a brief introduction to tuning, numbers, history

so i’ve decided to write a blog about something i’m passionate about. it’s not so obvious what i should post here, who my intended audience is and all that so i guess i’ll start with some basics.

intonation in music had been a defining feature for millenia, before music could play played one had to ask, with what pitches? before one can think about melody, or harmony, or voice leading, or voicing, one must first decide on pitch.

it seems in our modern world pitch is given too little attention, with the most obvious step being selecting one from twelve, the twelve notes found on almost all instruments and in most musicians’ heads as the only notes, the ‘right’ notes, handed down through the ages from the hands of the musical gods. this is rubbish.

those twelve notes, twelve relationships, that have come to define so much about modern music, are an invention, one simple cold calculated solution to the problem of which notes to choose, and where to stop, in defining a musical universe that is practical to play around in. twelve equal divisions of the octave: 12EDO.

all in all it’s a rather good system for its simplicity, with twelve equally spaced pitches we get only twelve unique interval classes: a minor second, a major second, a minor third, a major third, a very nearly perfect fourth, an augmented fourth, a very nearly perfect fifth, a minor sixth, major sixth, minor seventh, major seventh, and perfect octave. we are taught in modern theory that these twelve intervals are enharmonically equivalent to other intervals, that for example an augmented fourth is ‘the same’ as a diminished fifth, because they share the same key on the instrument, they are the same pitch even though we might call them different names or give them a different symbol on the page. that we should try to spell according to the classical rules of harmony and voice leading, but that in practice it’s all the same, a diminished fourth IS a major third, and an augmented unison is a minor second, and so on and so forth…

but where did these interval classes come from? how long have we had this conflation of perhaps twenty-one different categories (or more) being crammed into a grid of twelve notes?

there was a time only a few hundred years ago when a G# could never be an Ab, where no two notes were ‘equivalent’ in the modern sense.  if you wanted an augmented fourth and not a diminished fifth, that’s exactly what you got, not some middle-of-the-road approximation of both.

12EDO is a tuning that optimises the tuning of fifths, their sums, and their octave inversion, fourths. its ancestor is a tuning known as Pythagorean, though Pythagoras was evidently not the first person to come up with it. The tuning uses one ratio as its generator, the ratio 3 to 2, meaning when you go up a 3/2 the higher note’s frequency is 3/2 times that of the lower note. in musical terms, 3/2 is a perfect fifth. Really perfect. beatless. beautiful. every third harmonic of the lower tone lining up with every second of the higher note, it’s a focussed powerful sound, very familiar to most traditional musics, and our modern 12EDO version of 7 ‘semitones’ is pretty darn close, with an error of less than 0.28%. if we take a chain of fifths downwards and upwards from any arbitrary pitch, we end up with a chain like this, each step to the right going up a perfect fifth:

32/243-16/81-8/27-4/9-2/3-1/1-3/2-9/4-27/8-81/16-243/32-729/64

or in musical notation, taking the starting frequency 1/1 as C, we have:

Db-Ab-Eb-Bb-F-C-G-D-A-E-B-F#

although Db is now like six octaves below F#, so let’s bring the whole scale within one octave [doubling a frequency brings it up an octave, halving it brings it down] so that each ratio is between 1 (the initial frequency) and 2 (the octave above)

256/243-128/81-32/27-16/9-4/3-1/1-3/2-9/8-27/16-81/64-243/128-729/512, rearranging from smallest to largest (i.e. lowest to highest in pitch):

1/1-256/243-9/8-32/27-81/64-4/3-729/512-3/2-128/81-27/16-16/9-243/128

C-Db-D-Eb-E-F#-G-Ab-A-Bb-B.

Look familiar? twelve tone chromatic scale. the original, formalised actually well before Pythagoras in Ancient China (read the intro to Harry Partch’s Genesis of a Music for a more detailed and possibly humorous run-down on the history of tuning).

here though, the steps are Not equal, the augmented unison steps are larger than the minor seconds. and we can’t change the spelling willy-nilly without changing the ratios. F# means 6 fifths up from C, or 729/512, whereas Gb would be 6 fifths down, or 1024/729. different notes, Gb being less than a quarter of a modern semitone lower than its enharmonic partner. if we want to play F# with a fifth above it, and we’ve only got the twelve keys above, we have to use Db a the top, which at almost a quarter of a semitone flatter than a perfect fifth, doesn’t sound too pretty.

