Peppermint 24 lattice -- F on upper chain as 1/1
Organized around final F for _O Europae_
+0.6 -2.1 0 +2.1 -0.6 +1.5 -1.1 +1.0 =1.8 +0.4 +2.5 +0.7
991.8 495.9 0 704.1 208.2 912.3 416.4 1120.5 624.6 128.7 832.8 336.9
Eb-----Bb-----F-----C-----G-----D-----A------E------B------F#------C#------G#
39/22 4/3 1/1 3/2 44/39 22/13 14/11 21/11 56/39 14/13 21/13 17/14
991.2 498.0 0 702.0 208.8 910.8 417.5 1119.5 626.3 128.3 830.3 336.1
0 -2.1 -0.6 -3.3 -1.1 +1.0 -1.8 +0.4 +2.5 +0.1 +1.9 -0.7
933.1 437.2 1141.3 645.4 149.5 853.6 357.7 1061.8 565.9 70.0 774.1 278.2
Ebv---Bbv-----Fv----Cv----Gv----Dv----Av----Ev------Bv-----F#v----C#v-----G#v
12/7 9/7 176/91 16/11 12/11 18/11 16/13 24/13 18/13 126/121 264/169 168/143
933.1 435.1 1142.0 648.7 150.6 852.6 359.5 1061.4 563.4 70.1 772.2 278.9
Note: In my PDF score notation, the note Bbv is called A|\, which means it is
conceptually a comma above A. If F-A in this temperament is taken as
representing 14/11 (417.508 cents), and F-Bb\|/ or F-A|\ as 9/7 (435.084 cents),
then the comma between them would be 99/98 (17.576 cents). In the Peppermint
temperament, the actual difference between F-A (416.382 cents) and F-A|\
(437.225 cents) is 20.842 cents.
In this lattice diagram, I have chosen JI ratios so as to keep all regular
fifths in the Peppermint temperament within 5 cents of pure in a JI mapping baed
on this lattice.
Note that all regular fifths are at 704.096 cents, with a spacing between
corresponding notes on the two 12-note manuals at 58.680 cents, so that in this
temperament 7/6 and 12/7 are just. The tempered spacing of 58.680 cents, as will
be seen by comparing the just ratios on the lattice, most often represents such
ratios as 28:27 (62.961 cents, e.g. Bbv or A|\ at 9/7 versus Bb at 4/3); 121:117
(58.198 cents, e.g. Dv at 18/11 versus D at 22/13); 91:88 (58.036 cents, e.g. Av
at 16/13 versus A at 14/11); or 33/32 (53.273 cents, e.g. Cv at 16/11 versus C
at 3/2).
Generally the suggested ratios are based on primes 2-3-7-11-13, with the
exception of F-G# at 17/14, chosen in part because 14:17:21 is the simplest
division of a 3:2 fifth into two middle or neutral thirds. For F-G#v at 278.181
cents, I have chosen 168/143 (278.935 cents) as the product of the two
superparticular ratios of 12:11 (150.637 cents) and 14:13 (128.298 cents),
although, for example, 27/23 (277.591 cents) would be both simpler and closer to
the tempered value.
Margo Schulter
6 June 2017