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.

the third(s). vertical harmony

we hear major and minor thirds every day. almost any harmonised melody in existence is bound to contain a bunch of thirds or sixths (their inverse), and as people we seem to like the sound of them, gentle, easy, warm. if you get a few musicians or singers in a room together and give them a simple tune to play around with, one of the most natural things to do would be to stick some parallel thirds or sixths above it,or below, sometimes major, sometimes minor, to fit the tonality.

when these kinds of things happen, when musicians just go with the flow and whip some harmony out, blending melody and harmony into something really tasty, tuning happens almost by accident. good tuning. easy tuning. natural tuning. if there’s a major third, especially if there’s only two parts, it’ll be very close to a 5/4 (386 cents). it’s just what happens. it’s how most of us sing, especially when we’re not thinking too hard. it’s what happens when a string player double-stops when playing solo, because any ‘unnatural’ third will grate on the ears.
most of the time, smooth harmony means smooth, small numbers. and 5/4 definitely fits that bill.

while most string players’ perception may be that the ‘correct’ intonation of a major third might be at or very close to 81/64 (around 408 cents), four perfect fifths up of 3/2 each (minus two perfect octaves), and most instrumentalists who don’t think too much about tuning might say 400c (the one major third available in 12EDO, 4 semitones up) is the correct, or ‘only’ major third, the reality of music practice begs to differ. even within ‘classical’ music circles major thirds are frequently lowered to 390 cents or so, and if a stable major third or triad is needed, especially when sustained, even a third of 390 cents will stick out as ‘wrong’, the 5/4 comes into play. when you have a vertical major third, it’s very easy to play 5/4, but playing 400c or 81/64 or an ‘expressive’ 410 or 415 cent third is remarkably difficult.

it’s especially true of singing. the 5/4 is to most ears the only natural major third that comes to mind, though many might not have thought about it, the note on that piano or guitar you’re practicing with is the wrong note. try this:

play or get someone else to sing a drone on a pitch from 12EDO. find something mid-range. listen to the drone. now sing a major third above it, and hold it there. is it stable? is it warm? is it the right third? if you like the sound of it, check it against the major third on your 12EDO instrument, piano, guitar (as long as it’s been carefully tuned), or electronic instrument. what you should notice, is that the third on the instrument is not the third you want, not the one you’ve been singing, but around 1/7 of a semitone sharp. this is an experiment i should have got you to do right from my first post, before i delved into numbers and history and terminology. something fundamental to a lot of tuning practice. much of the reason many shy away from 12EDO, it just doesn’t have all of the right notes. 50, on the other hand, has really close approximations to quite a few of them.

[bonus experiment: try a minor seventh over a drone. see if you can get it to lock into tune, a powerful, almost otherworldly sound. a little lower than you might expect. this time you’ll find your 12EDO instrument is 31 cents (very nearly 1/3 of a semitone) sharp of the 7/4 you’re singing. owwww. although the temperament on 5 might go unnoticed by some, 12EDO really does not represent intervals of 7.]

ok so back to the thirds. if we agree on 5/4 being appropriate in most situations as the ideal major third (with 50EDO’s approximation @ 384c, from now on, simply the major third, only just over 2 cents flat), then what’s our ideal minor third? this question is perhaps a little harder to give one answer, there are three simple minor thirds that could all be useful in different contexts, as well as a few other far more complex ones. 6/5, 7/6, 19/16, 13/11, 32/27, 20/17…

6/5 (around 316c) is the classic choice, with an odd-limit of 5 (nice and simple). it is the 3/2 complement of 5/4, meaning if you set your fifth to the simplest ratio possible (3/2), and divide that into a 3/2, the remainder will be 6/5, either at the top (major chord, harmonics 4:5:6), or at the bottom (minor triad, harmonics 10:12:15 or subharmonics 6:5:4) of the major third. this minor third is brilliant when accurate, but tends to be  little discordant when not approximated well (e.g. off by more than 5-10 cents or so). in 50EDO, the minor third @312c is within our tolerance, and sounds pretty great.

