66edo: Difference between revisions
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In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO. | In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO. | ||
== Octave stretch or compression == | |||
66edo tunes multiple-of-3 [[harmonics]] quite sharp, but most other simple harmonics quite flat. | |||
Slight [[octave stretching]] turns 66edo into a [[dual-fifth]] tuning, sharing error near-equally between its two [[mapping]]s of 3, while improving most other simple harmonics. Slightly stretched tunings of 66edo include [[zpi|340zpi]], [[ed10|219ed10]] & [[ed9|209ed9]]. | |||
Substantial octave stretching will flip the mapping of many harmonics including 3, but will reduce the error of most of them. Though it will introduce more mapping inconsistencies. Such tunings of 66edo include [[zpi|338zpi]] and [[104edt]]. | |||
Moderate ''[[octave shrinking]]'' will instead improve upon 66edo's best mapping of prime 3 dramatically. It reduce the error on most other simple harmonics. It flips the mapping of many of them, but introduces less mapping inconsistencies than stretching. Moderately octave-compressed tunings for 66edo include [[105edt]] and [[zpi|342zpi]]. | |||
== Instruments == | == Instruments == | ||
Revision as of 05:51, 19 September 2025
| ← 65edo | 66edo | 67edo → |
66 equal divisions of the octave (abbreviated 66edo or 66ed2), also called 66-tone equal temperament (66tet) or 66 equal temperament (66et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 66 equal parts of about 18.2 ¢ each. Each step represents a frequency ratio of 21/66, or the 66th root of 2.
Theory
The patent val of 66edo is contorted in the 5-limit, tempering out the same commas (250/243, 2048/2025, 3125/3072, etc.) as 22edo. In the 7-limit it tempers out 686/675 and 1029/1024, in the 11-limit 55/54, 100/99 and 121/120, in the 13-limit 91/90, 169/168, 196/195 and in the 17-limit 136/135 and 256/255. It provides the optimal patent val for the 11- and 13-limit ammonite temperament. Otherwise, 66edo is not exceptional when it comes to approximating prime harmonics; however, it contains a quite accurate approximation to the 5:7:9:11:13 chord and can therefore be used for various primodal over-5 scales.
The 66b val tempers out 16875/16384 in the 5-limit, 126/125, 1728/1715 and 2401/2400 in the 7-limit, 99/98 and 385/384 in the 11-limit, and 105/104, 144/143 and 847/845 in the 13-limit.
109 steps of 66edo is extremely close to the acoustic pi with only +0.023 ¢ of error.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +7.14 | -4.50 | -5.19 | -3.91 | -5.86 | -4.16 | +2.64 | +4.14 | -6.60 | +1.95 | +8.09 |
| Relative (%) | +39.2 | -24.7 | -28.5 | -21.5 | -32.2 | -22.9 | +14.5 | +22.7 | -36.3 | +10.7 | +44.5 | |
| Steps (reduced) |
105 (39) |
153 (21) |
185 (53) |
209 (11) |
228 (30) |
244 (46) |
258 (60) |
270 (6) |
280 (16) |
290 (26) |
299 (35) | |
Subsets and supersets
Since 66 factors into 2 × 3 × 11, 66edo has subset edos 2, 3, 6, 11, 22, and 33. 198edo, which triples it, corrects its approximation to many of the lower harmonics.
