275edo: Difference between revisions

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Created page with "{{Infobox ET}} {{EDO intro|275}} == Theory == If harmonic 5 is used, 275et tends very sharp. It tempers out {{monzo| 24 -21 4 }} (vulture comma) and {{monzo| 19 10 -15 }}..."
 
+RTT table and rank-2 temperaments
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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 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.  


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]], 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]].  
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=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|275|intervals=prime|columns=11}}
{{Harmonics in equal|275|intervals=prime|columns=11}}
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 436 -275 }}
| [{{val| 275 436 }}]
| -0.1863
| 0.1862
| 4.27
|-
| 2.3.5
| {{monzo| 24 -21 4 }}, {{monzo| 19 10 -15 }}
| [{{val| 275 436 639 }}]
| -0.4184
| 0.3618
| 8.29
|-
| 2.3.5.7
| 6144/6125, 10976/10935, 9882516/9765625
| [{{val| 275 436 639 772 }}]
| -0.3051
| 0.3698
| 8.48
|-
| 2.3.5.7.11
| 441/440, 4000/3993, 6144/6125, 10976/10935
| [{{val| 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
| [{{val| 275 436 639 772 952 1018 }}] (275e)
| -0.4158
| 0.3574
| 8.19
|}
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Cents<br>(Reduced)
! Associated<br>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<br>(11\275)
| 497.45<br>(48.00)
| 4/3<br>(36/35)
| [[Hendecatonic]]
|}


[[Category:Equal divisions of the octave]]
[[Category:Equal divisions of the octave]]

Revision as of 18:25, 4 November 2022

← 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

Template:EDO intro

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)

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
(Reduced) 		Cents
(Reduced) 		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
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