275edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|275}}
{{ED intro}}


== Theory ==
== Theory ==
If harmonic 5 is used, 275et tends very sharp. It tempers out {{monzo| 24 -21 4 }} ([[vulture comma]]) and {{monzo| 19 10 -15 }} ([[trisedodge comma]]) in the 5-limit; [[6144/6125]] and [[10976/10935]] in the 7-limit.  
If [[harmonic]] [[5/1|5]] is used, 275et tends very sharp. It [[tempering out|tempers out]] {{monzo| 24 -21 4 }} ([[vulture comma]]) and {{monzo| 19 10 -15 }} (trisedodge comma) in the 5-limit; [[6144/6125]] and [[10976/10935]] in the 7-limit.  


The 275e val {{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 275e val {{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 {{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]].  
The 275 val {{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 ===
=== Prime harmonics ===
{{Harmonics in equal|275|intervals=prime|columns=11}}
{{Harmonics in equal|275|intervals=prime}}


=== Subsets and supersets ===
=== Subsets and supersets ===
Since 275 factors into 5<sup>2</sup> × 11, 275edo has subset edos {{EDOs| 5, 11, 25 and 55 }}.  
Since 275 factors into {{factorisation|275}}, 275edo has subset edos {{EDOs| 5, 11, 25 and 55 }}.  


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" | Tuning error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
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| 2.3
| 2.3
| {{monzo| 436 -275 }}
| {{monzo| 436 -275 }}
| [{{val| 275 436 }}]
| {{mapping| 275 436 }}
| -0.1863
| −0.1863
| 0.1862
| 0.1862
| 4.27
| 4.27
Line 35: Line 36:
| 2.3.5
| 2.3.5
| {{monzo| 24 -21 4 }}, {{monzo| 19 10 -15 }}
| {{monzo| 24 -21 4 }}, {{monzo| 19 10 -15 }}
| [{{val| 275 436 639 }}]
| {{mapping| 275 436 639 }}
| -0.4184
| −0.4184
| 0.3618
| 0.3618
| 8.29
| 8.29
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| 2.3.5.7
| 2.3.5.7
| 6144/6125, 10976/10935, 9882516/9765625
| 6144/6125, 10976/10935, 9882516/9765625
| [{{val| 275 436 639 772 }}]
| {{mapping| 275 436 639 772 }}
| -0.3051
| −0.3051
| 0.3698
| 0.3698
| 8.48
| 8.48
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| 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
| [{{val| 275 436 639 772 952 }}] (275e)
| {{mapping| 275 436 639 772 952 }} (275e)
| -0.4096
| −0.4096
| 0.3912
| 0.3912
| 8.97
| 8.97
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| 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
| [{{val| 275 436 639 772 952 1018 }}] (275e)
| {{mapping| 275 436 639 772 952 1018 }} (275e)
| -0.4158
| −0.4158
| 0.3574
| 0.3574
| 8.19
| 8.19
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=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per 8ve
|-
! Generator<br>(Reduced)
! Periods<br />per 8ve
! Cents<br>(Reduced)
! Generator*
! Associated<br>Ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
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|-
|-
| 11
| 11
| 114\275<br>(11\275)
| 114\275<br />(11\275)
| 497.45<br>(48.00)
| 497.45<br />(48.00)
| 4/3<br>(36/35)
| 4/3<br />(36/35)
| [[Hendecatonic]]
| [[Hendecatonic]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

Latest revision as of 23:09, 20 February 2025

← 274edo 275edo 276edo →
Prime factorization 52 × 11
Step size 4.36364 ¢ 
Fifth 161\275 (702.545 ¢)
Semitones (A1:m2) 27:20 (117.8 ¢ : 87.27 ¢)
Consistency limit 9
Distinct consistency limit 9

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

Approximation of prime harmonics in 275edo
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

Table of rank-2 temperaments by generator
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

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