Lesser tendoneutralisma
| Interval information |
reduced subharmonic
(Shannon, [math]\sqrt{nd}[/math])
The lesser tendoneutralisma is a small comma of the 2.3.13 subgroup which is the amount by which a stack of twelve 16/13's minus three octaves exceeds 3/2; that is, it is equal to (16/13)12 / (3/2) / (2/1)3 and so equivalently also to (16/13)11 / (39/4). Because the ~11 ¢ of error is distributed over twelve 16/13's in the pure-3's tuning, it is a very accurate way of connecting a chain of 13's to prime 3. A slightly less accurate but still good way of doing this is using the greater tendoneutralisma. Very importantly, both are distinct ways of mapping 2.3.13, so that you can't combine them unless you want to use the trivial tuning of 10edo, so that edos > 10 which have a good 3 and 13 will usually pick between one of these two mappings. (A much simpler but (relatively) much higher error way of mapping 3 for those that prefer sharp fifths is by tempering (16/13)2/(3/2) = 512/507.)
Temperaments
Lesser Tendoneutralic
Tempering the lesser tendoneutralisma in 2.3.13 leads to the highly notable 10 & 77 temperament, where 10edo is the trivial tuning approximately equal to the pure-13's tuning and 77edo can be used to approximate the pure-3's tuning, although 67edo is an interesting choice for combining this temperament with meantone and perhaps more notably 87edo is a very good choice for combining this temperament with parapyth and 13-limit harmony generally (although it doesn't appear in the optimal ET sequence).
Subgroup: 2.3.13
Comma list: [46 -1 (0 0 0) -12⟩
Sval mapping: [⟨1 10 3], ⟨0 -12 1]]
- sval mapping generators: 2, ~13
Optimal tuning (CTE): ~13/8 = 841.503 ¢
Optimal ET sequence: 10, 47, 57, 67, 77, 164, 241, 405, 646f
Badness (Dirichlet): 3.930