Greater tendoneutralisma

The greater tendoneutralisma is a small comma of the 2.3.13 subgroup which is the amount by which a stack of eight 16/13's minus two octaves falls short of 4/3; that is, it is equal to (16/3)/(16/13)⁸ and so equivalently also to (13/3)/(16/13)⁷. Although the comma is similar in size to something like 81/80, it is quite accurate because the error can be split evenly over eight 16/13's, so that the pure-3's tuning (very close to 53edo) has 13 off by only 2.78 ¢.

Interval information

Ratio	815730721/805306368
Factorization	2-28 × 3-1 × 138
Monzo	[-28 -1 0 0 0 8⟩
Size in cents	22.26629¢
Name	greater tendoneutralisma
FJS name	[math]\displaystyle{ \text{dddd2}^{13,13,13,13,13,13,13,13} }[/math]
Special properties	reduced
Tenney norm (log2 nd)	59.1885
Weil norm (log2 max(n, d))	59.207
Wilson norm (sopfr(nd))	163
Comma size	small
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Temperaments

Greater Tendoneutralic

Tempering the greater tendoneutralisma in 2.3.13 leads to the highly notable 10 & 53 temperament, where 10edo is the trivial tuning approximately equal to the pure-13's tuning and 53edo is the tuning practically equal to the pure-3's tuning, although 43edo is an interesting choice for combining this temperament with meantone and 63edo is an interesting choice if you prefer slightly sharp fifths.

Subgroup: 2.3.13

Comma list: 815730721/805306368

Mapping: [⟨1 4 4], ⟨0 -8 -1]]

Optimal tuning (CTE): ~16/13 = 362.248 ¢

Optimal ET sequence: 10, 33, 43, 53, 202, 255f, 308f, 361f

Badness (Dirichlet): 3.037

See also

• Lesser tendoneutralisma
• Tridecapyth comma