Context and Concept

Summary: It makes no difference how well you can sing if you don’t have anything to say.

Well, I’ve put this essay off for a good while now. I wasn’t looking forward to writing it, as it’s 1) not an easy topic to discuss in clear terms and b) it ties together several high-level concepts about art. Nevertheless, it’s come down to the wire. No more beating about the bush. It’s do or die. It’s high time. Let me give you a piece of my mind. Warning: this writing presupposes some basic familiarity with music notation.

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First, a lengthy example. Take any old string, or just imagine a string, which will probably work better. Stretch it and pluck it. It makes a certain tone, which varies as a function of the length of the string (shorter:higher::longer:lower). We’ll give this tone a name: Note I. Now cut the string in half, and pluck one of its halves. This will make a higher tone, which we’ll call Note II. The ratio of the frequencies of II:I will always be 2:1, because the fundamental frequency at which a string resonates is directly related to its length. In turn, the human ear perceives frequencies that have this ratio as octaves, or the “same note”. A higher note with the ratio 4:1 will be perceived as two octaves higher, for example.

Now take the original string and cut it into three parts of equal size. Pluck the new note, Note III. It follows, logically, that the ratio of the frequencies of III:I is 3:1, and the ratio of frequencies of III:II is 3:2. The ratio of 3:2 is the next simplest ratio besides that of 2:1, and is perceived by the human ear as a perfect fifth, another pleasantly consonant sound. (Play a C and a G on a piano for an example of this.) Some other simple ratios are 4:3 (the perfect fourth), 5:4 (the major third), 6:5 (the minor third), 5:3 (the major sixth), 8:5 (the minor sixth), 9:8 (the major second), & c. You get the idea.

Now, back before Bach, Western scales were based on these intervals. Gregorian chant, Palestrina, Machaut [1]; basically anytime before the 16th century, these ratios appeared in stringed instruments and the like. Pleasant sounding? You bet. This intervalic kind of tuning is called, appropriately, just intonation. As music became more and more complex, however, this approach became a problem. To wit:

C D E F G A B C

This is Western notation for a major musical scale. The C on the right is twice the frequency of the C on the left. I’m representing an octave here, after which the pattern repeats, ad nauseum. G is the fifth note, so the ratio of a perfect fifth is G:C, or 3:2. A perfect fourth: F:C, or 4:3. You get the idea. So, for example, what’s the ratio between E and G?

It is G:E, or (3:2, a perfect fifth):(5:4, a major third), or 3/2 divided by 5/4, or 3/2 multiplied by 4/5 (remember high-school math?) or 12/10, which reduces to 6/5. So G:E is 6:5, a simple ratio. G:E is a minor third (see above).

Now, what is the ratio between F and D? You would expect it to be a major third or minor third. Let’s find out. The ratio between F and C (perfect fourth) is 4:3. The ratio between D and C (major second) is 9:8. So take 4:3 divided by 9:8, or 4/3 multiplied by 8/9.

The answer? 32/27. And that’s not reducible. In decimal form 32/27 is representable as 1.185, repeating. A minor third (as in G:E, or C:A) is 6/5, or 1.2. A major third (as in E:C) is 5/4, or 1.25. The ratio of F:D in such a framework would sound strangely dissonant (slightly ‘flat’, as the musical parlance goes), and that’s not the only interval with a non-reducible fraction. The result of such tuning problems meant that when an instrument was tuned for a particular key, it sounded a little off when playing in all other keys. Quite a challenge for the musician with a large repertoire.

The solution: Equal temperament tuning. But first, a short aside!

In Western scales, *all* (yes, all) the notes are representable in this framework:

C C# D D# E F F# G G# A A# B C

The # key means that the note is ‘sharp’, or above the frequency of the lettered note indicated [2]. So, if I’m in the key of C, then the F is a perfect fourth above the C, or five semi-tones above the C (count ’em! C#, D, D#, E, F. Five tones.). If I’m in the key of G#, then the perfect fourth would be C# (five semi-tones: A, A#, B, C, C#). Hopefully that makes sense.

Now, it also makes sense that if such a pattern is transposable across the Western scale, then the ratios between the intervals must also be transposable. And that’s where equal temperament comes into play.

