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