192edo
| ← 191edo | 192edo | 193edo → |
192 equal divisions of the octave (abbreviated 192edo or 192ed2), also called 192-tone equal temperament (192tet) or 192 equal temperament (192et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 192 equal parts of exactly 6.25 ¢ each. Each step represents a frequency ratio of 21/192, or the 192nd root of 2.
Theory
192edo is closely related to 96edo, but the patent vals differ on the mapping for harmonic 7. Using the patent val, it tempers out the pythagorean comma, 531441/524288 in the 3-limit; the würschmidt comma, 393216/390625 and the unicorn comma, 1594323/1562500 in the 5-limit; 2401/2400, 3136/3125, and 1594323/1568000 in the 7-limit; 243/242, 441/440, 5632/5625, and 131769/131072 in the 11-limit; 676/675, 1287/1280, 1573/1568, and 2704/2695 in the 13-limit. Using the alternative 192f val, 351/350, 1001/1000, 1575/1573, 3584/3575, and 5445/5408 are tempered out in the 13-limit.
As a tuning standard
A step of 192edo is known as a tetramu (fourth MIDI-resolution unit, 4mu, 24 = 16 equal divisions of the 12edo semitone). The internal data structure of the 4mu requires one byte, with the first two bits reserved as flags, one to indicate the byte's status as data, and one to indicate the sign (+ or −) showing the direction of the pitch-bend up or down, and two other bits which are not used.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | -1.96 | +1.19 | -0.08 | +2.34 | -1.32 | -3.03 | -0.77 | +1.29 | +2.49 | -2.03 | +2.98 |
| relative (%) | -31 | +19 | -1 | +37 | -21 | -48 | -12 | +21 | +40 | -32 | +48 | |
| Steps (reduced) |
304 (112) |
446 (62) |
539 (155) |
609 (33) |
664 (88) |
710 (134) |
750 (174) |
785 (17) |
816 (48) |
843 (75) |
869 (101) | |
Subsets and supersets
Since 192 factors into 26 × 3, 192edo has subset edos 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, and 96.
See also
- Equal multiplications of MIDI-resolution units
- Intercom scales (the scales listed are tuned to 192edo)