
© 2004 Tonalsoft Inc. 
Lattice diagrams comparing rational implications
of various meantone chains
©2001 by Joe Monzo
In this webpage, I present several different lattice diagrams which plot various fractionofacomma meantonelike systems on a JI lattice with primefactor axes of 3 and 5.
The meantone chain forms its own linear axis on the lattice, and makes it easy to visualize which JI ratios are acoustically the most closely implied by the meantone pitches. The meantone pitchclasses are called by their generator number. These lattices are "8ve"invariant, that is, primefactor 2 is ignored.
CAVEAT: Compositional practice may insist that a meantone pitch implies a different JI ratio than the one which is acoustically the closest.
These lattices are given with the reference pitch of C = n^{0}.
In every case, the list of implied JI ratios includes both of the two nearest, except for instances where the meantone tuning gives one exact ratio (as, for example, the +3, +6, +9 generators A, F#, D# of 1/3comma; the +7 generator C# of 2/7comma; the +4, +8, +12 generators E, G#, B# of 1/4comma; the +5, +10 generators B, A# of 1/5comma; etc.).
The lattice diagrams, however, only give two ratios in cases where the meantone is exactly at the midpoint between them (as with the +2 generator "D" in 1/4comma meantone, and with the +3 generator "A" in 1/6comma meantone); otherwise only the nearest JI implied ratio is given.
The meantone chains could be extended beyond my diagrams; I chose a 27note chain of +/ 13 generators (Gbb to Fx) as an arbitrary limit in every example.
In the 1/3comma quasimeantone, the generator is 1/3comma narrower than 3:2. Thus, the +1 generator is 1/3comma flatter than G 3:2 (= 3^{1}) and 2/3comma sharper than G 40:27 (= 3^{3}5^{1}).
The next note in the chain (the +2 generator) is 2/3comma flatter than D 9:8 (= 3^{2}). Since 2/3comma is more than 1/2comma, it is evident that the +2 generator will be closer to the note a comma flatter than 9:8 (= 10:9 ratio = 3^{2}5^{1}), namely, 1/3comma sharper than D 10:9.
The +3 generator is exactly the JI ratio A 5:3 (= 3^{1}5^{1}).
In the 2/7comma quasimeantone, the generator is 2/7comma narrower than 3:2. Thus, the +1 generator is 2/7comma flatter than G 3:2 (= 3^{1}) and 5/7comma sharper than G 40:27 (= 3^{3}5^{1}).
The next note in the chain (the +2 generator) is 4/7comma flatter than 9:8 (= 3^{2}). Since 4/7comma is more than 1/2comma, it is evident that the +2 generator will be closer to the note a comma flatter than 9:8 (= 10:9 ratio = (= 3^{2}5^{1}), namely, 3/7comma sharper.
The +3 generator is 6/7comma flatter than A 27:16 (= 3^{3}) and only 1/7comma sharper than A 5:3 (= 3^{1}5^{1}).
The +4 generator is 1/7comma flatter than E 5:4 (= 5^{1})and 6/7comma sharper than E 100:81 (= 3^{4}5^{2}).
The +5 generator is 3/7comma flatter than B 15:8 (= 3^{1}5^{1}) and 4/7comma sharper than B 50:27 (= 3^{3}5^{2}).
The +6 generator is 5/7comma flatter than F# 45:32 (= 3^{2}5^{1}) and 2/7comma sharper than F# 25:18 (= 3^{2}5^{2}).
The +7 generator is exactly the JI ratio C# 25:24 (= 3^{1}5^{2}).
In the 1/4comma meantone, the generator is 1/4comma narrower than 3:2. Thus, the +1 generator is 1/4comma flatter than G 3:2 (= 3^{1}) and 3/4comma sharper than G 40:27 (= 3^{3}5^{1}).
The next note in the chain (the +2 generator) is 1/2comma flatter than D 9:8 (= 3^{2}) and 1/2comma sharper than D 10:9 (= 3^{2}5^{1}). Thus, this is exactly the "mean tone" between the two JI "wholetones".
The +3 generator is 3/4comma flatter than A 27:16 (= 3^{3}) and 1/4comma sharper than A 5:3 (= 3^{1}5^{1}) .
The +4 generator is exactly the JI ratio E 5:4 (= 5^{1}).
In the 1/5comma quasimeantone, the generator is 1/5comma narrower than 3:2. Thus, the +1 generator is 1/5comma flatter than G 3:2 (= 3^{1}) and 4/5comma sharper than G 40:27 (= 3^{3}5^{1}).
