94edo: Difference between revisions
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The '''94 equal temperament''', often abbreviated '''94-tET''', '''94-[[EDO]]''', or '''94-ET''', is the scale derived by dividing the [[octave]] into 94 equally-sized steps, where each step | The '''94 equal temperament''', often abbreviated '''94-tET''', '''94-[[EDO]]''', or '''94-ET''', is the scale derived by dividing the [[octave]] into 94 equally-sized steps, where each step is 12.766 [[cent|cents]]. | ||
== Theory == | |||
94edo is a remarkable all-around utility temperament, good from low [[prime limit]] to very high prime limit situations. It is the first equal temperament to be [[consistent]] through the [[23-limit]], and no other equal temperament is so consistent until [[282edo|282]] and [[311edo|311]] make their appearance. | |||
The list of 23-limit commas it tempers out is huge, but it's worth noting that it tempers out [[32805/32768]] and is thus a [[schismatic]] system, that it tempers out [[225/224]] and [[385/384]] and so is a [[marvel]] system, and that it also tempers out [[3125/3087]], [[4000/3969]], [[5120/5103]] and [[540/539]]. It provides the [[optimal patent val]] for the rank five temperament tempering out [[275/273]], and for a number of other temperaments, such as [[isis]]. | |||
== Approximations to prime harmonics== | :''See also: [[Table of 94edo intervals]]'' | ||
== Approximations to prime harmonics == | |||
{| class="wikitable center-all" | {| class="wikitable center-all" | ||
! colspan="2" | | ! colspan="2" | | ||
| Line 21: | Line 24: | ||
|- | |- | ||
! rowspan="2" |Error | ! rowspan="2" |Error | ||
!absolute (¢) | ! absolute (¢) | ||
| 0.0 | | 0.0 | ||
| +0.17 | | +0.17 | ||
| Line 34: | Line 37: | ||
| +3.90 | | +3.90 | ||
|- | |- | ||
!relative (%) | ! relative (%) | ||
|0.0 | | 0.0 | ||
| +1.4 | | +1.4 | ||
| -26.1 | | -26.1 | ||
| Line 50: | Line 53: | ||
== Rank two temperaments == | == Rank two temperaments == | ||
{| class="wikitable | {| class="wikitable center-all right-3 left-5" | ||
! Periods <br> per | ! Periods<br>per octave | ||
! Generator | ! Generator | ||
! Cents | ! Cents | ||
! Associated <br> ratio | ! Associated<br>ratio | ||
! Temperament | ! Temperament | ||
|- | |- | ||
| Line 92: | Line 95: | ||
| 64/63 | | 64/63 | ||
| [[Ketchup]] | | [[Ketchup]] | ||
|- | |- | ||
| 2 | | 2 | ||
| Line 104: | Line 101: | ||
| 27/25 | | 27/25 | ||
| [[Fifive]] | | [[Fifive]] | ||
|- | |||
| 2 | |||
| 30\94 | |||
| 382.979 | |||
| 5/4 | |||
| [[Wizard]] / [[gizzard]] | |||
|- | |- | ||
| 2 | | 2 | ||
| Line 112: | Line 115: | ||
|- | |- | ||
| 2 | | 2 | ||
| | | 43\94 | ||
| | | 548.936 | ||
| | | 11/8 | ||
| [[ | | [[Kleischismic]] | ||
|} | |} | ||
Below are some 23-limit temperaments supported by 94et. It might be noted that 94, a very good tuning for [[ | Below are some 23-limit temperaments supported by 94et. It might be noted that 94, a very good tuning for [[garibaldi temperament]], shows us how to extend it to the 23-limit. | ||
* 46&94 <<8 30 -18 -4 -28 8 -24 2 ... || | * 46&94 <<8 30 -18 -4 -28 8 -24 2 ... || | ||
| Line 132: | Line 135: | ||
* 94 solo <<12 -2 20 -6 -42 12 -36 -44 ... || (a rank one temperament!) | * 94 solo <<12 -2 20 -6 -42 12 -36 -44 ... || (a rank one temperament!) | ||
Temperaments for which 94 is a [[ | Temperaments for which 94 is a [[MOS]]: | ||
* 311&94 <<3 70 -42 69 -34 50 85 83...|| | * 311&94 <<3 70 -42 69 -34 50 85 83...|| | ||
Revision as of 05:37, 27 August 2020
The 94 equal temperament, often abbreviated 94-tET, 94-EDO, or 94-ET, is the scale derived by dividing the octave into 94 equally-sized steps, where each step is 12.766 cents.
