This is my first post in a rather long time. But since that last post I’ve been spending most of my time dealing with tuning systems that don’t support meantone.
What does it mean for a tuning not to support meantone? Simply, that four perfect fifths up doesn’t give you the best approximation to the fifth harmonic 5/1 (or octave reduced to 5/4). In a system of Pythagorean tuning (with pure 3/2 fifths, 701.955c), four fifths up gives you (3/2)^4=81/16, which though fairly close to 5/1, isn’t the best approximation if you have a system of 9 or more tones. In fact, a regular 12-tone chain of pure fifths will offer 4 sweet low thirds much closer to the 5/4 thirds of meantone, but they are spelled as diminished fourths, tuned as 8 fifths down, instead of four fifth up. How close to 5/4? Just a tiny comma of 32805/32768, or 1.95c, called the schisma, away from 5/4. In these cases where eight fifths down gives the best approximation to 5/4, the temperament is called “schismatic” as it tempers out this schisma.
A 12-note chain from Eb up to G# will give these diminished fourths at B-Eb, F#-Bb, C#-F and G#-C, with the remaining 8 major thirds tuned as 81/64s (also called ‘ditones’, as they are double the regular 9/8 tone). But if we extend our pythagorean cycle out to 53 tones, we can close the spiral of fifths without any wolf intervals. Once again, like we did with a 31-tone meantone circle, we can equally temper our 53-tone Pythagorean circle without much damage at all, giving very near pure thirds in all 53 keys, as well as pure fifths, and now very close to pure 6/5 minor thirds too. Because we have a pure octave, virtually pure third harmonic and very-close-to-pure fifth harmonic, 53EDO is a brilliant closed system that can represent almost all ratios in the 5-limit. Though as it also contains very good thirteenth and nineteenth harmonics, and a seventh harmonic that is only 4.76c sharp, it can be thought of as representing the full 2.3.5.7.13.19 subgroup with very good accuracy.
However 53 is a lot of tones, so we should go back to our pure fifth chain and find other places we can close the circle without much damage. As it turns out, 12, 29, and 41 show up as great options before we hit 53.
We already know about 12, but as it tempers 4 fifths up the same as 8 fifths down (i.e. equates major third and diminished fourth), it won’t give us a better approximation of 5/4 than the regular third, and so it works in practice very similar to most meantones.
29 has great fifths just slightly wide of pure, but this leads to an eightfold flattening of the syntonic third (=diminished fourth), and so instead of 5/4 (386.31c) we end up with 372.41c, a large middle third very close to 26/21, but really too small to hear it as a major third, or at least one drastically flat.
41, however, provides with a great system that excels in its approximations to ratios of both 3 and 7, as well as a very acceptable 5, 11 and 13. However, it goes even further to give 19, 29 and 31 with great accuracy. 17 and 23 only have errors of 12.12 and 13.64c respectively, both less than the error on 5/4 in well-loved 12EDO! We could probably go even higher, but a 31-limit seems high enough for intervals that are really appreciable to the ear.
The only thing that bugs me a little about 41 is the ~6c error on the schismatic major and minor thirds. Sure, 6c might not seem like much (“6% of a semitone?…!”) but it’s enough to give a wee bit of tension, a push towards the “middle third” region, and create a wee bit of beating that can be a little unpleasant in certain timbres. Still, in terms of efficiency and sheer cool, 41 is a great tuning. Probably my favourite non-meantone EDO of practical size (as in, all notes being available/playable on instruments). The ratios of seven sound particularly good, with the 1-step comma ~= the septimal comma 64/63, the 2-step small minor second a near-28/27, a wonderful small semitone, and 7/6, 9/7, 21/16, 7/5 and their inverses, all great sounding intervals in their own right.
Where 5/4 is reached by tuning 8 fifths down and 7/4 is reached by 14 fifths down, we get the Garibaldi temperament (named after Eduardo Sábat-Garibaldi, who promoted a schismatic-style 53-tone system where major thirds are lowered, minor raised a comma from their Pythagorean positions, and sevenths reached by similar means, lowering the regular minor seventh by a comma, building several “Dinarra” guitars on which to showcase this tuning system). Both 41 and 53EDO support Garibaldi, and it’s a very simple bridge into the 7-limit using essentially a 3-limit (pure fifths-based) system. However, it’s not the only [simple] solution outside of JI and meantone.
46EDO provides a very interesting alternative, where fifths are only 2.39c sharp of pure, and all tones can be reached by chaining these fifths. Looks very much like our Pythagorean circle tunings above right? This time, however, 8 fifths down gets us a neutral third around 21/17, and 14 fifths down gets us to 939.13c, or the best approximation of 12/7. As much better approximations to both 5/4 and 7/4 exist, this is not Garibaldi, nor schismatic at all. 7/4 is in fact reached by 15 fifths UP, or as a double augmented fifth if tuning by fifths, while 5/4 is 21 fifths up… However, 46EDO also affords simpler routes to the higher partials, with 13/8 tuned as a regular augmented fifth (8 fifths up), 11/8 tuned as an augmented third (11 fifths up) and 23/16 tuned as an augmented fourth (6 fifths up), which makes playing in the 2.3.(7.)11.13.23 subgroup pretty easy in a Bosanquet-like setup.
Of course there are an infinite number of non-meantone systems worthy of explorations, some without even a proper perfect fifth (such as 11, 13, 16, 18, etc EDOs), or at least on the borders of the usual acceptable range (5, 7 EDOs and their multiples), but some of the EDO systems where, though meantone isn’t supported, theory is fairly transferable between them and a system like 12EDO or Pythagorean, include 17, 22, 27, 29, 34, 36, 41, 46, 53.
I’ll try and getting round to posting some links soon instead of just spilling out words and numbers…
Comments