405/392

Interval information

Ratio	405/392
Factorization	2-3 × 34 × 5 × 7-2
Monzo	[-3 4 1 -2⟩
Size in cents	56.4819¢
Name	greenwoodma
Color name	rry-2, ruruyo negative 2nd
FJS name	[math]\text{d}{-2}^{5}_{7,7}[/math]
Special properties	reduced
Tenney height (log2 nd)	17.2765
Weil height (log2 max(n, d))	17.3236
Wilson height (sopfr(nd))	37
Harmonic entropy (Shannon, [math]\sqrt{nd}[/math])	~4.41802 bits
Comma size	medium
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405/392, the greenwoodma, is a medium 7-limit comma. It is the difference between 8/5 and a stack of two 9/7's. In terms of commas, it is the sum of the syntonic comma (81/80) and the jubilisma (50/49).

Temperaments

Tempering out this comma leads to the rank-3 greenwoodmic temperament. This temperament splits a ~5/2 in two, each representing ~14/9. See Greenwoodmic temperaments for a collection of rank-2 temperaments where it is tempered out.

Greenwoodmic

Subgroup: 2.3.5.7

Comma list: 405/392

Mapping: [⟨1 0 1 -1], ⟨0 1 0 2], ⟨0 0 2 1]]

mapping generators: ~2, ~3, ~14/9

Optimal tunings:

• CTE: ~2 = 1200.000, ~3/2 = 694.595, ~14/9 = 791.182

error map: ⟨0.000 -7.360 -3.949 +11.546]

• CWE: ~2 = 1200.000, ~3/2 = 693.344, ~14/9 = 790.072

error map: ⟨0.000 -8.611 -6.169 +7.935]

Optimal ET sequence: 9, 12, 26, 38, 73bc

Badness (Smith): 0.412 × 10^-3

Notation

Sagittal notation

In the Sagittal system, the downward version of this comma (possibly tempered) is represented by the sagittal ⁠ ⁠ and is called the 49/5 medium diesis, or 49/5M for short, because the simplest interval it notates is 49/5 = 7×7/5 (equiv. 49/40), as for example in C-E⁠ ⁠⁠ ⁠. The upward version is called 5/49M or 49/5M up and is represented by ⁠ ⁠.