4225/4224
| Ratio | 4225/4224 |
| Factorization | 2-7 × 3-1 × 52 × 11-1 × 132 |
| Monzo | [-7 -1 2 0 -1 2⟩ |
| Size in cents | 0.40980804¢ |
| Name | leprechaun comma |
| Color name | 3oo1uyy1, thotholuyoyo unison |
| FJS name | [math]\text{P1}^{5,5,13,13}_{11}[/math] |
| Special properties | square superparticular, reduced |
| Tenney height (log2 nd) | 24.0891 |
| Weil height (log2 max(n, d)) | 24.0895 |
| Wilson height (sopfr (nd)) | 64 |
| Harmonic entropy (Shannon, [math]\sqrt{n\cdot d}[/math]) |
~2.39821 bits |
| Comma size | unnoticeable |
| S-expression | S65 |
| open this interval in xen-calc | |
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
| [⟨ | 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 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
Etymology
The leprechaun comma was named by Margo Schulter in 2012[1].