76edo: Difference between revisions
→Theory: +subsets and supersets section |
Dave Keenan (talk | contribs) added Sagittal notation |
||
| Line 15: | Line 15: | ||
== Intervals == | == Intervals == | ||
{{Interval table}} | {{Interval table}} | ||
== Notation == | |||
===Sagittal notation=== | |||
This notation uses the same sagittal sequence as EDOs [[62edo#Sagittal notation|62]] and [[69edo#Sagittal notation|69]], and is a superset of the notations for EDOs [[38edo#Sagittal notation|38]] and [[19edo#Sagittal notation|19]]. | |||
====Evo flavor==== | |||
<imagemap> | |||
File:76-EDO_Evo_Sagittal.svg | |||
desc none | |||
rect 80 0 300 50 [[Sagittal_notation]] | |||
rect 300 0 575 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation] | |||
rect 20 80 170 106 [[1053/1024]] | |||
rect 170 80 330 106 [[567/550]] | |||
default [[File:76-EDO_Evo_Sagittal.svg]] | |||
</imagemap> | |||
====Revo flavor==== | |||
<imagemap> | |||
File:76-EDO_Revo_Sagittal.svg | |||
desc none | |||
rect 80 0 300 50 [[Sagittal_notation]] | |||
rect 300 0 604 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation] | |||
rect 20 80 170 106 [[1053/1024]] | |||
rect 170 80 330 106 [[567/550]] | |||
default [[File:76-EDO_Revo_Sagittal.svg]] | |||
</imagemap> | |||
====Evo-SZ flavor==== | |||
<imagemap> | |||
File:76-EDO_Evo-SZ_Sagittal.svg | |||
desc none | |||
rect 80 0 300 50 [[Sagittal_notation]] | |||
rect 300 0 553 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation] | |||
rect 20 80 170 106 [[1053/1024]] | |||
rect 170 80 330 106 [[567/550]] | |||
default [[File:76-EDO_Evo-SZ_Sagittal.svg]] | |||
</imagemap> | |||
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. | |||
Revision as of 11:46, 30 December 2024
| ← 75edo | 76edo | 77edo → |
Theory
76edo's patent val is contorted in the 5-limit, reflecting the fact that 76 = 4 × 19. In the 7-limit it tempers out 2401/2400 in addition to 81/80, and so supports the squares temperament. In the 11-limit, it tempers out 245/242 and 385/384, and supports pombe, the 24 & 26 temperament. In the 13-limit, it tempers out 105/104, 144/143, 351/350 and 364/363. While the 44\76 = 11\19 fifth is already flat, the 43\76 fifth, even flatter, is an almost perfect approximation to the hornbostel temperament's POTE fifth, whereas its sharp fifth, 45\76, makes for an excellent superpyth fifth. Hence you can do hornbostel/mavila, squares/meantone, and superpyth all with the same equal division.
Using the 76dgh val, 76edo provides an excellent tuning for teff temperament, a low-complexity, medium-accuracy, and high-limit (17- or 19-limit) temperament.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -7.22 | -7.37 | -5.67 | +1.35 | +1.31 | -3.69 | +1.20 | +5.57 | +2.49 | +2.90 | +3.30 |
| Relative (%) | -45.7 | -46.7 | -35.9 | +8.6 | +8.3 | -23.3 | +7.6 | +35.3 | +15.8 | +18.4 | +20.9 | |
| Steps (reduced) |
120 (44) |
176 (24) |
213 (61) |
241 (13) |
263 (35) |
281 (53) |
297 (69) |
311 (7) |
323 (19) |
334 (30) |
344 (40) | |
Subsets and supersets
Since 76 factors into 22 × 19, 76edo has subset edos 2, 4, 19, and 38. 152edo, which doubles it, is a zeta peak edo.
Intervals
| Steps | Cents | Approximate ratios | Ups and downs notation (Dual flat fifth 44\76) |
Ups and downs notation (Dual sharp fifth 45\76) |
|---|---|---|---|---|
| 0 | 0 | 1/1 | D | D |
| 1 | 15.8 | ^D, ^E♭♭♭ | ^D, vvE♭ | |
