27720edo: Difference between revisions
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The '''27720 equal divisions of the octave''' ('''27720edo'''), or the '''27720(-tone) equal temperament''' ('''27720tet''', '''27720et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 27720 [[equal]] parts of about {{ | {{Infobox ET}} | ||
The '''27720 equal divisions of the octave''' ('''27720edo'''), or the '''27720(-tone) equal temperament''' ('''27720tet''', '''27720et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 27720 [[equal]] parts of about 43 [[cent|millicent]]s, or exactly 10/231 of a cent each. | |||
== Theory == | |||
27720edo is the 23rd superabundant EDO, counting 95 proper divisors, and 25th highly composite EDO, with a proper index of about 3.05. 27720 is the least common multiple of integers of 1 through 12, with a large jump from [[2520edo]] caused by the prime factor 11. | |||
The prime subgroups best represented by this EDO are 2, 3, 5, 7, 13, 23, 37, 43, 53, 59, 61, 67, 71, 73, 87. The mapping for 3/2 in 27720edo derives from [[1848edo]]. | |||
=== Prime harmonics === | |||
{{Harmonics in equal|27720}} | |||
== Contorsion table == | |||
{| class="wikitable" | |||
|+For 2.prime subgroups | |||
!Prime p | |||
!Contorsion order | |||
for 2.p subgroup | |||
!Meaning that | |||
the mapping derives from | |||
|- | |||
|3 | |||
|15 | |||
|[[1848edo]] | |||
|- | |||
|5 | |||
|4 | |||
|6930edo | |||
|- | |||
|7 | |||
|60 | |||
|462edo | |||
|- | |||
|11 | |||
|45 | |||
|616edo | |||
|- | |||
|13 | |||
|24 | |||
|1155edo | |||
|- | |||
|17 | |||
|24 | |||
|1155edo | |||
|} | |||
Latest revision as of 16:03, 20 March 2023
| ← 27719edo | 27720edo | 27721edo → |
The 27720 equal divisions of the octave (27720edo), or the 27720(-tone) equal temperament (27720tet, 27720et) when viewed from a regular temperament perspective, divides the octave into 27720 equal parts of about 43 millicents, or exactly 10/231 of a cent each.
Theory
27720edo is the 23rd superabundant EDO, counting 95 proper divisors, and 25th highly composite EDO, with a proper index of about 3.05. 27720 is the least common multiple of integers of 1 through 12, with a large jump from 2520edo caused by the prime factor 11.
The prime subgroups best represented by this EDO are 2, 3, 5, 7, 13, 23, 37, 43, 53, 59, 61, 67, 71, 73, 87. The mapping for 3/2 in 27720edo derives from 1848edo.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0000 | -0.0069 | +0.0066 | +0.0053 | -0.0192 | -0.0082 | -0.0203 | +0.0195 | -0.0059 | -0.0101 | -0.0139 |
| Relative (%) | +0.0 | -16.1 | +15.3 | +12.2 | -44.4 | -18.9 | -47.0 | +44.9 | -13.7 | -23.3 | -32.2 | |
| Steps (reduced) |
27720 (0) |
43935 (16215) |
64364 (8924) |
77820 (22380) |
95895 (12735) |
102576 (19416) |
113304 (2424) |
117753 (6873) |
125393 (14513) |
134663 (23783) |
137330 (26450) | |
Contorsion table
| Prime p | Contorsion order
for 2.p subgroup |
Meaning that
the mapping derives from |
|---|---|---|
| 3 | 15 | 1848edo |
| 5 | 4 | 6930edo |
| 7 | 60 | 462edo |
| 11 | 45 | 616edo |
| 13 | 24 | 1155edo |
| 17 | 24 | 1155edo |