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

leading tones. semitones and other divisions.

all regular meantones have rather low major sevenths, in comparison to our modern 12EDO or Pythagorean tuning, as a result of flattened fifths combined with low (closer to optimal 5/4) major thirds, or alternatively, because the flatness of the the fifth is multiplied 5 times, e.g. F-C-G-D-A-E. many people see this as a problem, a reason to choose a well-temperament or closer-to-pure-fifth tuning over meantone, so that those major sevenths can be closer to the octave, a smaller leading tone providing not only a bit of tension to keep things moving but a rather nice close to phrases, small melodic minor seconds (100c or less, say) contrasting possibly with a reasonably large whole step of perhaps 200 or 210 or even 220 cents.

small step resolutions are very inviting, heightening dissonance before a cadence, and bending the listener’s ear just a little before resolving. historically many sizes of minor second have been used, and even today with 12EDO as a standard there is some debate on how high leading tones should be to provide just the right feel.

my view is that it depends on context. most people have overlooked – or simply not had enough experience playing and listening to – proper low leading tones, being the default option in meantones. those of 1/5-comma meantone at a pure 15/8 (1088.3c), 1/4-comma at 1082.9c, or 2/7-comma at 1079.1c, all rather different from say 12EDO’s semitone. to my ears, these are just right as major sevenths in most music without too much movement, a lazy large minor second up to the octave is rather restful, warm and means a pretty well-tuned dominant chord if resolving a perfect cadence. a lot of musicians argue that dominant chords and especially dominant sevenths should be anything but restful, with a sharpened third and possibly even seventh heightening tension, an active dominant being more effective than a static one. but i myself love the sound of leading tones in the above meantone systems, with the 120c seconds of 50EDO being particularly effective, especially if the tonic is sounded against it or in close proximity (e.g. in a pre-emptive resolution ala Purcell, or in a trill)

but of course this isn’t just about harmony, the high leading-tone aesthetic has a lot to do with melodic intonation, where a large minor second (say, 110c or more) will supposedly ‘stick out’ and weigh the phrase down. larger minor seconds are perhaps heavier in that respect. but how small is a good small second? The 100c semitones of 12EDO? 90c Pythagorean minor seconds? 84c (21/20)? 71c (25/24, or roughly the minor second of 17EDO)? 63c (28/27, really beautiful septimal minor second, made much use of in that La Monte Young piece i posted the other day)? 56c (22EDO’s minor second)? surely we’ve already surpassed what we can call a minor second, right? not so. i’ve found even 48c makes a great small second, just less than a quarter tone with regards to 12EDO. where does this interval show up? in 50EDO of course, along with a great selection of other minor seconds.

50EDO’s intervals in the minor second range are the following:

48c (arto or diminished second), 72c (sub second), 96c (lesser second), 120c (minor second), and if you’re adventurous, 144c (lesser middle second)

one really cool thing about meantones is that while their regular minor second will be on the large side, they will always have a smaller interval available in the form of the augmented unison, which often also represents the septimal minor or sub second. so although 120c is considerably large, we can go for a more regular semitone (this time it is actually half of the mean tone) at 96c, or use the sub second to give that 28/27 feeling <3, or we can actually use the diesis at 48c as a leading tone, rather effectively.

which means if we are resolving some kind of dominant cadence, a 120c minor second will give us a very close to pure major chord on the V, the 96c lesser second will give us a Pythagorean or greater third (408c, very close to 12EDO at 400, and very active in comparison), the sub second at 72c will give us 432c (a super third representing 9/7, an altogether different mood, which some might call ‘high tension’, though it’s actually much more stable than the greater third), and finally the diminished second, when used as a second, will mean we get an augmented third on the V chord, looking rather like 10:13:15, a tendo triad rather than a usual major.

of course anther option is to change the position of the dominant, so that placing it on the greater fifth (720c) will provide a pure third when we have a melodic semitone, and a super third where we have the melodic diminished/arto second in the melody. one more degree up at 744c almost doesn’t look like a fifth (being about 1/2 way in between 12EDO’s fifth and minor sixth), but it represents the regular wolf in meantone, e.g. G# to Eb, or B to Gb, and in 50EDO, this interval is actually a lot more consonant (at least to my ears) than it is in usual varieties like 1/4 comma or 31EDO. a cadence of the kind Vicentino was very fond of (in, or very close to, quarter-comma) might mean the bass descending from the ‘wolf’ super fifth to the tonic, while the leading tone on the super major seventh provides a pure third to the bass, while at the same time keeping the voice leading smooth with only a small 72c sub second. a very nice melodic minor second indeed, 3/8 of a 50EDO mean tone. in my 50EDO notation, we might have (in Bb major):

Cantus: Bb—-A^-Bb
Bass:     Bb-G-F^-Bb

or in four parts, a little more elaborate possibly not quite Vicentino’s style any more):

Cantus: Bb-Bb—–A^———-Bb
Alto:        D–D–Eb-D#^-E^-F^-F
Tenor:     F–G——A^—C^—–D
Bass:      Bb-G——F^———–Bb

although this is just a I-vi-V(4-3)-I progression, it’s got a bit of attitude. the dominant chord is raised by the diesis which means when we tie the Eb (minor sixth above G) over to the dominant, we get a sub seventh instead of a minor seventh, which being enharmonically equivalent to an augmented sixth (respelled F^-D#^) we can resolve it upwards just for fun (historically incorrect voice-leading but this is just a [bad] experiment really), the very smooth passing major seventh to octave on F^ leading us not to Bb^ but to Bb, possibly a very unexpected jump, though back right where we started. good luck singing that alto part…

a better example might be a modernised medieval style cadence to an open fifth and octave, with ‘double leading tones’: sharpened fourth and seventh ascending to the fifth and octave, and a lower voice moving from second down to the tonic. on D we might have:

Cantus: D—–C#^(tr.)-D
Alto:       G-A–G#^——A
Bass:      G-Fv-E———D

the only problem with this kind of voicing is that the fifth at 696c is a little flat when left open (without a third in the chord), the ear notices it’s not quite a perfect fifth in certain timbres, so either choose your timbres carefully, or maybe just don’t use it as a closing sonority, unless you want that hint of instability. filling in the chord would give us a more stable sound, but it would kind of kill the style we’ve got going.

but back to minor seconds, if you want to do something like 12EDO, or want the ability to be a little ambiguous, use the (here 96c) semitone, which can stand in for a whole bunch of different ratios, just like in 12EDO, it can mean the ability to go in just about any direction, it’s a bit tense (because it’s not really close to simple consonant ratios). we still haven’t had an example with the semitone so why don’t we try a more harmonically adventurous one like this:
F>Fsub7(b9)>Eb-m7>D+M13>D+sub7(b9)>Eb/G>F/G>G

F Gb-               >Gb-            >F#+                       G    A  B
A     C Db- Ebv Eb-     Db->C#+  D+      Eb >Eb F   G
C     Ebv    C      Db- C Bb-   B+ A+ Bb^ C-    Bb C  D
F                          Eb-               D+                         G      >G

ok that got way outta hand. but hopefully it at least helped demonstrate the ability of semitones to be ambiguous, and allow not only semitonal but greater tone modulations, as the semitone is a comma-inflected interval (either a comma-raised augmented unison or a comma-lowered minor second), comma-modulations are very easy, and I could just have easily moved to G+ instead of G (as the bass would suggest perhaps I should have, instead i held over the Eb, the b9 of D+ to give a minor sixth against G). there are just so many possibilities.

to be honest i’m usually much more concerned with harmonic events than melodic voice-leading, so i might use all of the different intervals in the ‘minor second’ range at different times, for different purposes, when i’m playing particularly involved passages. for me, the most obvious uses would be:

minor second: as  regular diatonic step, whenever playing a modal line

lesser second (semitone): as a trill, usually above the target pitch but can work from below too, and the most obvious ‘flat nine’, closest to the 17th harmonic

sub second: whenever resolving sub and super intervals: e.g. a sub seventh to a major sixth, a sub sixth to a fifth, a super seventh to an octave, a super third to a fourth.

arto second: when resolving arto and tendo intervals, or when doing something whacky, like moving between sub and super intervals- super sixth to sub seventh, sub third to super second, or when moving between regular and sub/super- fourth to super fourth (as in 4/3-11/8), or minor third to sub third (as in 6/5 to 7/6).

 

hardly any of this is going to make much sense without hearing it. for those of you bearing with me, thank you. but do some experimenting, find a way to play these things for yourself (my favourite is the programs Scala + pianoteq). i’ll try and make some demos in the next wee while to show off different kinds of resolutions using these steps and other intervals, but my main point in this post was that there are always options, and having many intonational possibilities even with just one chord sequence makes the world of harmony that much more exciting.

50EDO and why it kicks ass, types of thirds, a meantone series

For a year or so, I got heavily into meantone tunings, tuning my piano in a twelve note subset Eb to G#, realising with a set like that one can not only play beautifully rich major, minor, diminished and augmented chords in a number of keys but also those fun septimal chords, sub(minor) and super(major) triads and sevenths, and even making sense of the wolf

Doing a really DIY job of refretting an acoustic guitar to 31EDO and playing around with a 19 tone subset on keyboard (Gb to B#) made me realise that a lot of what makes meantone great happens over great distances in the spiral (or 31-tone circle), and that 12 notes didn’t really cut it if you wanted to do interesting stuff in several keys, without running into wolves by playing the ‘wrong’ enharmonic.

Falling in love with 31EDO and the closely related 31-tone quarter-comma meantone, I started feeling like this was a system I could settle down in so to speak, to musically exist within 31EDO as most musicians live within the bounds of 12EDO, it has a larger variety of concordant intervals than 12, and represents the 7 prime-limit incredibly well for its size, as well as a few good ratios of 11.

It’s hard to ask much more from a meantone. We have perfect or near-perfect 5 and 7 identities (the major third and sub seventh or augmented sixth), a fifth which is not too flat (19EDO’s fifth is about the limit before I start disiliking fifths altogether), and multiple different interval categories.

As well as minor and major we get sub (lower than minor or perfect, higher than diminished) and super (higher than major or perfect, lower than augmented), as well as middle, what a lot of musicians call the ‘neutral’ intervals, represented by quarter tones in standard current music practice

I really should have gone with an ascending sequence of meantones, starting with 12EDO (or 7 then 12 if you’d like), and ascending through 19 before we get to 31, 50, 81… This is part of what’s known as Kornerup’s sequence, which, adding the previous numbers gets us closer and closer to a Golden meantone division, where several intervals are in the Golden Ratio with one another. But it’s also a good way to build up an understanding of larger and larger meantone systems, through diatonic, chromatic, enharmonic and beyond.

12 we know (I hope), with 2 kinds of third-based tonalities in the basic sense, minor and major, although they are respectively 16 and 14 cents away from what we might call ideal thirds (6/5 and 5/4). 19 separates the enharmonic pairs of 12, provides distinct 1/3 and 2/3 tones for the augmented unison and minor second respectively, and allows us new sub and super sounds, which being closer to ratios of 13 than 7 sound pretty whacky, but are fun nevertheless. Major thirds are just a little too flat for my liking, minor thirds are pretty much bang on 6/5. 31 gives us very accurate ratios of 5 and 7, some of 11, and represents 5 different tonalities: sub(minor) [rather accurate 7/6], minor [1/4 comma flat of a 6/5], middle [very close 11/9], major [pretty much bang on], super(major) [not quite so accurate but still a similar flavour to 9/7]. It does an ok job at being an 11-limit system, but most of the time it’s better treated as staying within the 7-limit, where it does an excellent job.

So what more could we want? Why go past 31?

Because 50 has things that 12, 19 and 31 can’t do. It’s a meantone that can represent any interval at all with a maximum error of 12c, but represents a good majority of important intervals even up into much higher primes 11, 13, 17, 19, 23, etc, with very good accuracy. It’s a meantone that can almost sound like just intonation if you want it to, or can sound rather unfamiliar, yet with mostly very logical tonal connections.

Chords have a lovely warmth to it due to rather good the accuracy on all thirds, it is ideal for a lot of early music, modal or otherwise, and might provide a rather good way to analyse a large number of intervals within a meantone context, a master system if you will, with which to learn about and use new interval classes, as well as give flexibility to transform pre-existing structures and create entirely new ones within a 50-tone intonational universe.

I’m not saying it’s the be all and end all of tunings, but if meantone logic appeals to you, you couldn’t do too much better.

Here is a table of intervals, including cent values, interval names and note names. I don’t think the accidental symbols display properly on smartphones but hopefully it’s sweet on computers.

One important thing is out selection of thirds, and how close they are to certain ratios. As well as minor and major within about 4 and 2 cents of 6/5 and 5/4 respectively, and sub and super thirds within a couple cents of the septimal 7/6 and 9/7, we have a lesser and greater third in between, which might well be more familiar to those practiced in Pythagorean or expressive (slightly raised from Pythagorean) intonation, the kind practiced by most string soloists and generally thought of as ideal melodic intonation. But now instead of one middle third we get two, one closer to minor and one closer to major, both of which are familiar to the musics of Persia and the Middle East, but less so to much of western music, and might be viewed at first as a supraminor and submajor. Finally, we have two outliers which don’t at first look or sound like thirds, the diminished and augmented, or to use some newly coined terms, arto and tendo (from the Latin contract and expand), lower than sub and higher than super. I was skeptical of these two at first (they’re actually also enharmonically equivalent to the super second and sub fourth respectively!), but now really playing around with them and using them in the harmonic context of 15/13 and 13/10, I’ve realised what great little intervals they are.

Which all means we have a grand total of 10 types of thirds, 10 basic tonalities with which to build chords, scales, and music of many moods. [I made this little demo track, overdriven to demonstrate how different types of thirds in 50EDO can give us a strong difference tone bass line. Here I’m using 8 out of 10 different thirds once each in a chord sequence] This of course also holds for their inverse sixths, and since all thirds and sixths are rather close to simple just ratios, we get lovely in-tune 2 part harmony in thirds and sixths [with 3-part harmony often happening as a result of strong difference tones]. After getting really familiar with the thirds and sixths of 50, 31’s minor third seems pretty sour and 19’s major third just a bit wrong…

As 50EDO is pretty much the musical universe I’m operating in at the moment, there’s a lot that I want to do with it. But I wanted to introduce its features and get you all accustomed to it little by little first, before trying to explain the complex stuff like enharmonic modulations, irregular mappings to imitate other regular tunings, hyperchromatic and commatic voice leading, as well as something I’d like to focus on in future: bringing jazz and gospel harmony rooted in 12EDO into the universe of 50, and seeing what magic might happen

fifths. the spiral. circles. temperament.

just for anyone skeptical that a system of thirty one tones to the octave is ridiculous or impractical, keyboard (and other) instruments were built in the 16th, 20th and 21st century, and the Fokker organ in Amsterdam continues to see the performance of new compositions and old, and stands as a symbol of the thirty one tone movement. http://www.huygens-fokker.org/instruments/fokkerorgan.html

some other keyboard instruments to tickle your fancy
http://www.h-pi.com/eop-keyboards.html

and of course guitars
https://www.facebook.com/Swordguitars

thirty one tone music is really out there, you just have to know where to look…

anyway, on to today’s main topic: the spiral of fifths.

so, i’m guessing, if you’re a modern musician, you’ve been taught that fifths are arranged in a circle of twelve notes, and if you go up or down twelve fifths from any note you end up back where you started right? twelve fifths up from Ab is Ab? right?

this is a modern take on ancient theory, to suit the 12-notes-per-octave agenda, equal or otherwise. in truth fifths should be arranged in a spiral, regardless of how they are tempered.

as we saw in the thirty one tone post yesterday, the more fifths we go up, the more sharps in the key, and the more fifths we go down, the more flat. we should name the keys accordingly, and not assume that any pitch in the spiral is enharmonically equivalent to any other pitch.

so if you wanted to go up and down in fifths from C, you’d have:
…Fbb Cbb Gbb Dbb Abb Ebb Bbb Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# Fx Cx Gx Dx Ax Ex Bx…

this spiral continues infinitely in both directions, at least in theory, regardless of the size of the fifths. it might help to associate the height and length of the spiral with pitch at that point, so that if we travel up one node, we are always going a fifth higher in pitch, and down three nodes means going down three fifths, or a major thirteenth.

if fifths are equal to 3/2, we have a Pythagorean system with each 12 fifths taking us up around 23c in pitch, so that in ‘enharmonic pairs’, the ‘sharper’ of the two will be higher in pitch when both notes are taken within the same octave. e.g. B# higher than C, G# higher than Ab.

if fifths are equal to 700c, as in 12EDO, then each twelve fifths we go up will take us up 0 cents in pitch, i.e. we ain’t goin’ anywhere honey . but i said the spiral was infinite didn’t i? well, the note names should take in to account where you are in the spiral, so a fifth up from D# has to be A#, not suddenly Bb. even though 12EDO tempers the two notes to the same pitch, we still spell according to the spiral. we still spell the scale of F# major as F# G# A# B C# D# E# F#, though many musicians would much prefer seeing the more familiar Bb, Eb and F substituted for those i ain’t got no damn E# key on my piano notes.

so spelling is according to the spiral, but most of the time we’re actually hearing a circle of twelve fifths right? just like the textbook theory? well yeah, in what are called circulating systems the spiral has been cut off at some point (enough double sharps already!), and turned into a circle.

in 12 note circulating tunings, including 12EDO and any of the many well-temperaments, we cut the spiral off at 12 pitches, effectively making 12 fifths up exactly the same as 0 fifths up. which means probably flattening at least a few of them a little bit from 3/2 in order to avoid a wolf between the top and bottom of the spiral, e.g. between what would have been G# and Eb, a diminished sixth, if we take the 12 note chain Eb Bb F C G D A E B F# C# G#.

this alteration of a perfect interval, in this case 3/2, in order to merge two notes into one, is called temperament, and we see it all the time. almost all instruments in the modern world are tempered, mostly equally. we can flatten fifths slightly so that 12 fifths make seven perfect octaves (as in 12EDO), or a little bit more so that four fifths makes a double compound 5/4 major third (as in quarter-comma meantone, with 31EDO very close), or even more so that three make a compound 5/3 major sixth (as in 1/3-comma meantone, 19EDO very close). or we can widen intervals, widen our fifths rather dramatically so that four make a double compound 9/7 super third (22EDO gets close to this).

of course, temperament doesn’t just apply to fifths. we’re used to 3 major thirds adding up to an octave, but this is a particular tempered worldview. in regular theory, three major thirds makes an augmented seventh. which is only ever equal to an octave when major thirds are tempered to 400c each. in most systems three major thirds will either come out flatter (if major thirds are less than 400c) or sharper (when they’re bigger) than an octave. if they’re just 5/4s, then we end up 41 cents below an octave, just like in 1/4 comma meantone. which makes progressions like Ab-C-E-G# rather exciting!!!

four minor thirds? a diminished ninth! four 6/5 thirds gets us to (6/5)^4=1296/625, or 62.5 cents sharper than an octave. ouch. in diatonic tunings (built in chains of fifths), this means when fifths are equal to 700c, augmented sevenths and diminished ninths are equal to octaves, when fifths are less than 700c (think meantones), augmented sevenths are lower and diminished ninths are higher, and when fifths are greater than 700c (for example Pythagorean or super-Pythagorean, superpyth for short), augmented sevenths are higher and diminished ninths are lower. phew.

so why is this important?

the concept of a chain of a single generator plus the period of an octave (known as linear [rank-two] tunings) is pretty fundamental to scale building . a lot of the time this means generating scales by fifths as we’ve done here, known as syntonic tuning systems, where the fifths could be 700c or 703c or 696c or 720c or 686c (if you just love that howl).

of course we could have three generators (rank-three), for example 2/1, 3/2 and 5/4, which would give us an infinite lattice of what is known as 5-limit just intonation, all ratios containing the products and divisors of primes 2, 3 and 5. but most of the time we’d choose a set of them as our scale, like this twelve note scale i posted earlier:
1/1 16/15 9/8 6/5 5/4 4/3 45/32 3/2 8/5 5/3 9/5 15/8, or the 22 shrutis of the Indian Classical tradition:
1/1 256/243 16/15 10/9 9/8 32/27 6/5 5/4 81/64 4/3 27/20 45/32 729/512 3/2 128/81 8/5 5/3 27/16 16/9 9/5 15/8 243/128 2/1

or rank-four, with say, 2/1, 3/2, 7/4 and 11/8, to give us a rather interesting set of pitches, or rank n, but rank one should do for now. and in any case, syntonic tunings are probably the best way to explain important families of tuning suitable for almost all musics of the western tradition, and to build on what many of you already know about music theory.

if you’re wondering what some good circulating fifth systems are, you can start by looking at equal temperaments which have a useable fifth and approximate other colourful intervals along the chain. or simply take a chain of fifths of any size you like and see what you find.

5 and 7 EDO, with fifths of 720c and 686c respectively, will probably be at the limits of tolerance to what you might call a useable fifth. so they might provide an upper and lower bound on some tunings to explore.

check out this page too
http://en.wikipedia.org/wiki/Syntonic_temperament

from smallest to biggest, a good selection of equal temperaments [along with a good selection of musical tastings] might be:

685.7c: 7EDO [this album is one of the coolest things i’ve heard, definitely not an ideal tuning but very interesting]
https://archive.org/details/Knowsur-NanaWodori

690.9c: 33EDO [not really recommended, but what the hell]
http://soonlabel.com/xenharmonic/archives/1767

692.3c: 26EDO [i’m not really familiar with this one, a little strange]
http://midiguru.wordpress.com/2013/01/06/public-rituals/

694.7c: 19EDO [this one’s become rather popular, great for guitar, close to 1/3 comma]
http://www.broadlandsmedia.com/microstick/music/

696.0c: 50EDO [just amazing, my favourite, close to the 2/7 comma of Zarlino]

696.8c: 31EDO [funkadelic, popular, about as many frets as you can fit on a guitar, close to 1/4 comma]

697.7c: 43EDO [decent meantone, very close to 1/5 comma]

698.2c: 55EDO [Mozart’s tuning, great Baroque meantone close to 1/6 comma]

http://tonalsoft.com/monzo/55edo/55edo.aspx

700.0c: 12EDO [your old friend] [no links necessary]

701.9c: 53EDO [absolutely wonderful for primes 2.3.5.13, also as a master system for Indian musics]

702.4c: 41EDO [like Pythagorean, but with more attitude]

[also the basic layout for this beast of a keyboard http://www.h-pi.com/TPX28buy.html]

704.3c: 46EDO [great fun, just a little twisted]

705.8c: 17EDO [awesome as a guitar tuning]
http://micro.soonlabel.com/0-hosted-albums/heptadecaphilia.html

709.1c: 22EDO [another rather popular one, very odd at first, but a gem]

711.1c: 27EDO [interesting…]

720.0c: 5EDO [great pentatonic, just a bit bizarre]
https://soundcloud.com/search?q=5EDO

fwoah! that should do it for now. enjoy the links. next time i might spend a bit of time talking about my favourite of these, 50EDO, and why it’s so damn cool.

meantones. thirty one. primes.

ok so the first post i was thinking about introducing a bunch of theory from scratch but now i’m thinking, that’ll take way too long, and i’ll probably explain it poorly and it won’t be too much use anyhow. so now i’m going to try and talk about what i really want to talk about. meantones.

meantone tunings have probably been the most popular family of tunings in western music history. almost all music from the renaissance period was composed and performed with meantone theory in mind, and they continued behind the well-temperament scenes into the 19th century, being fairly recently revived by a push towards emphasis on early music and period practice, not just pretending everything was always in 12EDO.

but meantones are ancient history right? wrong. although they’ve fallen out of favour to 12EDO as the ideal for fixed-pitch instruments, meantone thinking still defines how the majority of western music operates, even today, and it is still seen in practice, used possibly unknowingly by singers, string players, and anyone else who can bend their notes into the sweet spots near 5-limit intervals like 16/15 (minor second), 6/5 (minor third), 5/4 (major third), 8/5 (minor sixth), 5/3 (major sixth) and 15/8 (major seventh)

so what is a meantone tuning? any tuning where the major third is cut into two identical tones, each half of the major third. hence the name mean. and that meantone is two fifths up, so, like Pythagorean, the tuning is generated in a chain of fifths. though this time, the fifths are slightly flatter than 3/2 in order to make the sum of four fifths as low as the major third.

historically, this meant using a very accurate major third at or close to 5/4, and if 5/4 was used, the meantones would also be the mean of the historically named minor and major tones at 10/9 and 9/8, so that the meantones represented both ratios at the same time, without need for extra keys or the wolf intervals that occurred when using the ‘wrong’ tone. e.g. a 9/8 was needed to get a perfect fifth over 3/4, but 10/9 was needed to get a perfect fifth below 5/3, or a minor third below 4/3, or a major third above 8/9.

so meantone solved this problem by not needing to choose which tone to choose, conflating the ratios 10/9 and 9/8, and 16/9 and 9/5, and by the same logic any two tones 81/80 (the so-called ‘syntonic’ comma) apart. the meantone family tempers out this comma so it can’t “cause a problem” and can’t show up anywhere in meantone-generated scales.

meantone tunings optimise not fifths and fourths like Pythagorean, but thirds and sixths, meaning two part harmony will sound pretty darn good and triads will be much warmer now that they’ve been ‘5-ified’. however, one must be careful in choosing which sharps and flats one wants if limited, say, to 12 notes per octave. historically, the most popular layouts were chains of fifths from Eb up to G# or Ab up to D#, but if the music required more sharps or flats the instruments were tuned accordingly. however, instrument innovation especially in the sixteenth century led to keyboards with split keys, where part of the key sounded the flat note and the other the sharp, and keyboards with 14, 17, 19, 24, and 31 keys were built, and meant not only more freedom in modulation to more distant keys in the spiral of fifths [“wait ‘spiral’, you mean circle don’t you?” i hear you asking, but no, the spiral concept is rather important, and i might get to that in another post], but new sounds, new intervals, and the ability to emulate even the scales of the Ancient Greeks (see Vicentino, Ancient Music Adapted to Modern Practice)

just a sidenote here to introduce something i’ll be using from now on, a very handy unit to measure intervals logarithmically, so we can just add interval sizes instead of multiplying ratios (or when we don’t actually have rational numbers) – the cent

first, divide the octave into 1200 equal pieces. then, …. wait were done. each piece is 1 cent. which means a conventional 12EDO semitone is 100 cents, a quarter tone is 50 cents, and the difference between enharmonic pairs of sharps and flats in this particular meantone is about 41 cents!…

ok so where were we? ah yes, meantone sets for instruments of fixed pitch. if we had the chain of fifths from Db to F# (not the most popular choice but never-the-less), we would have the notes:

C Db D Eb E F F# G Ab A Bb B. Look familiar?

here, though, the minor seconds (C-Db, D-Eb, E-F, F#-G, G-Ab, A-Bb, B-C) are around 117 cents, larger than the augmented unisons (the other pairs) at 76c, a reversal of what we had in Pythagorean. comparing the size of the minor seconds with the augmented unisons, they are approximately in the ratio 3/2

Here we have perfect 5/4 major thirds on C, Db, D, Eb, F, C, Ab and Bb (8 out of 12, not bad), and good (if slightly flat) fifths on all of the keys except F#, where we have a diminished sixth F#-Db, 738c, about a fifth-tone sharp of a fifth… owwwww. this is what’s known as a wolf fifth. because, you know, it makes you howl.

the meantone tuning with perfect 5/4s is called quarter-comma meantone, because it flattens each fifth by ~5.4c= 1/4 of the syntonic comma (81/80) to correct the major thirds (and minor sixths). it also provides pretty good minor thirds, with an error of only 5.4 cents(1/4 syntonic comma, the same as the error on the fifths!) from the simplest minor third ratio 6/5. so it’s pretty good for harmonising anything in thirds right?

there’s more though, 1/4 comma meantone also provides some great intervals where you might not expect. The augmented seconds at 269 cents are incredibly close to the just ratio 7/6 (267c), a really cool low bluesy minor third, that’s actually used in barbershop, jazz, blues, etc. familiar but not too obvious. also the augmented sixths from Db-B and Ab-F# are just flat of 7/4, the simplest and most consonant minor seventh, and may be more familiar to many of you.

but wait, … 7 is a prime, what is a new prime doing here? before we were only talking about 2, 3 and 5. well well, they pop up from time to time, and 7’s another goody, overlooked in the majority of musical writings but still used knowingly or not in a variety of musics.

(If you want to hear ratios of 7 in practice, check some of this. La Monte Young. Well. Tuned. Piano. Kinda a sidenote because this piece has nothing to do with meantone tuning. But it might help to introduce you to the sounds of 7, as opposed to 5, as well as some whacky stuff that happens when you combine 3 and 7. Could be important later. The piano here is tuned in septimal (7-limit) just intonation, with all intervals used being ratios of only 2, 3, and 7. Really amazing piece. Hope you’ve got some time

but what if we want more than 12 keys? just extend the chain of fifths.
from C upwards we get G-D-A-E-B-F#-C#-G#-D#-A#-E#, or we could keep going: B#-Fx-Cx-Gx-Dx-Ax, and downwards we get F-Bb-Eb-Ab-Db-Gb-Cb-Fb-Bbb-Ebb-Dbb-Gbb.
31 tones. too many?
maybe. but 31’s a good number. why?

because if we go one more fifth up from Ax, Ex ends up almost being the same pitch as Gbb, and we have a perfectly usable fifth of 702.6c from Ax up to Gbb (less than a cent sharp of a pure 3/2) so we have created not a circle of 12 fifths but a circle of 31 fifths. pretty crazy huh? http://upload.wikimedia.org/wikipedia/commons/3/3a/31-TET_circle_of_fifths.png

if you wanted to totally iron out the differences and have each fifth exactly the same then you’d have 31EDO, a brilliant system very similar to 31 tone 1/4-comma meantone, though with only 31 unique intervals.

i know this post is getting long again, but here are the intervals, most are very nice indeed. printout from the tuning program Scala [http://www.huygens-fokker.org/scala/] with my own names and notation

0: 1/1 C perfect unison
1: 38.710 cents C^ Dbb diesis, super unison, diminished second
2: 77.419 cents C# Dbv augmented unison, sub second
3: 116.129 cents Db minor second
4: 154.839 cents Dv middle second
5: 193.548 cents D major second
6: 232.258 cents D^ Ebb super second, diminished third
7: 270.968 cents D# Ebv augmented second, sub third
8: 309.677 cents Eb minor third
9: 348.387 cents Ev middle third
10: 387.097 cents E major third
11: 425.806 cents E^ Fb super third, diminished fourth
12: 464.516 cents E# Fv augmented third, sub fourth
13: 503.226 cents F fourth
14: 541.935 cents F^ super fourth
15: 580.645 cents F# augmented fourth
16: 619.355 cents Gb diminished fifth
17: 658.065 cents Gv sub fifth
18: 696.774 cents G fifth
19: 735.484 cents G^ Abb super fifth, diminished sixth
20: 774.194 cents G# Abv augmented fifth, sub sixth
21: 812.903 cents Ab minor sixth
22: 851.613 cents Av middle sixth
23: 890.323 cents A major sixth
24: 929.032 cents A^ Bbb super sixth, diminished seventh
25: 967.742 cents A# Bbv augmented sixth, sub seventh
26: 1006.452 cents Bb minor seventh
27: 1045.161 cents Bv middle seventh
28: 1083.871 cents B major seventh
29: 1122.581 cents B^ Cb super seventh, diminished octave
30: 1161.290 cents B# Cv augmented seventh, sub octave
31: 2/1 C octave

so in 31EDO, the diesis (distance between enharmonic pairs, smallest unit) is one step (^), augmented unison is two steps (#), minor second is 3 steps and meantone is 5 steps. Plus we also get a bunch of middle/’neutral’ intervals to play around with, as well as our usual major and minor plus our “septimal” (7-y) ‘sub’ and ‘super’ categories. fun fun fun

31EDO is a pretty damn good tuning. in my opinion miles better than 12EDO and capable of very different things. but it’s still not my favourite. more on that next time.

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