196edo: Difference between revisions
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[[ | == Theory == | ||
The equal temperament [[tempering out|tempers out]] 9765625/9565938 (fifive comma) and {{monzo| 32 -7 -9 }} ([[escapade comma]]) in the 5-limit. Using the [[patent val]], it tempers out [[245/243]], [[65625/65536]], and 235298/234375 in the 7-limit; [[385/384]], [[896/891]], [[3388/3375]], and 117649/117128 in the 11-limit; [[352/351]], [[364/363]], [[625/624]], [[1001/1000]], and 9295/9261 in the 13-limit; [[289/288]], [[442/441]], [[715/714]], and [[1156/1155]] in the 17-limit. | |||
196edo can also treated as a 2.9.5.7.11.13.17 [[subgroup]] temperament (with the patent 9), providing a distinct flat tendency for [[harmonic]]s [[5/1|5]], [[7/1|7]], [[9/1|9]], [[11/1|11]], [[13/1|13]], and [[17/1|17]]. With the patent 9, it tempers out 321489/320000, [[420175/419904]], and [[703125/702464]] in the 2.9.5.7 subgroup; [[441/440]], [[8019/8000]], [[41503/41472]], and 9453125/9437184 in the 2.9.5.7.11 subgroup; [[729/728]], [[1001/1000]], [[1575/1573]], [[6656/6655]], and [[10985/10976]] in the 2.9.5.7.11.13 subgroup; [[833/832]], [[936/935]], [[1089/1088]], [[1225/1224]], [[2025/2023]], and 14365/14336 in the 2.9.5.7.11.13.17 subgroup. | |||
Since it is part of the Fibonacci sequence beginning with 5 and 12, it closely approximates [[peppermint]] temperament. | |||
=== Odd harmonics === | |||
{{Harmonics in equal|196}} | |||
=== Subsets and supersets === | |||
Since 196 factors into {{factorization|196}}, 196edo has subset edos {{EDOs| 2, 4, 7, 14, 28, 49, and 98 }}. | |||
== Intervals == | |||
See: [[Table of 196edo intervals]] | |||
Latest revision as of 19:30, 20 February 2025
| ← 195edo | 196edo | 197edo → |
196 equal divisions of the octave (abbreviated 196edo or 196ed2), also called 196-tone equal temperament (196tet) or 196 equal temperament (196et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 196 equal parts of about 6.12 ¢ each. Each step represents a frequency ratio of 21/196, or the 196th root of 2.
Theory
The equal temperament tempers out 9765625/9565938 (fifive comma) and [32 -7 -9⟩ (escapade comma) in the 5-limit. Using the patent val, it tempers out 245/243, 65625/65536, and 235298/234375 in the 7-limit; 385/384, 896/891, 3388/3375, and 117649/117128 in the 11-limit; 352/351, 364/363, 625/624, 1001/1000, and 9295/9261 in the 13-limit; 289/288, 442/441, 715/714, and 1156/1155 in the 17-limit.
196edo can also treated as a 2.9.5.7.11.13.17 subgroup temperament (with the patent 9), providing a distinct flat tendency for harmonics 5, 7, 9, 11, 13, and 17. With the patent 9, it tempers out 321489/320000, 420175/419904, and 703125/702464 in the 2.9.5.7 subgroup; 441/440, 8019/8000, 41503/41472, and 9453125/9437184 in the 2.9.5.7.11 subgroup; 729/728, 1001/1000, 1575/1573, 6656/6655, and 10985/10976 in the 2.9.5.7.11.13 subgroup; 833/832, 936/935, 1089/1088, 1225/1224, 2025/2023, and 14365/14336 in the 2.9.5.7.11.13.17 subgroup.
Since it is part of the Fibonacci sequence beginning with 5 and 12, it closely approximates peppermint temperament.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +2.13 | -0.60 | -1.48 | -1.87 | -0.30 | -1.75 | +1.53 | -0.87 | +2.49 | +0.65 | +2.34 |
| Relative (%) | +34.7 | -9.8 | -24.2 | -30.5 | -4.9 | -28.6 | +24.9 | -14.3 | +40.6 | +10.6 | +38.2 | |
| Steps (reduced) |
311 (115) |
455 (63) |
550 (158) |
621 (33) |
678 (90) |
725 (137) |
766 (178) |
801 (17) |
833 (49) |
861 (77) |
887 (103) | |
Subsets and supersets
Since 196 factors into 22 × 72, 196edo has subset edos 2, 4, 7, 14, 28, 49, and 98.