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, we're 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.
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