275edo: Difference between revisions
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" |
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! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br | ! rowspan="2" | Optimal<br>8ve stretch (¢) | ||
! colspan="2" | Tuning error | ! colspan="2" | Tuning error | ||
|- | |- | ||
| Line 28: | Line 28: | ||
|- | |- | ||
| 2.3 | | 2.3 | ||
| {{ | | {{Monzo| 436 -275 }} | ||
| {{ | | {{Mapping| 275 436 }} | ||
| −0.1863 | | −0.1863 | ||
| 0.1862 | | 0.1862 | ||
| Line 35: | Line 35: | ||
|- | |- | ||
| 2.3.5 | | 2.3.5 | ||
| {{ | | {{Monzo| 24 -21 4 }}, {{monzo| 19 10 -15 }} | ||
| {{ | | {{Mapping| 275 436 639 }} | ||
| −0.4184 | | −0.4184 | ||
| 0.3618 | | 0.3618 | ||
| Line 43: | Line 43: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 6144/6125, 10976/10935, 9882516/9765625 | | 6144/6125, 10976/10935, 9882516/9765625 | ||
| {{ | | {{Mapping| 275 436 639 772 }} | ||
| −0.3051 | | −0.3051 | ||
| 0.3698 | | 0.3698 | ||
| Line 50: | Line 50: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 441/440, 4000/3993, 6144/6125, 10976/10935 | | 441/440, 4000/3993, 6144/6125, 10976/10935 | ||
| {{ | | {{Mapping| 275 436 639 772 952 }} (275e) | ||
| −0.4096 | | −0.4096 | ||
| 0.3912 | | 0.3912 | ||
| Line 57: | Line 57: | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 364/363, 441/440, 676/675, 6144/6125, 10976/10935 | | 364/363, 441/440, 676/675, 6144/6125, 10976/10935 | ||
| {{ | | {{Mapping| 275 436 639 772 952 1018 }} (275e) | ||
| −0.4158 | | −0.4158 | ||
| 0.3574 | | 0.3574 | ||
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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
|- | |- | ||
! Periods<br | ! Periods<br>per 8ve | ||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br | ! Associated<br>ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 98: | Line 98: | ||
|- | |- | ||
| 11 | | 11 | ||
| 114\275<br | | 114\275<br>(11\275) | ||
| 497.45<br | | 497.45<br>(48.00) | ||
| 4/3<br | | 4/3<br>(36/35) | ||
| [[Hendecatonic]] | | [[Hendecatonic (temperament)|Hendecatonic]] | ||
|} | |} | ||
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct | <nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct | ||
Latest revision as of 12:20, 5 April 2026
| ← 274edo | 275edo | 276edo → |
275 equal divisions of the octave (abbreviated 275edo or 275ed2), also called 275-tone equal temperament (275tet) or 275 equal temperament (275et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 275 equal parts of about 4.36 ¢ each. Each step represents a frequency ratio of 21/275, or the 275th root of 2.
Theory
If harmonic 5 is used, 275et tends very sharp. It tempers out [24 -21 4⟩ (vulture comma) and [19 10 -15⟩ (trisedodge comma) in the 5-limit; 6144/6125 and 10976/10935 in the 7-limit.
The 275e val ⟨275 436 639 772 952] being the best, tempers out 441/440, 4000/3993, 14700/14641, and 19712/19683. This can be extended to the 13-limit through 364/363, 676/675, 1001/1000, 1575/1573 and 2080/2079.
The 275 val ⟨275 436 639 772 951] tempers out 3025/3024, 3773/3750, 8019/8000. This can be extended to the 13-limit through 352/351, 676/675, 1716/1715, 2200/2197, and 3584/3575.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.59 | +2.05 | -0.10 | -1.50 | +1.65 | -0.23 | -0.79 | +0.09 | +0.24 | -1.76 |
| Relative (%) | +0.0 | +13.5 | +47.0 | -2.3 | -34.4 | +37.9 | -5.2 | -18.0 | +2.0 | +5.5 | -40.4 | |
| Steps (reduced) |
275 (0) |
436 (161) |
639 (89) |
772 (222) |
951 (126) |
1018 (193) |
1124 (24) |
1168 (68) |
1244 (144) |
1336 (236) |
1362 (262) | |
Subsets and supersets
Since 275 factors into 52 × 11, 275edo has subset edos 5, 11, 25 and 55.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [436 -275⟩ | [⟨275 436]] | −0.1863 | 0.1862 | 4.27 |
| 2.3.5 | [24 -21 4⟩, [19 10 -15⟩ | [⟨275 436 639]] | −0.4184 | 0.3618 | 8.29 |
| 2.3.5.7 | 6144/6125, 10976/10935, 9882516/9765625 | [⟨275 436 639 772]] | −0.3051 | 0.3698 | 8.48 |
| 2.3.5.7.11 | 441/440, 4000/3993, 6144/6125, 10976/10935 | [⟨275 436 639 772 952]] (275e) | −0.4096 | 0.3912 | 8.97 |
| 2.3.5.7.11.13 | 364/363, 441/440, 676/675, 6144/6125, 10976/10935 | [⟨275 436 639 772 952 1018]] (275e) | −0.4158 | 0.3574 | 8.19 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 6\275 | 26.18 | 1594323/1562500 | Sfourth (5-limit) |
| 1 | 109\275 | 485.64 | 320/243 | Vulture (5-limit) |
| 1 | 128\275 | 558.55 | 112/81 | Condor (275e) |
| 5 | 17\275 | 74.18 | 25/24 | Countdown (275e) |
| 11 | 114\275 (11\275) |
497.45 (48.00) |
4/3 (36/35) |
Hendecatonic |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct