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] http://micro.soonlabel.com/gene_ward_smith/Others/Kosmorsky/43%20edo%20counterpoint.mp3
698.2c: 55EDO [Mozart’s tuning, great Baroque meantone close to 1/6 comma] http://www.seraph.it/dep/int/AdagioKV540.mp3 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] http://micro.soonlabel.com/41edo/20130910_magic%5b19%5dor_41_the_magic_of_belief.mp3 [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.
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