561/560

Interval information

Ratio	561/560
Factorization	2-4 × 3 × 5-1 × 7-1 × 11 × 17
Monzo	[-4 1 -1 -1 1 0 1⟩
Size in cents	3.088732¢
Name	monardisma
Color name	17o1org1, solorugu unison, Solorugu comma
FJS name	[math]\displaystyle{ \text{d1}^{11,17}_{5,7} }[/math]
Special properties	superparticular, reduced
Tenney norm (log2 nd)	18.2611
Weil norm (log2 max(n, d))	18.2637
Wilson norm (sopfr(nd))	51
Comma size	unnoticeable
S-expression	S33⋅S34
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561/560, the monardisma is an unnoticeable 17-limit superparticular comma of about 3.09 cents. It is the difference between the pentacircle minor third, 33/28, and the diatismic minor third, 20/17, on account of it being the difference between the diatisma, 136/135, and the pentacircle comma, 896/891. It also represents the difference between the supraminor thirds of 17/14 and 40/33, and between the quartertones of 33/32 and 35/34.

Temperaments

Tempering out this comma in the full 17-limit results in the rank-6 monardismic temperament, or in the 2.3.5.7.11.17 subgroup, the rank-5 monardic temperament, enabling monardismic chords. This temperament has a number of notable properties:

• 14/11 and 20/17 become fifth complements;
• 11/10 and 17/14 become fourth complements.

Monardic

Subgroup: 2.3.5.7.11.17

Subgroup-val mapping:

[⟨	1	0	0	0	0	4	],
⟨	0	1	0	0	0	-1	],
⟨	0	0	1	0	0	1	],
⟨	0	0	0	1	0	1	],
⟨	0	0	0	0	1	-1	]]

mapping generators: ~2, ~3, ~5, ~7, ~11

Optimal tunings:

• WE: ~2 = 1200.2042 ¢, ~3/2 = 701.6222 ¢, ~5/4 = 386.1804 ¢, ~7/4 = 968.8197 ¢, ~11/8 = 550.0934 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.6204 ¢, ~5/4 = 386.3575 ¢, ~7/4 = 968.9591 ¢, ~11/8 = 550.2059 ¢

Optimal ET sequence: 12, 14c, 15, 17cg, 19g, 22, 31, 41g, 43, 46, 58, 68, 72, 118, 183, 255, 301, 373g, 674eg, 768egg, 1290beeeggg

Badness (Sintel): 0.204

Monardismic

Subgroup: 2.3.5.7.11.13.17

Mapping:

[⟨	1	0	0	0	0	0	4	],
⟨	0	1	0	0	0	0	-1	],
⟨	0	0	1	0	0	0	1	],
⟨	0	0	0	1	0	0	1	],
⟨	0	0	0	0	1	0	-1	],
⟨	0	0	0	0	0	1	0	]]

mapping generators: ~2, ~3, ~5, ~7, ~11, ~13

Optimal tunings:

• WE: ~2 = 1200.2042 ¢, ~3/2 = 701.6222 ¢, ~5/4 = 386.1804 ¢, ~7/4 = 968.8197 ¢, ~11/8 = 550.0934 ¢, ~13/8 = 839.9145 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.6204 ¢, ~5/4 = 386.3575 ¢, ~7/4 = 968.9591 ¢, ~11/8 = 550.2059 ¢, ~13/8 = 840.0917 ¢

Optimal ET sequence: 12f, 14cf, 15, 17cg, 19g, 22, 26, 29g, 31, 38df, 41g, 43, 46, 58, 72, 103, 111, 130, 140, 183, 243e, 301, 373g, 886eefgg, 1362bdeeefgggg *

* optimal patent val: 484

Badness (Sintel): 0.398

Etymology

This comma was named by Scott Dakota in 2023 after the Monarda scale, a 12-tone scale of tannic, where this is one of the commas being tempered out.

See also

• List of superparticular intervals