Lesser tendoneutralisma

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Interval information
Ratio 70368744177664/69894255367443
Factorization 246 × 3-1 × 13-12
Monzo [46 -1 0 0 0 -12
Size in cents 11.713058¢
Name lesser tendoneutralisma
Color name Sasa-quadtrithu comma
FJS name [math]\text{6d}{-2}_{13,13,13,13,13,13,13,13,13,13,13,13}[/math]
Special properties reduced,
reduced subharmonic
Tenney height (log2 nd) 91.9902
Weil height (log2 max(n, d)) 92
Wilson height (sopfr(nd)) 251
Harmonic entropy
(Shannon, [math]\sqrt{nd}[/math])
~2.37235 bits
Comma size small
open this interval in xen-calc

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 sequence10, 47, 57, 67, 77, 164, 241, 405, 646f

Badness (Dirichlet): 3.930

See also