The next note in the chain (the +2 generator) is 2/5comma flatter than 9:8 (= 3^{2}). and 3/5comma sharper than 10:9 (= 3^{2}5^{1}).
The +3 generator is 3/5comma flatter than A 27:16 (= 3^{3}) and 2/5comma sharper than A 5:3 (= 3^{1}5^{1}) .
The +4 generator is 4/5comma flatter than E 81:64 (= 3^{4})and 1/5comma sharper than E 5:4 (= 5^{1}).
The +5 generator is exactly the JI ratio B 15:8 (= 3^{1}5^{1}).
In the 1/6comma quasimeantone, the generator is 1/6comma narrower than 3:2. Thus, the +1 generator is 1/6comma flatter than G 3:2 (= 3^{1}) and 5/6comma sharper than G 40:27 (= 3^{3}5^{1}).
The next note in the chain (the +2 generator) is 1/3comma flatter than 9:8 (= 3^{2}) and 2/3comma sharper than 10:9 (= 3^{2}5^{1}).
The +3 generator is 1/2comma flatter than A 27:16 (= 3^{3}) and 1/2comma sharper than A 5:3 (= 3^{1}5^{1}). Thus, it is precisely the mean "6th" between the two JI ratios.
The +4 generator is 2/3comma flatter than E 81:64 (= 3^{4})and 1/3comma sharper than E 5:4 (= 5^{1}).
The +5 generator is 5/6comma flatter than B 243:128 (= 3^{5})and 1/6comma sharper than B 15:8 (= 3^{1}5^{1}).
The +6 generator is exactly the JI ratio F# 45:32 (= 3^{2}5^{1}).
And here is a more accurate lattice of the above, showing a closed 55tone 1/6comma meantone chain and its implied pitches, all enclosed within a complete periodicityblock defined by the two unisonvectors 81:80 = [4 4 1] (the syntonic comma, the shorter boundary extending from southwest to northeast on this diagram) and [51 19 9] (the long nearly vertical boundary), portrayed here as the white area.
For the bounding corners of the periodicityblock, I arbitrarily chose the lattice coordinates [7.5 5] for the northwest corner, [11.5 4] for northeast, [11.5 4] for southwest, and [7.5 5] for southeast. This produces a 55tone system centered on n^{0}.
The grey area represents the part of the JI lattice outside the defined periodicityblock (and thus, with each of those pitchclasses in its own periodicityblock), and the lattice should be imagined as extending infinitely in all four directions. The other periodicityblocks, all identical to this one, can be tiled against it to cover the entire space.
In the 1/11comma quasimeantone, which is audibly indistinguishable from the usual 12EDO tuning, the generator is 1/11comma narrower than 3:2. Thus, the +1 generator is 1/11comma flatter than G 3:2 (= 3^{1}) and 10/11comma sharper than G 40:27 (= 3^{3}5^{1}).
The +2 generator is 2/11comma flatter than 9:8 (= 3^{2}) and 9/11comma sharper than 10:9 (= 3^{2}5^{1}).
The +3 generator is 3/11comma flatter than A 27:16 (= 3^{3}) and only 8/11comma sharper than A 5:3 (= 3^{1}5^{1}).
The +4 generator is 4/11comma flatter than E 81:64 (= 3^{4})and 7/11comma sharper than E 5:4 (= 5^{1}).
The +5 generator is 5/11comma flatter than B 243:128 (= 3^{5}) and 6/11comma sharper than B 15:8 (= 3^{1}5^{1}).
The +6 generator is 6/11comma flatter than F# 729:512 (= 3^{6}) and 5/11comma sharper than F# 45:32 (= 3^{2}5^{1}).
The +7 generator is 7/11comma flatter than C# 2187:2048 (= 3^{7}) and 4/11comma sharper than C# 135:128 (= 3^{3}5^{1}).
The +8 generator is 8/11comma flatter than G# 6561:4096 (= 3^{8}) and 3/11comma sharper than G# 405:256 (= 3^{4}5^{1}).
The +9 generator is 9/11comma flatter than D# 19683:16384 (= 3^{9}) and 2/11comma sharper than D# 1215:1024 (= 3^{5}5^{1}).
The +10 generator is 10/11comma flatter than A# 59049:32768 (= 3^{10}) and 1/11comma sharper than A# 3645:2048 (= 3^{6}5^{1}).
The +11 generator is exactly the JI ratio E# 10935:8192 (= 3^{7}5^{1}).
Note that because the JI lattice implied by 1/11comma meantone includes the skhisma, the chain effectively closes at 12 tones; a typical version could be 3 Eb to +8 G#.
See also: Ellis's Duodene and a "bestfit" meantone.
updated:
2000.12.09