46032edo
| ← 46031edo | 46032edo | 46033edo → |
46032 equal divisions of the octave (abbreviated 46032edo or 46032ed2), also called 46032-tone equal temperament (46032tet) or 46032 equal temperament (46032et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 46032 equal parts of about 0.0261 ¢ each. Each step represents a frequency ratio of 21/46032, or the 46032nd root of 2.
When used as an interval size unit, 46032edo's step size is called a flu (from FLU, "five-limit unit"), in terms of which the cent is approximately 38.36 flus. The FLU is useful when crafting irregular or circulating temperaments with twelve notes to the octave with an eye to its 5-limit approximations.
Theory
In the 5-limit, the 46032 division tempers out Kirnberger's atom, [161 -84 -12⟩ as well as [454 427 -487⟩. It maps the syntonic comma, 81/80, to 825 flus, and the Pythagorean comma, 531441/524228, to 900 flus. Both of these are composite numbers divisible by 3 and 5, which is convenient for the purpose of twelve-note tempering, and 825/11 = 900/12 = 75 flus, which is the measure in flus of the schisma, 32805/32768.
46032edo marks the optimal patent val for atomic in the 5-limit. The size of the atom, however, does exceed half of the edostep's size.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0000 | +0.0002 | +0.0002 | -0.0042 | +0.0116 | +0.0094 | -0.0023 | +0.0104 | +0.0103 | +0.0109 | +0.0114 |
| Relative (%) | +0.0 | +0.6 | +0.6 | -16.2 | +44.4 | +35.9 | -9.0 | +40.1 | +39.6 | +41.9 | +43.5 | |
| Steps (reduced) |
46032 (0) |
72959 (26927) |
106883 (14819) |
129228 (37164) |
159245 (21149) |
170339 (32243) |
188154 (4026) |
195541 (11413) |
208229 (24101) |
223623 (39495) |
228052 (43924) | |