if we continue this chain we run into bundles of these close pairs, which means we either need an instrument with 53 keys (where the notes almost exatcly line up – see http://en.wikipedia.org/wiki/53_equal_temperament), or we need to stick to a set of twelve notes so we don’t wind up with any pairs. the clever thing is, if we narrow each fifth just a tad, the sharps will drop and the flats raise, so that if we set our fifths to the twelfth root of 2, as they are in 12EDO, all of the pairings merge to a single note

the perfect fifth is what gives most modern harmony its power, the reason power chords in rock and pop and fourth harmonisations in jazz are popular today is probably because the 4/3 fourth and 3/2 fifth are very well represented in 12EDO, just as the fifth was given extreme importance prior to the renaissance period. 3/2 is a beautiful stable consonance.

but it is not the only beautiful stable interval. ratios of the magic numbers 2 and 3 reigned supreme throughout a lot of early musical history, but 5 soon came to be seen as warm, emotive, and important in expressing human drama in a way ratios of 3 and 2 couldn’t do so easily. what’s special about 2, 3, and 5? they’re all primes, and though there are differing opinions on their fundamentality, primes are important building blocks for rational harmony.

if you’ve ever heard overtone singing in a tonal context, or tried tuning a fretless instrument playing solo, or heard any good a capella quartets or choirs or small string groups, you probably already know the sound of 5. 5/4 is the major third, the Perfect major third, the one most people will sing or play instinctively because in most contexts it sounds miles better than the 81/64 major third you get by going up four perfect fifths, or even the major third in 12EDO, which people keep telling us is ‘right’ and ‘in tune’…

so how about we substitute all those big ugly numbers out of our old scale with some simpler warmer harmonies? let’s draw up a little lattice with 3/2 fifths going horizontally and 5/4 major thirds vertically:

5/3   5/4  15/8 45/32
A-       E-    B-      F#-
4/3    1/1  3/2    9/8
F         C      G       D
16/15 8/5 6/5  9/5
Db+   Ab+ Eb+ Bb+        or, arranged by pitch:

1/1-16/15-9/8-6/5-5/4-4/3-45/32-3/2-8/5-5/3-9/5-15/8
C       Db+    D    Eb+ E-    F      F#-     G    Ab+  A-   Bb+  B-

still a 12-tone chromatic scale, but less even, with 34 different intervals…! a whole bunch of flavours to try!

i feel like i should stop here, didn’t quite expect to write that long of a first post intro, but hopefully it gives you a little taste of what numbers might mean in music, and perhaps the inkling that 12EDO might not be the ideal musical universe after all

2 thoughts on “a brief introduction to tuning, numbers, history

  1. If an orchestra or chamber group sounds in tune, they are not playing with equal temperament. The musicians are correcting away from equal temperament to bring the vertical chords into tune. Melodic or linear intonation has different requirements and is exploited for emotional effect (tension and release). A pure major third which sounds great in a major chord may not be the best choice if that note is needing some melodic “charge”. Then it instinctively gets raised – possibly higher than equal temperament. Only pianos and other fixed pitch instruments have to contend with the 12 note limitation. The rest of us have infinite nuances available.
    It’s also interesting that when trying to define a new set of pitches based on simpler ratios, if you start with C to generate the rest of the notes you get different notes than if, for example, you start with Db. etc etc.
    Good point about the a capella groups – but we don’t sing “Row Row Row Your Boat” with a flattened third.

    • I do, and I’m sure most singers instinctively do use the 5/4 third or something nearby a lot more than they use the 400c third or Pythagorean 81/64 ditone!

      Leading tones was a point I’m intending to write on soon, high leading tones have been in high demand for centuries, and are ideal in many situations, higher than 12EDO’s 1100c or even the Pythagorean major seventh. A leading tone as high as 27/14 (giving a lovely sub second at 28/27, 63c) or even 22EDO’s ridiculously high 1145c may be just right in different contexts.

      However, most [classical] musicians bag on low leading tones, when really they haven’t had much experience at all with them. low leading tones were the shizzle in the 16th and 17th centuries, and I prefer them most of the time if I want rich harmony, a sense of stability, and an interesting melodic contour.

      When I write an entry on ‘semitones’, chromatic and diatonic seconds, and leading tones, I’ll show how even syntonic systems with low major sevenths can support high leading tones and a variety of voice leading options.

      The ‘rest of us’ (as well as us microtonalists) Have infinite nuances available, but not too many of them are recognised in the current theory. A point I’m very keen to make in upcoming posts.

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