The interval between the octave and the fundamental frequency is kept the same, at 2:1. The rest are divided equally. There are 12 semi-tones separating the two octaves, so the ratio between adjacent semitones must equal the twelfth root of two. This way, the ratio between different notes becomes an abstraction, and unrelated to the actual key chosen, or the type of instrument played. All scales can be played on a single instrument. Progress [3]!

It also follows, logically, that there are many other systems of tuning. A brief overview from Wikipedia follows:

“Many systems that divide the octave equally can be considered relative to other systems of temperament (Writer’s context addendum for the reader: 12 tone equal temperament is often referred to as 12-TET, of which there are many others). 19-TET and especially 31-TET are extended varieties of and approximate most just intonation intervals considerably better than 12-TET. They have been used sporadically since the 16th century, with 31-TET particularly popular in Holland, there advocated by Christiaan Huygens and Adriaan Fokker. 31-TET, like most non-12-tone temperaments, has a less accurate fifth than 12-TET. It has been used in Indonesian music.

There are in fact five numbers by which the octave can be equally divided to give progressively smaller total mistuning of thirds, fifths and sixths (and hence minor sixths, fourths and minor thirds): 12, 19, 31, 34 and 53. The sequence continues with 118, 441, 612…, but these finer divisions produce improvements that are not audible. The explanation for this curious series of numbers lies in the denominators of fractions that approximate the logarithm to base 2 of the frequency ratios of the consonant intervals.

In the 20th century, standardized Western pitch and notation practices having been placed on a 12-TET foundation made the quarter tone scale (or 24-TET) a popular microtonal tuning. Though it only improved non-traditional consonances, such as 11/4, 24-TET can be easily constructed by superimposing two 12-TET systems tuned half a semitone apart.”

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Okay. Deep breath. The point of all this exposition is to display the many forms of tuning available in Western music. Is equal temperament less “true” than just intonation because it uses a power series to determine the frequencies of its constituency instead of a set of simple ratios? No, of course not. Each of these systems of tunings came into being because it has a function. The just intonation tuning is convenient for harmonically simple music and easy tuning. The 12-tone equal temperament tuning is convenient for harmonically complex music, as well as music that changes keys constantly. For more subtle melodic possibilities, perhaps a 24-tone equal temperament tuning is called for.

And I haven’t even considered non-Western tunings. A popular Indonesian scale, for example, uses a 7-TET scale. No perfect fifth, no melodies or harmonies that can be directly understood by Western ears that are used to Western scales. What about a scale that uses unwieldy ratios like 10/9, or a 15-TET tuning that only repeats every two octaves? To say that such a scale is wrong is as silly as saying the number 12 is better than the number 13. After all, any scale or system of tunings can ultimately be reduced to mathematical relationships; and anyone having heard African or Indonesian music for the first time can attest to difficulty of “learning to hear” the novel relationships between frequencies.

The type of tuning used in a particular piece of music is part of the context surrounding the piece. The type and number of instruments, the electronic or sound effects used, the length of the piece, and the use or lack of use of certain melodies and harmonies are also all context. Ultimately all sound can be reduced to a waveform, anyway, if you want to get really reductionist, and who’s to say that my waveform is intrinsically better or worse than yours? They are equivalent, valueless. Context has no inherent meaning or value; it is a system of choices that frame the concept, the intellectual drive behind a piece, the reason for its existence.

The concept exists as an abstraction, apart from the piece of music. It also exists separately in the mind of the listener and the composer. For example, take Cage’s notorious 4:33. The context is not important – it can be played in a concert hall full of listeners, or by oneself in a forest grove. The only context, it seems, is that of intent, that the piece is indeed going to be played.

The concept in Cage’s mind (from Cage’s Silence, 1961):

In 1951, Cage visited the anechoic chamber at Harvard University. An anechoic chamber is a room designed in such a way that the walls, ceiling and floor absorb all sounds made in the room, rather than reflecting them as echoes. They are also externally sound-proofed. Cage entered the chamber expecting to hear silence, but he wrote later, “I heard two sounds, one high and one low. When I described them to the engineer in charge, he informed me that the high one was my nervous system in operation, the low one my blood in circulation.”