Theory
94edo is a remarkable all-around utility temperament, good from low prime limit to very high prime limit situations. It is the first equal temperament to be consistent through the 23-limit, and no other equal temperament is so consistent until 282 and 311 make their appearance.
The list of 23-limit commas it tempers out is huge, but it's worth noting that it tempers out 32805/32768 and is thus a schismatic system, that it tempers out 225/224 and 385/384 and so is a marvel system, and that it also tempers out 3125/3087, 4000/3969, 5120/5103 and 540/539. It provides the optimal patent val for the rank five temperament tempering out 275/273, and for a number of other temperaments, such as isis.
- See also: Table of 94edo intervals
Approximations to prime harmonics
| prime 2 | prime 3 | prime 5 | prime 7 | prime 11 | prime 13 | prime 17 | prime 19 | prime 23 | prime 29 | prime 31 | ||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | 0.0 | +0.17 | -3.33 | +1.39 | -2.38 | +2.03 | -2.83 | -3.90 | -2.74 | +4.47 | +3.90 |
| relative (%) | 0.0 | +1.4 | -26.1 | +10.9 | -18.7 | +15.9 | -22.2 | -30.5 | -21.5 | +35.0 | +30.6 | |
Rank two temperaments
| Periods per octave |
Generator | Cents | Associated ratio |
Temperament |
|---|---|---|---|---|
| 1 | 3\94 | 38.298 | 49/48 | Slender |
| 1 | 5\94 | 63.830 | 25/24 | Sycamore / betic |
| 1 | 11\94 | 140.426 | 243/224 13/12 |
Tsaharuk Quanic |
| 1 | 19\94 | 242.553 | 147/128 | Septiquarter |
| 1 | 39\94 | 497.872 | 4/3 | Schismatic / Garibaldi |
| 2 | 2\94 | 25.532 | 64/63 | Ketchup |
| 2 | 11\94 | 140.426 | 27/25 | Fifive |
| 2 | 30\94 | 382.979 | 5/4 | Wizard / gizzard |
| 2 | 34\94 | 434.043 | 9/7 | Pogo / supers |
| 2 | 43\94 | 548.936 | 11/8 | Kleischismic |
Below are some 23-limit temperaments supported by 94et. It might be noted that 94, a very good tuning for garibaldi temperament, shows us how to extend it to the 23-limit.
- 46&94 <<8 30 -18 -4 -28 8 -24 2 ... ||
- 68&94 <<20 28 2 -10 24 20 34 52 ... ||
- 53&94 <<1 -8 -14 23 20 -46 -3 -35 ... || (one garibaldi)
- 41&94 <<1 -8 -14 23 20 48 -3 -35 ... || (another garibaldi, only differing in the mappings of 17 and 23)
- 135&94 <<1 -8 -14 23 20 48 -3 59 ... || (another garibaldi)
- 130&94 <<6 -48 10 -50 26 6 -18 -22 ... || (a pogo extension)
- 58&94 <<6 46 10 44 26 6 -18 -22 ... || (a supers extension)
- 50&94 <<24 -4 40 -12 10 24 22 6 ... ||
- 72&94 <<12 -2 20 -6 52 12 -36 -44 ... || (a gizzard extension)
- 80&94 <<18 44 30 38 -16 18 40 28 ... ||
- 94 solo <<12 -2 20 -6 -42 12 -36 -44 ... || (a rank one temperament!)
Temperaments for which 94 is a MOS:
- 311&94 <<3 70 -42 69 -34 50 85 83...||
- 422&94 <<8 124 -18 90 -28 102 164 96 ... ||