Interval table
| Steps | Cents | Approximate ratios | Ups and downs notation (Dual flat fifth 38\66) |
Ups and downs notation (Dual sharp fifth 39\66) |
|---|---|---|---|---|
| 0 | 0 | 1/1 | D | D |
| 1 | 18.2 | ^D, vE♭♭♭♭ | ^D, vvE♭ | |
| 2 | 36.4 | D♯, E♭♭♭♭ | ^^D, vE♭ | |
| 3 | 54.5 | 31/30, 32/31, 33/32, 34/33 | ^D♯, vE♭♭♭ | ^3D, E♭ |
| 4 | 72.7 | 24/23 | D𝄪, E♭♭♭ | ^4D, ^E♭ |
| 5 | 90.9 | 20/19 | ^D𝄪, vE♭♭ | v4D♯, ^^E♭ |
| 6 | 109.1 | 16/15, 33/31 | D♯𝄪, E♭♭ | v3D♯, ^3E♭ |
| 7 | 127.3 | 14/13 | ^D♯𝄪, vE♭ | vvD♯, ^4E♭ |
| 8 | 145.5 | D𝄪𝄪, E♭ | vD♯, v4E | |
| 9 | 163.6 | 11/10, 34/31 | ^D𝄪𝄪, vE | D♯, v3E |
| 10 | 181.8 | E | ^D♯, vvE | |
| 11 | 200 | 28/25 | ^E, vF♭♭♭ | ^^D♯, vE |
| 12 | 218.2 | 17/15, 25/22 | E♯, F♭♭♭ | E |
| 13 | 236.4 | ^E♯, vF♭♭ | ^E, vvF | |
| 14 | 254.5 | 22/19 | E𝄪, F♭♭ | ^^E, vF |
| 15 | 272.7 | 34/29 | ^E𝄪, vF♭ | F |
| 16 | 290.9 | 13/11 | E♯𝄪, F♭ | ^F, vvG♭ |
| 17 | 309.1 | ^E♯𝄪, vF | ^^F, vG♭ | |
| 18 | 327.3 | 29/24 | F | ^3F, G♭ |
| 19 | 345.5 | ^F, vG♭♭♭♭ | ^4F, ^G♭ | |
| 20 | 363.6 | 21/17 | F♯, G♭♭♭♭ | v4F♯, ^^G♭ |
| 21 | 381.8 | ^F♯, vG♭♭♭ | v3F♯, ^3G♭ | |
| 22 | 400 | 29/23 | F𝄪, G♭♭♭ | vvF♯, ^4G♭ |
| 23 | 418.2 | 14/11 | ^F𝄪, vG♭♭ | vF♯, v4G |
| 24 | 436.4 | F♯𝄪, G♭♭ | F♯, v3G | |
| 25 | 454.5 | 13/10 | ^F♯𝄪, vG♭ | ^F♯, vvG |
| 26 | 472.7 | 21/16, 25/19 | F𝄪𝄪, G♭ | ^^F♯, vG |
| 27 | 490.9 | ^F𝄪𝄪, vG | G | |
| 28 | 509.1 | G | ^G, vvA♭ | |
| 29 | 527.3 | 19/14, 23/17 | ^G, vA♭♭♭♭ | ^^G, vA♭ |
| 30 | 545.5 | 26/19 | G♯, A♭♭♭♭ | ^3G, A♭ |
| 31 | 563.6 | ^G♯, vA♭♭♭ | ^4G, ^A♭ | |
| 32 | 581.8 | 7/5 | G𝄪, A♭♭♭ | v4G♯, ^^A♭ |
| 33 | 600 | 17/12, 24/17 | ^G𝄪, vA♭♭ | v3G♯, ^3A♭ |
| 34 | 618.2 | 10/7 | G♯𝄪, A♭♭ | vvG♯, ^4A♭ |
| 35 | 636.4 | ^G♯𝄪, vA♭ | vG♯, v4A | |
| 36 | 654.5 | 19/13 | G𝄪𝄪, A♭ | G♯, v3A |
| 37 | 672.7 | 28/19, 31/21, 34/23 | ^G𝄪𝄪, vA | ^G♯, vvA |
| 38 | 690.9 | A | ^^G♯, vA | |
| 39 | 709.1 | ^A, vB♭♭♭♭ | A | |
| 40 | 727.3 | 32/21 | A♯, B♭♭♭♭ | ^A, vvB♭ |
| 41 | 745.5 | 20/13 | ^A♯, vB♭♭♭ | ^^A, vB♭ |
| 42 | 763.6 | A𝄪, B♭♭♭ | ^3A, B♭ | |
| 43 | 781.8 | 11/7 | ^A𝄪, vB♭♭ | ^4A, ^B♭ |
| 44 | 800 | 35/22 | A♯𝄪, B♭♭ | v4A♯, ^^B♭ |
| 45 | 818.2 | ^A♯𝄪, vB♭ | v3A♯, ^3B♭ | |
| 46 | 836.4 | 34/21 | A𝄪𝄪, B♭ | vvA♯, ^4B♭ |
| 47 | 854.5 | ^A𝄪𝄪, vB | vA♯, v4B | |
| 48 | 872.7 | B | A♯, v3B | |
| 49 | 890.9 | ^B, vC♭♭♭ | ^A♯, vvB | |
| 50 | 909.1 | 22/13 | B♯, C♭♭♭ | ^^A♯, vB |
| 51 | 927.3 | 29/17 | ^B♯, vC♭♭ | B |
