275edo: Difference between revisions

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== 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 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 ===

=== Subsets and supersets ===

Since 275 factors into ~~52 × 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 ==

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list~~|Comma List~~]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve ~~Stretch~~ (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

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| 2.3

| {{monzo| 436 -275 }}

| [{{~~val~~| 275 436 }}]

| {{mapping| 275 436 }}

| -0.1863

| −0.1863

| 0.1862

| 4.27

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| 2.3.5

| {{monzo| 24 -21 4 }}, {{monzo| 19 10 -15 }}

| [{{~~val~~| 275 436 639 }}]

| {{mapping| 275 436 639 }}

| -0.4184

| −0.4184

| 0.3618

| 8.29

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| 2.3.5.7

| 6144/6125, 10976/10935, 9882516/9765625

| [{{~~val~~| 275 436 639 772 }}]

| {{mapping| 275 436 639 772 }}

| -0.3051

| −0.3051

| 0.3698

| 8.48

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| 2.3.5.7.11

| 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

| 8.97

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| 2.3.5.7.11.13

| 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

| 8.19

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=== Rank-2 temperaments ===

{| 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 per 8ve

|-

! Generator~~ (Reduced)~~

! Periods per 8ve

! Cents~~ (Reduced)~~

! Generator*

! Associated ~~Ratio~~

! Cents*

! Associated ratio*

! Temperaments

|-

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|-

| 11

| 114\275 (11\275)

| 114\275 (11\275)

| 497.45 (48.00)

| 497.45 (48.00)

| 4/3 (36/35)

| 4/3 (36/35)

| [[Hendecatonic]]

|}

<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|275}}
+{{ED intro}}
 == 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 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 ===
-{{Harmonics in equal|275|intervals=prime|columns=11}}
+{{Harmonics in equal|275|intervals=prime}}
 === 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 ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! 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]] (¢)
@@ Line 28: / Line 29: @@
 | 2.3
 | {{monzo| 436 -275 }}
-| [{{val| 275 436 }}]
+| {{mapping| 275 436 }}
-| -0.1863
+| −0.1863
 | 0.1862
 | 4.27
@@ Line 35: / Line 36: @@
 | 2.3.5
 | {{monzo| 24 -21 4 }}, {{monzo| 19 10 -15 }}
-| [{{val| 275 436 639 }}]
+| {{mapping| 275 436 639 }}
-| -0.4184
+| −0.4184
 | 0.3618
 | 8.29
@@ Line 42: / Line 43: @@
 | 2.3.5.7
 | 6144/6125, 10976/10935, 9882516/9765625
-| [{{val| 275 436 639 772 }}]
+| {{mapping| 275 436 639 772 }}
-| -0.3051
+| −0.3051
 | 0.3698
 | 8.48
@@ Line 49: / Line 50: @@
 | 2.3.5.7.11
 | 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
 | 8.97
@@ Line 56: / Line 57: @@
 | 2.3.5.7.11.13
 | 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
 | 8.19
@@ Line 64: / Line 65: @@
 === Rank-2 temperaments ===
 {| 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
 |-
@@ Line 96: / Line 98: @@
 |-
 | 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]]
 |}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct