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.

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.