but even though one might unconsciously go for the 5/4 major third even in an unaccompanied melody, minor thirds are often a bit lower than 6/5, which might have something to do with our familiarity and even preference for a tone close to 12EDO’s minor third. which is actually ridiculously close to 19/16. though people often try to explain 12EDO as a 5 prime-limit system with rather large deviations, it might be more helpful to think of the primes 2, 3, 17, 19 (with an absence of both 5 and 7):
1/1 17/16 [or more accurately, 18/17] 9/8 19/16 24/19 4/3 17/12 3/2 19/12 32/19 16/9 32/17 [17/9] 2/1, giving an error of between about 1 and 4.5 cents.

so why do we like 19/16? the numbers aren’t as small as 6/5 or 7/6 or even 11/9, and the prime limit is much higher (19)…
the answer is likely difference tones and what some call virtual fundamental.

difference tones are present every time two or more frequencies/pitches interfere, and are simply pitched at the difference between the pitches. if we have two tones at 440Hz (concert A4) and 528Hz (a little above concert C5), then the difference tone will be 528-440=112Hz, or around F2. in frequency ratios, we can represent 440:528 as 5:6, so the difference tone will be 6-5=1, or the fundamental of the series where 440Hz is the 5th partial and 528Hz the 6th. so the dyad 5:6 gives the triad [1:]5:6 in the first order, then [1:]{1:}{4:}{5:}5:6 when including second order difference tones. difference tones generally get fainter as order increases so there’s not usually much point looking beyond the first order.

so a 5:6 minor third actually has a virtual fundamental a major third below the lower tone, as it forms a full 4:5:6 major chord through its difference tone. which means when we have a full triad of 10:12:15, we get [2:3:5:]10:12:15, or octave reduced to [8]:10:12:15, a major 7th chord, because of the added difference tone a major third below. which means we have somewhat of a conflict of roots, e.g. if we had C minor we’d also be hearing a low Ab.

the 19:16 third, however, is a little more harmonically coherent if you will, with the lower pitch. [3:]16:19 means the difference tone is on a low dominant bass note, much like a classic V64 chord before a resolution to V53 — and we can do this if we move from 1/1 and 19/16 to 15/8 and 9/8, with the difference tone sustaining the bass, even though the dominant note is not present in either dyad!!! so when we add the fifth to our 19-limit minor third dyad, we reinforce that fifth already present way down at the bottom of the chord, and thus a minor triad like 16:19:24 is fairly strong [although we also get a bit of a clash of thirds with [3: fifth][5: major third][8: octave]16 octave:19 minor third:24 fifth

***NOTE: one of the few things about 50EDO i don’t like is its lack of an accurate 19:16. at 297.5c it can be represented almost equally well (or badly depending on your viewpoint) by the lesser third of 288c or the minor third of 312c, although other ratios of 19 such as 19/18 (as 96c), 24/19 (as 408c), 19/14 (as 528c), and their inverses,

the other obvious third is the sub or subminor third, at or nearby 7:6 (264c in 50EDO), which due to a first order difference tone a fifth below the ‘root’ of the chord leads to an implied [1:]6:7, or a 1-5-sub7 triad (perhaps we could call this a power seventh?).

finally the last simple third is the 9/7 (approximated well by 50EDO’s 432c), the super or supermajor. which can represent a rooted tonality but is much more intuitive to use as a harmonic extension on a different root, most obviously the sub seventh below the bottom of the dyad, as implied by difference tones, e.g. D and F#^ implying a possible root of E^ for a 1-sub7-9 triad, or the fifth above that for a sub triad, e.g. B^ D F#^

once we have these thirds we’ve at least got what we need for simple and strong vertical harmony, with each type of third pointing towards a certain ideal 3 part harmony based on difference tones: root position triads, 64 minors, major 7s, sub minors and sub7s, sub9s…

next time i’ll try and cover the not-so-obvious, or not-so-simple thirds in a bit more detail. these others aren’t particularly recognised in western theory, and we might have a tough time understanding or fighting with difference tones, but they’re a bunch of fun to play around with and allow us more tonal identities than we might be used to