A few "obscure" temperaments

Here are some temperaments built from "obscure" (non-patent) vals. I'll be using my letter notation, e.g. 45e differs from the patent val of 45-ET in that it uses the second best approximation of 11 (a=2, b=3, c=5, d=7, e=11).

11-limit squares, 31&45e
(31&45 is more complex for the same TOP error.)

31: [<31, 49, 72, 87, 107]>
45e: [<45, 71, 104, 126, 155]>
wedgie: <<4, 16, 9, 10, 16, 3, 2, -24, -32, -3]]
GM: [<1, 3, 8, 6, 7], <0, -4, -16, -9, -10]>
TOP: P = 1201.698520, G = 426.458163
TOP-RMS: P = 1201.674433, G = 426.551643

Unknown, 24d&31

24d: [<24, 38, 56, 68, 83]>
31: [<31, 49, 72, 87, 107]>
wedgie: <<2, 8, 20, 5, 8, 26, 1, 24, -16, -55]]
GM: [<1, 1, 0, -3, 2], <0, 2, 8, 20, 5]>
TOP: P = 1201.698520, G = 348.782195
TOP-RMS: P = 1201.750418, G = 348.690286

An 11-version of octacot, 27e&41
27e: [<27, 43, 63, 76, 94]>
41: [<41, 65, 95, 115, 142]>
wedgie: <<8, 18, 11, 20, 10, -5, 4, -25, -16, 18]]
GM: [<1, 1, 1, 2, 2], <0, 8, 18, 11, 20]>
TOP: P = 1198.375800, G = 88.009258
TOP-RMS: P = 1199.608944, G = 87.946449

An 11-limit version of magic, 19e&41
19e: [<19, 30, 44, 53, 65]>
41: [<41, 65, 95, 115, 142]>
wedgie: <<5, 1, 12, 33, -10, 5, 35, 25, 73, 51]]
GM: [<1, 0, 2, -1, -7], <0, 5, 1, 12, 33]>
TOP: P = 1201.287109, G = 380.750920
TOP-RMS: P = 1201.267491, G = 380.633430

Unknown, 38d&41
38d: [<38, 60, 88, 106, 131]>
41: [<41, 65, 95, 115, 142]>
wedgie: <<10, 2, 24, 25, -20, 10, 5, 50, 51, -13]]
GM: [<1, 5, 3, 11, 12], <0, -10, -2, -24, -25]>
TOP: P = 1201.328138, G = 410.398477
TOP-RMS: 1201.080965, G = 410.278975

Unknown, 46&60e
46: [<46, 73, 107, 129, 159]>
60e: [<60, 95, 139, 168, 207]>
wedgie: <<10, 26, 12, 18, 18, -9, -6, -45, -48, 9]]
GM: [<2, 1, -1, 3, 3], <0, 5, 13, 6, 9]>
TOP: P = 600.577603, G = 260.641671
TOP-RMS: P = 600.506244, G = 260.612918

Herman Miller wrote:
> Here are some temperaments built from "obscure" (non-patent) vals. I'll > be using my letter notation, e.g. 45e differs from the patent val of > 45-ET in that it uses the second best approximation of 11 (a=2, b=3, > c=5, d=7, e=11).

To show it can be done I've found ways of writing them all with optimal vals (by TOP-RMS I think).

> 11-limit squares, 31&45e
> (31&45 is more complex for the same TOP error.)
> > 31: [<31, 49, 72, 87, 107]>
> 45e: [<45, 71, 104, 126, 155]>
> wedgie: <<4, 16, 9, 10, 16, 3, 2, -24, -32, -3]]
> GM: [<1, 3, 8, 6, 7], <0, -4, -16, -9, -10]>
> TOP: P = 1201.698520, G = 426.458163
> TOP-RMS: P = 1201.674433, G = 426.551643

This kind of 45 is the optimal one.

> Unknown, 24d&31
> > 24d: [<24, 38, 56, 68, 83]>
> 31: [<31, 49, 72, 87, 107]>
> wedgie: <<2, 8, 20, 5, 8, 26, 1, 24, -16, -55]]
> GM: [<1, 1, 0, -3, 2], <0, 2, 8, 20, 5]>
> TOP: P = 1201.698520, G = 348.782195
> TOP-RMS: P = 1201.750418, G = 348.690286

I found these neutral-third meantones difficult to disambiguate before. There are different, reasonable ways of doing it. This one, anyway, uses the optimal mapping of 7.

> An 11-version of octacot, 27e&41
> 27e: [<27, 43, 63, 76, 94]>
> 41: [<41, 65, 95, 115, 142]>
> wedgie: <<8, 18, 11, 20, 10, -5, 4, -25, -16, 18]]
> GM: [<1, 1, 1, 2, 2], <0, 8, 18, 11, 20]>
> TOP: P = 1198.375800, G = 88.009258
> TOP-RMS: P = 1199.608944, G = 87.946449

This is the optimal mapping of 27.

> An 11-limit version of magic, 19e&41
> 19e: [<19, 30, 44, 53, 65]>
> 41: [<41, 65, 95, 115, 142]>
> wedgie: <<5, 1, 12, 33, -10, 5, 35, 25, 73, 51]]
> GM: [<1, 0, 2, -1, -7], <0, 5, 1, 12, 33]>
> TOP: P = 1201.287109, G = 380.750920
> TOP-RMS: P = 1201.267491, G = 380.633430

I think 19 and 22 already disagree in the 11-limit. In this case you have to go up to 60 for the right optimal val.

19 is notoriously ambiguous in higher limits, and often comes up multiple times in the searching with more than one mapping contributing to the best rank-2 results.

> Unknown, 38d&41
> 38d: [<38, 60, 88, 106, 131]>
> 41: [<41, 65, 95, 115, 142]>
> wedgie: <<10, 2, 24, 25, -20, 10, 5, 50, 51, -13]]
> GM: [<1, 5, 3, 11, 12], <0, -10, -2, -24, -25]>
> TOP: P = 1201.328138, G = 410.398477
> TOP-RMS: 1201.080965, G = 410.278975

Optimal mapping of 38.

> Unknown, 46&60e
> 46: [<46, 73, 107, 129, 159]>
> 60e: [<60, 95, 139, 168, 207]>
> wedgie: <<10, 26, 12, 18, 18, -9, -6, -45, -48, 9]]
> GM: [<2, 1, -1, 3, 3], <0, 5, 13, 6, 9]>
> TOP: P = 600.577603, G = 260.641671
> TOP-RMS: P = 600.506244, G = 260.612918

Optimal mapping of 60 (and of 46 in case you consider it ambiguous).

Graham