40/21

Interval information

Ratio	40/21
Factorization	23 × 3-1 × 5 × 7-1
Monzo	[3 -1 1 -1⟩
Size in cents	1115.533¢
Names	septimal acute major seventh, aberschismic major seventh
Color name	ry7, ruyo 7th
FJS name	[math]\displaystyle{ \text{M7}^{5}_{7} }[/math]
Special properties	reduced
Tenney norm (log2 nd)	9.71425
Weil norm (log2 max(n, d))	10.6439
Wilson norm (sopfr(nd))	21
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40/21, conventionally known as the septimal acute major seventh, is an interval in 7-limit just intonation measuring about 1115.5 cents. Despite being approximated by the diminished octave in systems like septimal meantone, and even being called the septimal diminished octave by some musicians due to analogy with 21/20 being called the "septimal chromatic semitone", 40/21 functions as a major seventh, as is supported by Sagittal notation, Helmholtz–Ellis notation and Functional Just System, viewed as the Pythagorean major seventh (243/128) altered by an aberschisma. This gives rise to the more precise name aberschismic major seventh.

Approximation

Edo approximations for 40/21 (1115.53 ¢)
≤ 80edo, relative error ≤ 10%

Edo	Step size	Cents (¢)	Absolute error (¢)	Relative error (%)
13	12\13	1107.69	-7.84	-8.49
14	13\14	1114.29	-1.25	-1.45
15	14\15	1120.00	+4.47	+5.58
27	25\27	1111.11	-4.42	-9.95
28	26\28	1114.29	-1.25	-2.91
29	27\29	1117.24	+1.71	+4.13
42	39\42	1114.29	-1.25	-4.36
43	40\43	1116.28	+0.75	+2.67
44	41\44	1118.18	+2.65	+9.71
56	52\56	1114.29	-1.25	-5.82
57	53\57	1115.79	+0.26	+1.22
58	54\58	1117.24	+1.71	+8.26
70	65\70	1114.29	-1.25	-7.27
71	66\71	1115.49	-0.04	-0.24
72	67\72	1116.67	+1.13	+6.80

See also

• 21/20 – its octave complement
• Gallery of just intervals