122edo: Difference between revisions
Wikispaces>genewardsmith **Imported revision 214063972 - Original comment: ** |
m →Subsets and supersets: 244 also corrects harmonic 7 |
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122 is flat in tendency, with the [[prime harmonic]]s from 3 to 13 tuned flat. As an equal temperament, it [[tempering out|tempers out]] 78732/78125 ([[sensipent comma]]) in the [[5-limit]]; [[225/224]] in the [[7-limit]]; [[385/384]] and [[4000/3993]] in the [[11-limit]]; and [[351/350]] and [[364/363]] in the [[13-limit]]. It provides the [[optimal patent val]] for the 7-limit [[tritonic]] temperament and the 11-limit [[Marvel temperaments #Tritoni|tritoni]] temperament, and the [[rank-3|planar]] temperament [[squalentine]]. | |||
122 = [[55edo|55]] + [[67edo|67]], and so using the 122c [[val]] it is the [[convergent]] towards [[1/6-comma meantone]], with a fifth just a hundredth of a cent flatter. | |||
=== Odd harmonics === | |||
{{Harmonics in equal|122}} | |||
Harmonic 25 (two 5/4 major thirds) is about halfway between 122edo's steps. | |||
=== Subsets and supersets === | |||
Since 122 factors into {{factorization|122}}, 122edo contains [[2edo]] and [[61edo]] as its subsets. [[244edo]] (double 122edo) provides a good correction to harmonics 7 and 25. | |||
[[Category:Tritonic]] | |||
[[Category:Meantone]] | |||
Latest revision as of 08:21, 18 March 2026
| ← 121edo | 122edo | 123edo → |
122 equal divisions of the octave (abbreviated 122edo or 122ed2), also called 122-tone equal temperament (122tet) or 122 equal temperament (122et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 122 equal parts of about 9.84 ¢ each. Each step represents a frequency ratio of 21/122, or the 122nd root of 2.
122 is flat in tendency, with the prime harmonics from 3 to 13 tuned flat. As an equal temperament, it tempers out 78732/78125 (sensipent comma) in the 5-limit; 225/224 in the 7-limit; 385/384 and 4000/3993 in the 11-limit; and 351/350 and 364/363 in the 13-limit. It provides the optimal patent val for the 7-limit tritonic temperament and the 11-limit tritoni temperament, and the planar temperament squalentine.
122 = 55 + 67, and so using the 122c val it is the convergent towards 1/6-comma meantone, with a fifth just a hundredth of a cent flatter.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -3.59 | -2.71 | -4.89 | +2.65 | -0.50 | -4.46 | +3.53 | +3.24 | -2.43 | +1.35 | +1.23 |
| Relative (%) | -36.5 | -27.5 | -49.7 | +26.9 | -5.1 | -45.4 | +35.9 | +33.0 | -24.7 | +13.7 | +12.5 | |
| Steps (reduced) |
193 (71) |
283 (39) |
342 (98) |
387 (21) |
422 (56) |
451 (85) |
477 (111) |
499 (11) |
518 (30) |
536 (48) |
552 (64) | |
Harmonic 25 (two 5/4 major thirds) is about halfway between 122edo's steps.
Subsets and supersets
Since 122 factors into 2 × 61, 122edo contains 2edo and 61edo as its subsets. 244edo (double 122edo) provides a good correction to harmonics 7 and 25.