There has been some skepticism about the accuracy of the engineer’s explanation, especially as to being able to hear one’s own nervous system. A mild case of tinnitus might cause one to hear a small, high-pitched sound, for example. Whatever the truth of these explanations, Cage had gone to a place where he expected total silence, and yet heard sound. “Until I die there will be sounds. And they will continue following my death. One need not fear about the future of music.” The realisation as he saw it of the impossibility of silence led to the composition of 4′33″.

The concept in my mind:

Fascinating. A treatise on the nature of what music can be. How many notes must one take away before the piece performed ceases to be music? 99% of the notes? Leaving just one note? Leaving no notes? The sounds of the other listeners/performers consist of breathing, rustling of clothes, shifting of feet. The play of my own heartbeat against a bird’s cry and the flag slapping against the flagpole in the high wind create an interesting rhythm, syncopated and dynamic. I close my eyes. A man and woman having a conversation grows and lessens in intensity. Still the bird calls. The flag has grown silent. And now a new player enters the piece. What is it? What is it? Ah, the muted percussion of a far-off helicopter. Perfect. This piece perfectly encompasses the serenity and tranquility of nature, the inestimable value of just listening.

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Naturally this is an extreme example, but the idea holds for all art. The context exists only to serve the concept. As a drummer, my physical skill and endurance exists meaningfully only as a function of the concept of the piece I’m playing. Does it matter whether I can play a double bass roll faster than 150 bpm? Only if the concept requires it. Does it matter whether I manage to not drop my stick or whether my cymbals are new, or whether my tom-toms are in tune? Only if the concept requires it.

Does it matter whether I can accurately describe the feeling of nostalgia, or happiness, or destitution by playing my drums and writing my songs?

Only if the concept requires it.

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CONTEXT IS NOTHING CONCEPT IS EVERYTHING

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Footnotes.

1. Ah, Machaut. The most well-known composer of the 13th century. What? You haven’t heard of him? Who cares! Seriously, though, if you’d like to know more, here’s a piece by him.

Quant es Moy – Machaut

2. Granted, the Western major scale can also be represented with ‘flats’, or notes below the lettered tone (where b = flat): C Db D Eb E F Gb G Ab A Bb B C. Db, for example, is the same frequency as C#. Whether a flat or a sharp is used is dependent on the context of the melody or harmony: it isn’t important.

3. The 12-TET is not a bad estimation by any means. A perfect fifth, represented by the ratio 3:2, can also be represented as a decimal, 1.5. The seventh twelfth-root of 2, or 2 to the seven-twelfths power, is 1.498. Pretty close. Similarly, a perfect fourth in just intonation is 4/3, or 1.333. In 12-TET, it is 1.335. (I still like just intonation more, not that it matters.)

The thing about “the thing about X”.

It’s notoriously hard to stay on topic during a given conversation. For example, I’ll start talking to a friend – quite vehemently mind you – about the ills of common society or insert tragically misunderstood topic here. And then they’ll misunderstand me. Note the gender neutral pronoun “they” which refers to “a friend” in the previous sentence. I believe that’s called an antecedent. Not that I’m particularly picky about grammar. I mean the sentence before this one isn’t even a full sentence, and this sentence is both unnecessarily colloquial and self-referential. But certain things I do dislike, such as the awkward he/she construction when referring to a person or golem of indeterminate gender. And the point here is that they as a means of covering up their either real or perceived misunderstanding will often launch into a new, wholly unrelated discussion about something that happened to them recently, or they’ll simply nod in acquiescence and move on to a series of complaints about things that piss them off. Which reminds me of another grammatical issue. It looks like “this.” Today’s title, “The thing about “the thing about X”.”, is an example of my preferred punctual (as in referring to punctuation) construction, which also occurs in this sentence. Consider it civil disobedience of the highest degree. “Yes,” he claims. “I’ve had it up to here with your grammatical “shenanigans”.” See? The punctuation goes outside of the quotation marks, unless it’s part of the quote. It makes sense to me, perfect logic, but the Grammarians of America refuse to bow to logic. As always.

I mean, come on, et al.