Girardisma

Interval information

Ratio	10450/10449
Subgroup monzo	2.3.5.11.19.43 [1 -5 2 1 1 -1⟩
Size in cents	0.1656763¢
Name	girardisma
Color name	43u19o1oyy1 fothunoloyoyo unison
FJS name	[math]\text{P1}^{5,5,11,19}_{43}[/math]
Special properties	superparticular, reduced
Tenney height (log2 nd)	26.7023
Weil height (log2 max(n, d))	26.7024
Wilson height (sopfr(nd))	100
Harmonic entropy (Shannon, [math]\sqrt{nd}[/math])	~1.19847 bits
Comma size	unnoticeable
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10450/10449, the girardisma, is a 43-limit superparticular comma of about 0.17 cents.

Commatic relationships

This comma is the difference between the superparticular pair 1936/1935 and 2376/2375.

It can be factored into the following superparticular commas:

• 10648/10647 and 561925/561924
• 10880/10879 and 264385/264384
• 14535/14534 and 37180/37179
• 17545/17544 and 25840/25839
• 20125/20124 and 21736/21735

Temperaments

Tempering out this comma in the full 43-limit leads to the rank-13 temperament girardismic. Using the 2.3.5.11.19.43 subgroup leads to the rank-5 temperament girardic.

Girardismic

Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41.43

Comma list: 10450/10449

Mapping:

[⟨	1	1	2	2	3	3	4	4	4	4	4	5	5	7	],
⟨	0	1	0	0	0	0	0	0	0	0	0	0	0	-5	],
⟨	0	0	1	0	0	0	0	0	0	0	0	0	0	2	],
⟨	0	0	0	1	0	0	0	0	0	0	0	0	0	0	],
⟨	0	0	0	0	1	0	0	0	0	0	0	0	0	1	],
⟨	0	0	0	0	0	1	0	0	0	0	0	0	0	0	],
⟨	0	0	0	0	0	0	1	0	0	0	0	0	0	0	],
⟨	0	0	0	0	0	0	0	1	0	0	0	0	0	1	],
⟨	0	0	0	0	0	0	0	0	1	0	0	0	0	0	],
⟨	0	0	0	0	0	0	0	0	0	1	0	0	0	0	],
⟨	0	0	0	0	0	0	0	0	0	0	1	0	0	0	],
⟨	0	0	0	0	0	0	0	0	0	0	0	1	0	0	],
⟨	0	0	0	0	0	0	0	0	0	0	0	0	1	0	]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.969, ~5/4 = 386.301, ~7/4 = 968.826, ~11/8 = 551.304, ~13/8 = 840.528, ~17/16 = 104.955, ~19/16 = 297.492, ~23/16 = 628.274, ~29/16 = 1029.577, ~31/16 = 1145.036, ~37/32 = 251.344, ~41/32 = 429.062

Optimal ET sequence: 5fi, 5f, 5, 5g, 7dfghijjkllmmmn, 7dfghijjkllmn, 7dfghijjkmn, 7l, 8dhhimmn, 8dhhijjmmn, 10h, 12del, 12el, 12fijkmn, 14cfjjkklmmn, 14cfjjlmmn, 17cgn, 17cgkn, 19eghijjkklmnn, 19eghikklmnn, 22h, 24dn, 26eiknn, 26ik, 27egj, 27egij, 31fghjk, 34dhkn, 34dhjkmn, 34dhjklmn, 38dfijlm, 39dfgijkl, 39dfgijklmm, 45efghhijkmmnn, 46kln, 46ln, 53gikn, 53gkn, 53kn, 53ej, 58, 60eln, 72hijkllmn, 72hijkmn, 77m, 80, 80k, 87, 99efkmn, 111jln, 113l, 113, 118fikl, 121hhiklm, 125fm, 137n, 140kn, 144ghj, 145jkl, 152fjm, 152fgjm, 171, 183k, 183, 193, 217n, 217ln, 236dghin, 243eklmn, 270m, 270, 270i, 296, 311n, 373g, 388, 422, 422l, 525, 540, 566gjl, 639hjmn, 653, 711, 718, 764hj, 836, 863efgjk, 882, 908m, 954hjln, 1065, 1164fimn, 1178j, 1178, …

Girardic

Subgroup: 2.3.5.11.19.43

Comma list: 10450/10449

Mapping:

[⟨	1	1	2	3	4	7	],
⟨	0	1	0	0	0	-5	],
⟨	0	0	1	0	0	2	],
⟨	0	0	0	1	0	1	],
⟨	0	0	0	0	1	1	]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.969, ~5/4 = 386.301, ~11/8 = 551.304, ~19/16 = 297.492

Optimal ET sequence: 5, 7hn, 7, 12e, 14cn, 15, 19ehnn, 19h, 22h, 24n, 26, 27e, 31h, 34hn, 38, 41, 53n, 65, 118, 152, 217n, 270, 335, 460, 487, 566, 612, 677, 718, 836, 988, 1053, 1323, 1513, 1783, 2501, 2836, 4284, 4349, 4501, 6284, 7120, 8785, 9120

Etymology

The name girardisma was named by Francium in 2024. It refers to 10450 Girard, the asteroid. Which was in turn named after an American astronomer.

See also

• Unnoticeable comma
• List of superparticular intervals