4225/4224

Interval information

Ratio	4225/4224
Factorization	2-7 × 3-1 × 52 × 11-1 × 132
Monzo	[-7 -1 2 0 -1 2⟩
Size in cents	0.409808¢
Name	leprechaun comma
Color name	3oo1uyy1, thotholuyoyo unison
FJS name	[math]\displaystyle{ \text{P1}^{5,5,13,13}_{11} }[/math]
Special properties	square superparticular, reduced
Tenney norm (log2 nd)	24.0891
Weil norm (log2 max(n, d))	24.0895
Wilson norm (sopfr(nd))	64
Comma size	unnoticeable
S-expression	S65
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4225/4224, the leprechaun comma, is an unnoticeable 13-limit superparticular interval. This comma is the difference between the following pairs:

• 65/64 and 66/65
• 169/168 and 176/175
• 325/324 and 352/351
• 2080/2079 and 4096/4095
• 3025/3024 and 10648/10647

It factors into 4375/4374 and 123201/123200.

Temperaments

Tempering out this comma in the full 13-limit leads to the rank-5 leprechaun temperament. It is also tempered out in lower-rank temperaments including abigail, newt, decoid, donar, etc. Tempering it out also splits the 33/32 Alpharabian quarter-tone in two, each part corresponding to 65/64 and 66/65. You may find a list of good equal temperaments that support this temperament below.

Subgroup: 2.3.5.7.11.13

Mapping:

[⟨	1	0	0	0	1	4	],
⟨	0	1	0	0	1	1	],
⟨	0	0	1	0	0	-1	],
⟨	0	0	0	1	0	0	],
⟨	0	0	0	0	2	1	]]

mapping generators: ~2, ~3, ~5, ~7, ~65/48

Optimal tunings:

• WE: ~2 = 1200.0205 ¢, ~3/2 = 701.9418 ¢, ~5/4 = 386.2411 ¢, ~7/4 = 968.7849 ¢, ~65/48 = 524.6851 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9442 ¢, ~5/4 = 386.2465 ¢, ~7/4 = 968.7910 ¢, ~65/48 = 524.6866 ¢

Optimal ET sequence: 41, 46, 53, 80, 87, 103, 121, 130, 183, 190, 217, 224, 270, 494, 684, 764, 935, 954, 1178, 1308, 1448, 1578, 1889, 2756, 2843, 3337, 4421f, 4915f, 7758cff, 10514cdff

Badness (Sintel): 0.244

Etymology

This comma was named by Margo Schulter in 2012 as a tribute to George Secor’s 2002 short story A Bit O' Blarney.

See also

• List of superparticular intervals