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