application

50EDO might seem like an impossibility, another musical universe with nothing tying it to the system most people claim to think in. it shares no common pitches with 12EDO, and has more than 4 times the number of intervals. which means we can’t apply what we already know right? we have to start learning from scratch?

nope. although we spend almost all of our time as musicians locked into something at least close to 12EDO, a lot of musical thinking follows the rules of meantone, and traditional music theory is pretty much based on it. although some music is very free with its enharmonic spelling (like spelling a C# major chord with F standing in for E#), or even benefits from this freedom (so, say, we don’t have to use key signatures with 7 sharps or more), and we can always count things in semitones – our atomic unit of pitch in 12EDO, the traditional theory says to spell correctly, so that a fifth up always looks like a fifth, and so while F is a minor seventh up from G, E# is an augmented sixth, an important difference in terms of tonal directionality, but also traditionally could have been a matter of tuning (although augmented sixth chords mostly came into use when meantone was fizzling out and being replaced by circulating well-temperaments)

‘flats’ being ‘higher than sharps’ in enharmonic pairs a diesis apart (e.g. Db vs C#, F vs E#, Dbb vs C) fits very well with our standard notation, so that ascending intervals almost always look ascending on paper (note however that the interval from Cb-B#, a semitone in 12EDO, is now only a coma of 24c, or half the diesis, which in 12EDO, is tempered out!)

the main thing about meantone though is its mean tones, right? if we accept the major third as needing to be around 5/4 (in our case a smidgen lower at 384c), then we aren’t going to divide it into a larger and smaller tone, but two even tones.

this may seem like common sense, and 12EDO also follows this pattern, but non-meantone systems don’t. Just Intonation, schismic tunings and superpythagorean tunings with a larger tone at 9/8 (204c) or higher will need the additional tone to be around 10/9 (182c) to balance things out. which is fine if you know your way around comma adjustments and make sure you don’t end up with wolves around the place. meaning ‘traditional’ (meantone) chord sequences like ii-V would often require a comma movement from the root of ii up to the fifth of V. the comma represented by “-” here might be between 17c [as in 72EDO] and 55c [as in 22EDO] for example.

a four part ii-V-I on C:

D- D  E-
A- B- C
F   G  G
D- G  C

in meantones of course, this comma is tempered out, so it certainly makes diatonic chord sequences a lot easier.

so far so easy right? but, i hear you saying, surely we’ll run into trouble when we play something which Assumes enharmonic equivalence, like a symmetrical pattern which divides the octave into any number of equal parts, notably 3 or 4 or 6 (factors of 12 which are not factors of 50).

let’s first try the division of the octave into thirds. in 12EDO this would be an augmented chord, which, when inverted retains exactly the same intervals.
with 50, it seems like this is impossible, but instead we just have a few different options.

if we want to repeat the same interval three times, then we won’t land on a perfect octave, but we can land close, or a bit further away and ‘bend the ear’ a little. a classic augmented triad would be a stack of major thirds (16\50 steps), so C, E, G#, leading to the next note being B# (48\50), 48 cents lower than we started. which means if we build major chords on each root, we would go around 25 of the 50 keys (all of the even steps) before getting back to where we started, instead of just three steps C-E-G#-C.

so we could use a slightly larger interval of 17\50, or 408 cents. very close to our 12EDO major third of 400c. but 3 times 17 is 51, so we end up modulating up a comma of 24 cents. this could be very interesting on the ears. C, E+, G#^(Ab), C+. micro-modulations, you might call these. obviously if we keep going, we will get through all 50 keys, since 17 and 50 are coprime.

but if we want to land up on the same C after just three steps (which might be a desired option in a more familiar style), we simply need major thirds of different sizes. which we have, 16 steps (major), 17 steps (greater) and 18 steps (super) [also possibly the greater middle and augmented thirds at 15 and 19 steps respectively if you’re getting really experimental]. once option is 16+16+18 and its permutations, two major thirds with the remaining being super, the only solution in regular 12 tone meantone subsets, e.g. C, E, G#(Abv), C. but of course we also have 16+17+17 and its permutations, which would be the smoothest option for those familiar with 12EDO and thirds close to 5/4. but it would mean if you were elaborating on this three tone structure by splitting it into a six tone [uneven] ‘whole-tone’ scale, or harmonising in triads, you would once again have to choose from a few options, which might be a rather good excercise.

my pick for a harmonically appealing ‘whole-tone’ scale might be C, D, E, F#+(Gb-), Ab, Bb, C, with steps of 8, 8, 9, 9, 8, 8, although anything from 7 to 10 steps might be ‘perceived as a kind of whole-tone’ in this context.

likewise diminished [4-step] and octatonic [8-step] patterns are also assymetrical, but beautiful to play around with the options, with different combinations benefiting from anything from strong harmonic foundation to a smooth melodic contour, or just something a little whacky.
C, D, Eb, F, Gb, Ab, A, B, C (‘classic’)
C, D, Eb-, F, Gb-, Ab-, A+, B+, C (‘quasi-Pythagorean’)
C, D, Ebv, F, Gbv, Abv, A, B^, C (‘sub’)
C, D-, Eb, Fv, Gb-, G#, A+, Bv, C (‘quasi-equal’)
are all lovely rather different octatonics. and there are many more, plus their modes, each has its own place in some musical style or other.

in fact there are multiple reinterpretations of every single scale in 12EDO, every chord, every musical entity. some of them may have to be bent to fit 50’s pitch grid if you like, but there are always options.

if theoretically correct spelling is preserved in a piece of 12EDO music, then there should be no problem whatsoever in adapting it to 50. the main consideration which might come up is which kind of augmented, diminished, and ‘dominant’ seventh chord to use.

due to harmonically consistent spelling we can assume the basic major and minor triads to be 0-384-696c (16,29 steps) and 0-312-696c (13,29 steps) respectively. but for dominant sevenths, we can choose 1 of 3 notes for the seventh if the chord, as well as alter the third, depending in how much tension or stability we want.

i would class dominant sevenths as chords of (16/17), 29, (40/41/42) steps, with other alterations making the chord a bit unique to just call a seventh. first we have the classic or diatonic seventh 0-384-696-1008, built from the classic diatonic scale. the seventh is rather tense and high (diminished fifth against the major third) in comparison to the smooth triad, so it’s great as a tense moment in diatonic contexts. particularly in earlier styles where smooth triads are the normal and sevenths are an occasional heightened dissonance. next we have the sub seventh, 0-384-696-960, the closest 50 has to 4:5:6:7, straight from the harmonic series, more concordant and rooted, without much need to resolve to anything. this is my favourite in most contexts, especially modern harmonic styles, but the seventh may be a little low melodically. third is the lesser seventh, 0-384-696-984, with the familiar symmetrical 600c ‘tritone’ between third and seventh. this would my pick for baroque and classical style repertoire (and possibly also more modern styles) where the seventh is a regular occurrence, but still needs to be resolved.

next time i’ll try and post a concrete example of this stuff in practice, by taking a 12EDO, Pythagorean or well-tempered piece and ’50-ifying’ it, where one can choose to preserve harmonic mood and shape or create a kind of parallel universe, where consonance, dissonance and voice leading can be very different indeed from how they were originally intended.

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