| 2 | 31.6 | ^^D, vvE♭♭ | ^^D, vE♭ | |
| 3 | 47.4 | 36/35, 38/37 | vD♯, vE♭♭ | ^3D, E♭ |
| 4 | 63.2 | 29/28 | D♯, E♭♭ | ^4D, ^E♭ |
| 5 | 78.9 | 23/22 | ^D♯, ^E♭♭ | ^5D, ^^E♭ |
| 6 | 94.7 | ^^D♯, vvE♭ | v5D♯, ^3E♭ | |
| 7 | 110.5 | vD𝄪, vE♭ | v4D♯, ^4E♭ | |
| 8 | 126.3 | 14/13 | D𝄪, E♭ | v3D♯, ^5E♭ |
| 9 | 142.1 | ^D𝄪, ^E♭ | vvD♯, v5E | |
| 10 | 157.9 | 34/31 | ^^D𝄪, vvE | vD♯, v4E |
| 11 | 173.7 | 32/29 | vD♯𝄪, vE | D♯, v3E |
| 12 | 189.5 | 19/17, 29/26, 39/35 | E | ^D♯, vvE |
| 13 | 205.3 | ^E, ^F♭♭ | ^^D♯, vE | |
| 14 | 221.1 | 33/29 | ^^E, vvF♭ | E |
| 15 | 236.8 | vE♯, vF♭ | ^E, vvF | |
| 16 | 252.6 | 22/19, 37/32 | E♯, F♭ | ^^E, vF |
| 17 | 268.4 | 7/6 | ^E♯, ^F♭ | F |
| 18 | 284.2 | 33/28 | ^^E♯, vvF | ^F, vvG♭ |
| 19 | 300 | 19/16, 25/21 | vE𝄪, vF | ^^F, vG♭ |
| 20 | 315.8 | 6/5 | F | ^3F, G♭ |
| 21 | 331.6 | 23/19 | ^F, ^G♭♭♭ | ^4F, ^G♭ |
| 22 | 347.4 | ^^F, vvG♭♭ | ^5F, ^^G♭ | |
| 23 | 363.2 | vF♯, vG♭♭ | v5F♯, ^3G♭ | |
| 24 | 378.9 | F♯, G♭♭ | v4F♯, ^4G♭ | |
| 25 | 394.7 | ^F♯, ^G♭♭ | v3F♯, ^5G♭ | |
| 26 | 410.5 | 33/26 | ^^F♯, vvG♭ | vvF♯, v5G |
| 27 | 426.3 | vF𝄪, vG♭ | vF♯, v4G | |
| 28 | 442.1 | F𝄪, G♭ | F♯, v3G | |
| 29 | 457.9 | ^F𝄪, ^G♭ | ^F♯, vvG | |
| 30 | 473.7 | ^^F𝄪, vvG | ^^F♯, vG | |
| 31 | 489.5 | vF♯𝄪, vG | G | |
| 32 | 505.3 | G | ^G, vvA♭ | |
| 33 | 521.1 | 23/17 | ^G, ^A♭♭♭ | ^^G, vA♭ |
| 34 | 536.8 | ^^G, vvA♭♭ | ^3G, A♭ | |
| 35 | 552.6 | 11/8 | vG♯, vA♭♭ | ^4G, ^A♭ |
| 36 | 568.4 | 25/18, 32/23 | G♯, A♭♭ | ^5G, ^^A♭ |
| 37 | 584.2 | 7/5 | ^G♯, ^A♭♭ | v5G♯, ^3A♭ |
| 38 | 600 | ^^G♯, vvA♭ | v4G♯, ^4A♭ | |
| 39 | 615.8 | 10/7 | vG𝄪, vA♭ | v3G♯, ^5A♭ |
| 40 | 631.6 | 23/16, 36/25 | G𝄪, A♭ | vvG♯, v5A |
| 41 | 647.4 | 16/11 | ^G𝄪, ^A♭ | vG♯, v4A |
| 42 | 663.2 | ^^G𝄪, vvA | G♯, v3A | |
| 43 | 678.9 | 34/23 | vG♯𝄪, vA | ^G♯, vvA |
| 44 | 694.7 | A | ^^G♯, vA | |
| 45 | 710.5 | ^A, ^B♭♭♭ | A | |
| 46 | 726.3 | ^^A, vvB♭♭ | ^A, vvB♭ | |
| 47 | 742.1 | vA♯, vB♭♭ | ^^A, vB♭ | |
| 48 | 757.9 | A♯, B♭♭ | ^3A, B♭ | |
| 49 | 773.7 | ^A♯, ^B♭♭ | ^4A, ^B♭ | |
| 50 | 789.5 | ^^A♯, vvB♭ | ^5A, ^^B♭ | |
| 51 | 805.3 | vA𝄪, vB♭ | v5A♯, ^3B♭ | |
| 52 | 821.1 | 37/23 | A𝄪, B♭ | v4A♯, ^4B♭ |
| 53 | 836.8 | ^A𝄪, ^B♭ | v3A♯, ^5B♭ | |
| 54 | 852.6 | ^^A𝄪, vvB | vvA♯, v5B | |
| 55 | 868.4 | 33/20, 38/23 | vA♯𝄪, vB | vA♯, v4B |
| 56 | 884.2 | 5/3 | B | A♯, v3B |
| 57 | 900 | 32/19, 37/22 | ^B, ^C♭♭ | ^A♯, vvB |
| 58 | 915.8 | ^^B, vvC♭ | ^^A♯, vB | |
| 59 | 931.6 | 12/7 | vB♯, vC♭ | B |
| 60 | 947.4 | 19/11 | B♯, C♭ | ^B, vvC |
| 61 | 963.2 | ^B♯, ^C♭ | ^^B, vC | |
| 62 | 978.9 | ^^B♯, vvC | C | |
| 63 | 994.7 | vB𝄪, vC | ^C, vvD♭ | |
| 64 | 1010.5 | 34/19 | C | ^^C, vD♭ |
| 65 | 1026.3 | 29/16 | ^C, ^D♭♭♭ | ^3C, D♭ |
| 66 | 1042.1 | 31/17 | ^^C, vvD♭♭ | ^4C, ^D♭ |
| 67 | 1057.9 | vC♯, vD♭♭ | ^5C, ^^D♭ | |
| 68 | 1073.7 | 13/7 | C♯, D♭♭ | v5C♯, ^3D♭ |
| 69 | 1089.5 | ^C♯, ^D♭♭ | v4C♯, ^4D♭ | |
| 70 | 1105.3 | ^^C♯, vvD♭ | v3C♯, ^5D♭ | |
| 71 | 1121.1 | vC𝄪, vD♭ | vvC♯, v5D | |
| 72 | 1136.8 | C𝄪, D♭ | vC♯, v4D | |
| 73 | 1152.6 | 35/18, 37/19 | ^C𝄪, ^D♭ | C♯, v3D |
| 74 | 1168.4 | ^^C𝄪, vvD | ^C♯, vvD | |
| 75 | 1184.2 | vC♯𝄪, vD | ^^C♯, vD | |
| 76 | 1200 | 2/1 | D | D |
Notation
Sagittal notation
This notation uses the same sagittal sequence as EDOs 62 and 69, and is a superset of the notations for EDOs 38 and 19.
Evo flavor
Revo flavor
Evo-SZ 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.