| 52 | 945.5 | 19/11 | B𝄪, C♭♭ | ^B, vvC |
| 53 | 963.6 | ^B𝄪, vC♭ | ^^B, vC | |
| 54 | 981.8 | 30/17 | B♯𝄪, C♭ | C |
| 55 | 1000 | 25/14 | ^B♯𝄪, vC | ^C, vvD♭ |
| 56 | 1018.2 | C | ^^C, vD♭ | |
| 57 | 1036.4 | 20/11, 31/17 | ^C, vD♭♭♭♭ | ^3C, D♭ |
| 58 | 1054.5 | 35/19 | C♯, D♭♭♭♭ | ^4C, ^D♭ |
| 59 | 1072.7 | 13/7 | ^C♯, vD♭♭♭ | v4C♯, ^^D♭ |
| 60 | 1090.9 | 15/8 | C𝄪, D♭♭♭ | v3C♯, ^3D♭ |
| 61 | 1109.1 | 19/10 | ^C𝄪, vD♭♭ | vvC♯, ^4D♭ |
| 62 | 1127.3 | 23/12 | C♯𝄪, D♭♭ | vC♯, v4D |
| 63 | 1145.5 | 31/16, 33/17 | ^C♯𝄪, vD♭ | C♯, v3D |
| 64 | 1163.6 | C𝄪𝄪, D♭ | ^C♯, vvD | |
| 65 | 1181.8 | ^C𝄪𝄪, vD | ^^C♯, vD | |
| 66 | 1200 | 2/1 | D | D |
Notation
Sagittal notation
This notation uses the same sagittal sequence as 59-EDO, and is a superset of the notations for EDOs 22 and 11.
Evo flavor
Revo flavor
In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's primary comma (the comma it exactly represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it approximately represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.
Octave stretch or compression
66edo tunes multiple-of-3 harmonics quite sharp, but most other simple harmonics quite flat.
Slight octave stretching turns 66edo into a dual-fifth tuning, sharing error near-equally between its two mappings of 3, while improving most other simple harmonics. Slightly stretched tunings of 66edo include 340zpi, 219ed10 & 209ed9.
Substantial octave stretching will flip the mapping of many harmonics including 3, but will reduce the error of most of them. Though it will introduce more mapping inconsistencies. Such tunings of 66edo include 338zpi and 104edt.
Moderate octave shrinking will instead improve upon 66edo's best mapping of prime 3 dramatically. It reduce the error on most other simple harmonics. It flips the mapping of many of them, but introduces less mapping inconsistencies than stretching. Moderately octave-compressed tunings for 66edo include 105edt and 342zpi.
Instruments
Lumatone
Skip fretting
Skip fretting system 66 7 11 is a skip fretting system for 66edo. All examples on this page are for 7-string guitar.
- Prime harmonics
1/1: string 2 open
2/1: string 1 fret 11 and string 7 fret 11
3/2: string 3 fret 4
5/4: string 2 fret 3
7/4: string 3 fret 6
11/8: string 5 fret 9
13/8: string 3 fret 5
17/16: string 6 fret 4
19/16: string 5 fret 7
23/16: string 2 fret 5
29/16: string 4 fret 5
31/16: string 2 fret 9