more complex than power chords[k] tend to sound muddled to most listener's ears. Most people don't play jazz standards with full-on crunch distortion, but maybe that'd be interesting.
[a] Guitar tuning has evolved over the years. While it's true that a lot of different chords and scale[aa] shapes are easily playable in the standard tuning (from lowest-to-highest: E, A, D, G, B, E), use of standard tuning is somewhat arbitrary. There is a nearly infinite number of ways to tune the guitar, such as DADGAD (self-referential), lute tuning (E, A, D, F#, B, E), etc -- there are too many to name. The use of standard tuning has the advantage of providing, in fact, a standard. If everyone agrees that the guitar will be tuned in this certain way, then techniques can be refined and built upon. A culture of guitar playing evolves around a particular tuning, and there begins to exist a collective library of theoretical knowledge. Since I play in DADGAD, no chord book, scale book, or fingerboard interval[bb] guide is useful to me -- I'm basically on my own. Of course, there is a culture surrounding DADGAD, but it isn't very theoretically involved, since DADGAD is largely associated with folk music. To help DADGAD theory along, I derived an interval chart for it. I've also discovered some fun chords, which I may post at some point.
[b] A note is an instance of sound of a particular, consistent pitch.
[c] Physically, a pitch can be defined as being of a particular frequency. Pitch is also determined, if not defined, aurally. Of two notes of significantly different pitch, one will sound 'higher' than another. In the Western musical lexicon, the pitch of notes have been fixed as standards. For instance, the specific note "A 440," which is used to tune orchestras, is an instance of the pitch that vibrates at the frequency of 440 times a second.
[d] Hz is an abbreviation for hertz (after Heinrich Rudolf Hertz), which is "a unit of frequency equal to one cycle per second" (dictionary.com). So, a note that sounds at 440 hertz vibrates 440 times a second. This is how sound waves are produced -- by an object vibrating in a medium, such as air or water. When the object moves, it disturbs the medium and causes a pressure wave to travel forth. When an object moves quickly enough and repeatedly in a similar way, several pressure waves of similar quality are produced in succession. Our ear drums are designed to detect a rapid succession of such pressure waves (specifically, those generated by vibrations of between 20 and 20,000 Hz), and our brains interpret those detected vibrations as "sound." If the vibrations are consistent enough in frequency (all being, say, roughly 440 Hz as opposed to ranging from 30-600 Hz), or brains interpret them as discreet "pitches." A guitar string, when it is plucked, vibrates in this rapid, consistent way.
[e] An octave is definable physically and describable aurally. A note that sounds one octave higher than 440Hz will be, by definition, 880Hz. so, doubling the frequency of a note produces another note one octave higher. Likewise, halving the frequency of that note would produce another note once octave lower. The quality of the sound of two notes exactly one octave apart is very similar -- in a certain sense, one might almost call them the same note. And indeed, a note sounding at 44Hz and one sounding at 880Hz are both referred to as an "A." furthermore, each octave is equally divided into increments such that there are 12 notes in each octave, including only one of the context-notes. For instance, in the context we were just discussing, there would only be one "A." the next "A" an octave higher would be considered part of the next octave, and so on. In the octave of "A 440," the next interval after that first "A" could be referred to as "A sharp" or "B flat." Whichever you pick is meaningless in describing the pitch of the note, but has rather to do with conventions of Western music; in short, it has to do with preserving the sequential alphabetic integrity of a series of notes. For instance, if our scale is an ''A melodic minor scale" (A, B, C, D, E, F#, G#, A), then we'd want to use "F#" to refer to the sixth note as opposed to "F-flat" just to preserve that scale's alphabetic integrity, even though in fact "F#" and "G-flat" are of the same pitch.
[f] Rhythm is a temporal quality of music. Imagine an event that occurs once every second (for one relatively static observer, or course). A note might sound only on that time-event, or once every half-event, and so on. Rhythm produces a sense of temporal vitality -- our memories of incrementally-occurring events that have passed give a sense of time to existence that wouldn't ordinarily be felt as strongly. It might be an interesting experiment to go through an entire day with while listening to a very simple rhythm (a single drum-beat every half-second, for instance). Rhythm can be thought of as a measuring device, and analogous to a ruler -- as a ruler increments units of space, rhythm increments units of time.
[g] "To strum" is to strike a few or all of the strings on a guitar in one upwards or downwards sweeping motion of the fingers or pick. a 'strum pattern' is a rhythmic, repeated sequence of strums.
