275edo: Difference between revisions

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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3
| 2.3
| {{monzo| 436 -275 }}
| {{monzo| 436 -275 }}
| {{mapping| 275 436 }}
| {{mapping| 275 436 }}
| -0.1863
| &minus;0.1863
| 0.1862
| 0.1862
| 4.27
| 4.27
Line 36: Line 28:
| {{monzo| 24 -21 4 }}, {{monzo| 19 10 -15 }}
| {{monzo| 24 -21 4 }}, {{monzo| 19 10 -15 }}
| {{mapping| 275 436 639 }}
| {{mapping| 275 436 639 }}
| -0.4184
| &minus;0.4184
| 0.3618
| 0.3618
| 8.29
| 8.29
Line 43: Line 35:
| 6144/6125, 10976/10935, 9882516/9765625
| 6144/6125, 10976/10935, 9882516/9765625
| {{mapping| 275 436 639 772 }}
| {{mapping| 275 436 639 772 }}
| -0.3051
| &minus;0.3051
| 0.3698
| 0.3698
| 8.48
| 8.48
Line 50: Line 42:
| 441/440, 4000/3993, 6144/6125, 10976/10935
| 441/440, 4000/3993, 6144/6125, 10976/10935
| {{mapping| 275 436 639 772 952 }} (275e)
| {{mapping| 275 436 639 772 952 }} (275e)
| -0.4096
| &minus;0.4096
| 0.3912
| 0.3912
| 8.97
| 8.97
Line 57: Line 49:
| 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)
| {{mapping| 275 436 639 772 952 1018 }} (275e)
| -0.4158
| &minus;0.4158
| 0.3574
| 0.3574
| 8.19
| 8.19
|}
{{comma basis end}}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
|-
| 1
| 1
Line 96: Line 82:
|-
|-
| 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]]
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}

Revision as of 04:00, 16 November 2024

← 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)

Subsets and supersets

Since 275 factors into 52 × 11, 275edo has subset edos 5, 11, 25 and 55.

Regular temperament properties

Template:Comma basis begin |- | 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 Template:Comma basis end

Rank-2 temperaments

Template:Rank-2 begin |- | 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 Template:Rank-2 end Template:Orf

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