[h] Vacuum-tubes produce amplification by generating electricity with heat in a vacuum. When a filament is heated up in an airless environment, a cloud of negatively-charged electrons is released, which are then drawn to a positively-charged metal plate. Between the filament and plate is a grid of wires to direct said electrons, and when an outside signal is sent to this grid (such as from an electric guitar), electrons produced by the heated filament follow the path of the unamplified signal from grid to plate. The amplified signal is then strong enough to power the magnets that cause the amp's speaker cone to vibrate, creating pressure waves in the air that the listener hears as sound.
[i] I can't figure out how a transistor amplifier works -- it's too complicated. But somehow, it does the same thing as a valve or tube amp, and manages to do it without any signal distortion. They must essentially work in the same way: there is a source of power, and somehow additional electrons are drawn from this source along the same path as an original signal. Furthermore, I believe that they are attracted along the path of the signal by something called an "electron field." Beyond that, I really don't know.
[j] A chord is any three or more notes sounding together. Usually, it's more specifically defined as three or more notes sounding together that also fall into something called "tertiary harmony," or the use of every other note as a chordal element. For example, a chord based on "C" might extend all the way up to "B" (C, E, G, B), and might even extend further (remember, the musical alphabet goes from "A-G" and then back to "A" again). This tertiary pattern can extend all the way into the next octave (notes in the next octave are capitalized): c, e, g, b, D, F, A. After that last "A," the tertiary pattern falls back to "C" again -- it can only span 2 octaves before becoming redundant. These chords whose upper elements fall into the next octave are known as "extended chords." However, every note in the extended chord does not have to be present for it to retain it's aural quality -- for example, a C13 chord, which by definition includes C, E, G, B-flat, D, F, and A, need only contain C, E, B-flat, and A to "sound right." The "B" is flat because a C13 chord is theoretically speaking most often used as the dominant chord in the key of F, in which the note "B" is always flattened to preserve the distinctive "major scale" sound, using "F" as a root note. If we didn't flat the B, and used only straight letter names (F, G, A, B, C, D, E), then that fourth note would be too high for that sequence of notes to sound like a major scale. Instead, we'd have something that sounded like an "F Lydian mode[cc]." Since chords are generated from scales, it's difficult to talk about one without going into some detail on the other. Every chord has a name, and some chords have more than one. This is because they can be correctly analyzed using different notes as the root. Consider the chord: D, F, A, C. This can be analyzed with "D" as the root note, in which case it's clearly a "D minor seven" (root note of D, a minor third from D-F, and a minor seventh from D to C). However, if you assume that F is the root note, then it's an F-major chord with an added 6th (F major 6). If you really wanted to get wild, you could even call it an "E-flat major 6/9 flat 5 with no root," but this probably wouldn't be contextually plausible. The more notes in a chord, the more plausible names it has.
[k] Power chords are really harmonic intervals (two notes played at once). The two notes are the "root" (or the "home base" note in whatever context we're talking about -- in this case, the context is the particular power chord) and the "fifth" (a note with a frequency a tiny bit less than 1.5 times the frequency of the root). Example: a "C" power chord would be the notes "C" and "G." One might also add another "C," one octave higher than the first "C." It's this mathematical simplicity (1:1.5) that creates matching vibrational patterns in the two sounding notes. Thus, the interval of the fifth being referred to as a "perfect" interval (along with the intervals of a fourth and an octave, which also have these very simple numeric relationships between their notes).
[aa] A scale is an incremented succession of notes progressing from one octave down or up to the next. For instance, from "C" to the next "C" up might be accomplished using all 12 tones: C, C#, D, E-flat, E, F, F#, G, G#, A, B-flat, B, C. Note I could have just as easily notated C# as "D-flat," "B-flat" as "A#," etc. The reason for my choices has to do with the most commonly used symbols in Western music -- one simply more often sees the note of a particular pitch written as a "B-flat" rather than "a#" because the keys in which this would make alphabetical sense are more commonly used. In turn, the reason these keys are more commonly used is partially that certain keys are easier to write music in because they have fewer accidentals (an accidental has "sharp" or "flat" following a letter-name of a note). This is clearly a circular process, but the Western musical notation system sometimes features strange logic. Getting back to our 12-tone scale, the note "A#" would interrupt the alphabetical integrity of a major scale rooted on E-flat, for instance -- in that case, the fifth note of that scale should be kept some kind of "B," rather than some kind of "A"). A scale can technically have as few as three notes, but the idea of a scale is to progress by intervals from one octave to another in such a way that progress is a notable quality, so this incremental approach is confounded by too few tones. The least number of tones regularly seen are five. These scales are logically called "pentatonic scales." Example: C, D, E, G, A, C (five notes, not including the octave). Every scale (just like every chord) has a name, but unlike ever chord, a scale only has one name. Here's one more example of a scale. D-flat harmonic minor: D-flat, E-flat, F-flat, G-flat, A-flat, B-double-flat, C, D-flat. Remember the concept of preserving the alphabetical integrity of a series of notes -- it's an important one. This is why F-flat and B-double-flat came up, even though the note "F" flattened by one semitone[aaa] sounds the same as the note "E," and "B-double-flat" sounds the same as an "A").
[bb] An interval is the name of the distance between two notes, and is referred to as a number. Consider the scale C, D, E, F, G, A, B (a C-major scale). There are seven tones there. It would stand to reason that they would be numbered according to their place, D being equal to 2, E being equal to 3, and so on. So, we end up translating our notation entirely from letters into numbers: 1, 2, 3, 4, 5, 6, 7. "1" is often referred to as "R," for "root." So, the distance from 1 (or R) to 3 is called a "third," from R to 4 is a"'fourth," and so on. One can also move the numeric designations to another scale, let's say a G-natural-minor scale (G, A, B-flat, C, D, E-flat, F, G). So, "G" becomes R, "A" becomes 2,"B-flat" becomes 3, and so on. But the distance from the root to the third note in this scale is different than the distance from the root to the third note in our C-major scale. So clearly, these numeric intervals can vary a little bit -- the third in our first scale is a bit bigger than the third in our second scale. We call this bigger third a "major third," and the smaller third a "minor third." Another way to look at it is without the imposition of 8-tone scale patterns -- let's just move from C to C in semitones, and then name the notes: C, C#, D, E-flat, E, F, F#, G, A-flat, A, B-flat, B. The pitch-distance from C to C# (we're no longer concerned with preserving alphabetical integrity using an interval system) is called a "minor second" (the same pitch difference as a semitone, but used in slightly different contexts). From C to D is a "major second" (also called a "full tone," or just "tone"). C to E-flat: minor third. C to E: major third. C to F: fourth, or perfect fourth. C to F#: sharp fourth or flat fifth. C to G: fifth or perfect fifth. C to A-flat: minor sixth. C to A: major sixth. C to B-flat: minor 7th. C to C: (perfect) octave. Actually, we can go on for one more octave, just like we did in our discussion on chords. We'll use differing cases to indicate that the note belongs in the next octave up. c to C#: minor ninth. c to D: major ninth (even though "major ninth" sometimes refers to a chord containing R, 3, 5, 7, and 9 -- it can be confusing). You can derive the rest yourself. The important thing to remember about intervals is that they're independent of notes. Any two notes have a nameable interval. For instance, from F# to B-flat is a major third (for it to make more sense alphabetically, one could say "from G-flat to B-flat" and be talking about the same pitches). From B-flat to F# is a minor sixth, which illustrates the principle of interval-inversion -- from the first note to the second note is equal to "interval x." Now, imagine a note an octave higher than the first note. The interval from the second note to this note one octave higher than the first is "interval y." For any three notes, the outermost being one octave apart, one can define an interval (from the first note to the second) and then an inverse interval (from the second note to the third). These relationships are consistent: the inverse of a major third is always a minor sixth. The inverse of a minor seventh is always a major second. The inverse of a perfect fourth is always a perfect fifth. The rest can be easily derived, but some memorization helps, too.
[cc] A mode is the particular sound quality of the progression of any natural (not sharpened or flattened) letter-name up or down to the next octave, moving sequentially from natural note to natural note. Consider this scalar progression: F, G, A, B, C, D, E. The quality of this particular sound is called a "Lydian mode," and is differentiable from a major scale by its sharpened fourth (if we were to make an F-major scale, we'd have to flatten the fourth, or make it into "B-flat"). This Lydian pattern of scalar sound can be applied to any note and it's octave. For instance, an E-flat Lydian mode would consist of the notes: E-flat, F, G, A, B-flat, C, D. The important thing to remember about the Lydian mode (and any other mode) is its interval pattern (little "m" means "minor" -- big "M" means "major"): R, M2, M3, #4, 5, M6, M7. Here are all the modes -- each corresponds to the natural notes from one up or down to the next octave. The modes, along with being referred to by Greek names, can also be referred to by numbers, the first being the mode that corresponds to the scalar-quality of "C" to the next "C" up or down. I'll give the number of the mode, then the Greek name, then the interval pattern. First (Ionian): R, M2, M3, 4, 5, M6, M7. Second (Dorian): R, M2, m3, 4, 5, M6, m7. Third (Phrygian): R, m2, m3, 4, 5, m6, m7. Fourth (Lydian): R, M2, M3, #4, 5, M6, M7. Fifth (mixoLydian): R, M2, M3, 4, 5, M6, m7. sixth (aeolian): R, M2, m3, 4, 5, m6, m7. seventh (locrian): R, m2, m3, 4, flat 5, m6, m7. a mode can in fact be derived from any scale -- just start on a different note than the usual root, and end one octave away. consider a harmonic minor scale, which sounds like an aeolian modal structure with a sharpened seventh tone, producing a minor third between the 6th and 7th: a, b, c, d, e, f, g#. so, a harmonic minor mode preserves this tonal structure but merely starts and stops at a different place. for example: c, d, e, f, g#, a, b. one can derive similarly analogous modes from any scalar pattern, and refer to them as an 'x mode' (modes derived from a melodic minor scale would be called melodic minor modes).
[aaa] A semitone is the smallest distance of motion permissible within Western scalar progression -- it is the smallest melodic interval, equivalent to the harmonic interval of a "minor second." The octave, from any note to the note double that first note's frequency is divided into 12 semitones, equidistant in pitch. From C to C# (or D-flat) is one semitone, as is from G to A-flat (or G#). Semitones (as opposed to whole tones, or just tones) naturally occur in a "C major" context between the third and fourth notes and between the seventh and eighth notes. The presence of only two semitones is, of course, preserved in all 7 modes -- they just occur in different places. Most scales are a combination of semitonal melodic intervals and full-tonal melodic intervals, with a few exceptions. Some exceptions are 1) the whole-tone scale divides the octave into 6 whole-tone intervals: C, D, E, F#, G#, A#, C (note that it's impossible to preserve perfect alphabetic integrity here -- we either have to call the sixth note A# or B-flat; even though the pitches are perfectly evenly spaced into six increments, the alphabet doesn't progress from letter to letter). 2) A chromatic scale from root to octave consists of all 12 semitones: C, C#, D, E-flat, F, F#, etc. 3) Any of the modal pentatonic scales: C, D, E, G, A. D, E, G, A, C. etc. 4) Any of the harmonic minor modes. 5) Any scale that doesn't consist of purely a mixture of tones and semitones -- you can make up your own and give it a name, although whatever you make up will likely have a name already.
How is music like math? In two ways:
1) The mechanism of music and sound is a physical process, and the phenomena of physics are elaborated upon with numerical and mathematical language. For instance, consider the note a tuning fork produces, an "A." this note sounds at 440Hz, or consists of pressure waves that hit the ear drum at a rate of 440 times a second, because they are produced by a tuning fork that vibrates in the medium of air 440 times a second.
2) Music theory is concerned with dividing up music into elements, naming those elements, analyzing relationships between those elements, naming those relationships, making analyses of the relationship between those relationships, etc. This is a mathematical process.
For example, let's look at a C9 chord. The notes in this chord are C, E, G, B-flat and D. This chord is used as the dominant for the key of F-major, because C is the fifth note in the scale of F, and the fifth is considered the dominant note of any root (in this case, C is the dominant of F). We use the notes C, E, G, B-flat and D because we use every-other-note in our system of tertiary harmony, or harmony that moves in thirds -- from C to E is a third (a major third), from E to G is a third (a minor third), from G to B-flat is a third (a minor third), etc. It's called a C9 chord because a note a third away from the seventh (B-flat) is a D in the next octave up. If you extend the numbering system past 8, that D is referred to as a 9th as opposed to a 2nd of the next scale, since our context is the original scale.
Furthermore, it is possible to derive relationships from simpler relationships in an axiomatic way. For instance, from pitch we derive the extant notes in the Western musical lexicon. From the extant notes and tertiary harmonic principles we derive chord theory. From chord theory, we can derive, identify and name complex, altered, and polyphonous chords. The progressive nature of music theoretical concepts (one relies on the one before it and so on) is analogous to axiomatic deduction in mathematics.
See? inherently mathematical.