730edo: Difference between revisions

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== Theory ==

730edo is a very strong 5-limit system, but is also distinctly consistent up to the [[15-odd-limit]]. It tempers out ~~the [[counterschisma~~]], {{monzo| -69 45 -1 }}~~, the minortone comma~~, {{monzo| -16 35 -17 }}~~, the kwazy~~ comma, {{monzo| -53 10 16 }}~~, the whoosh~~ comma, {{monzo| 37 25 -33 }}, and ~~the pirate comma,~~ {{monzo| -90 -15 49 }}. In the 7-limit it tempers out [[4375/4374]] and {{monzo| -21 0 3 5 }}, so that it [[support]]s the [[mitonic]] temperament. In the 11-limit, [[3025/3024]] and {{monzo| 4 -3 -6 4 1 }}, so that it supports the [[deca]] temperament. In the 13-limit, [[1001/1000]] and [[4225/4224]], supporting 13-limit deca.

730edo is a very strong 5-limit system, but is also [[consistency|distinctly consistent]] up to the [[15-odd-limit]]. As an equal temperament, it [[tempering out|tempers out]] the {{monzo| -69 45 -1 }} ([[counterschisma]]), {{monzo| -16 35 -17 }} (minortone comma), {{monzo| -53 10 16 }} ([[kwazy comma]]), {{monzo| 37 25 -33 }} (whoosh comma), and {{monzo| -90 -15 49 }} (pirate comma). In the 7-limit it tempers out [[4375/4374]] and {{monzo| -21 0 3 5 }}, so that it [[support]]s the [[mitonic]] temperament. In the 11-limit, [[3025/3024]] and {{monzo| 4 -3 -6 4 1 }}, so that it supports the [[deca]] temperament. In the 13-limit, [[1001/1000]] and [[4225/4224]], supporting 13-limit deca.

[[W. S. B. Woolhouse]] ~~proposed 730edo~~<ref name="summary">[https://www.webcitation.org/5zxZzQ3eS A summary of W. S. B. Woolhouse's Essay on musical intervals], 1999 by [[~~Joe~~ Monzo]]</ref> ~~as a [[Interval size measure|logarithmic measure of interval size]]~~, sometimes called the '''Woolhouse unit'''. While 730 is divisible by 2, 5, 10, 73, 146 and 365, it is not divisible by 12 and it is also deficient, with [[abundancy index]] of 0.82, which limits its application as an interval size measure.

{{W|W. S. B. Woolhouse}} proposed 730edo as a [[interval size measure|logarithmic measure of interval size]]<ref name="summary">[https://www.webcitation.org/5zxZzQ3eS A summary of W. S. B. Woolhouse's Essay on musical intervals], 1999 by [[Joseph Monzo]]</ref>, sometimes called the '''Woolhouse unit'''. While 730 is divisible by 2, 5, 10, 73, 146, and 365, it is not divisible by 12 and it is also deficient, with [[abundancy index]] of 0.82, which limits its application as an interval size measure.

=== Prime harmonics ===

=== Subsets and supersets ===

Since 730 factors into 2 × 5 × 73, 730edo has subset edos {{EDOs| 2, 5, 10, 73, 146, and 365 }}. 1460edo, which doubles it, gives alternative approximations to harmonics 7, 11, and 13. [[2190edo]], which triples it, corrects these harmonics to near-just levels of accuracy. [[4380edo]] gives a possible full 31-limit system.

== Intervals ==

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== Regular temperament properties ==

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

|-

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

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

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

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

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

! colspan="2" | Tuning error

|-

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|-

| 2.3

| {{~~monzo~~| -1157 730 }}

| {{Monzo| -1157 730 }}

| [{{~~val~~| 730 1157 }}]

| {{Mapping| 730 1157 }}

| +0.0117

| 0.0117

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|-

| 2.3.5

| {{~~monzo~~| -53 10 16 }}, {{monzo| -16 35 -17 }}

| {{Monzo| -53 10 16 }}, {{monzo| -16 35 -17 }}

| [{{~~val~~| 730 1157 1695 }}]

| {{Mapping| 730 1157 1695 }}

| +0.0096

| 0.0100

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| 2.3.5.7

| 4375/4374, 2100875/2097152, {{monzo| 12 -3 -14 9 }}

| [{{~~val~~| 730 1157 1695 2049 }}]

| {{Mapping| 730 1157 1695 2049 }}

| +0.0612

| 0.0899

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| 2.3.5.7.11

| 3025/3024, 4375/4374, 391314/390625, 2100875/2097152

| [{{~~val~~| 730 1157 1695 2049 2525 }}]

| {{Mapping| 730 1157 1695 2049 2525 }}

| +0.0856

| 0.0940

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| 2.3.5.7.11.13

| 1001/1000, 3025/3024, 4225/4224, 4375/4374, 2100875/2097152

| [{{~~val~~| 730 1157 1695 2049 2525 2701 }}]

| {{Mapping| 730 1157 1695 2049 2525 2701 }}

| +0.0951

| 0.0884

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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)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated ~~Ratio~~

! Associated ratio*

! Temperaments

|-

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| 162.74

| 1125/1024

| [[~~Kwazy~~]]

| [[Crazy]]

|-

| 1

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| 113\730

| 185.75

| {{~~monzo~~| 24 4 -13 }}

| {{Monzo| 24 4 -13 }}

| [[Pirate]]

|-

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| [[Deca]]

|}

<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

== Scales ==

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{| class="wikitable"

|+Woolhouse 730EDO diatonic scale

|+ style="font-size: 105%;" | Woolhouse 730EDO diatonic scale

|-

!Sequence!!Mode (suggested name)!!I1!!I2!!I3!!I4!!I5!!I6!!I7

! Sequence !! Mode (suggested name) !! I1 !! I2 !! I3 !! I4 !! I5 !! I6 !! I7

|-

|LMsLMLs||Woolhouse Ionian||P1||[[9/8|M2]]||[[5/4|M35]]||[[4/3|P4]]||[[3/2|P5]]||[[5/3|M65]]||[[15/8|M75]]

| LMsLMLs || Woolhouse Ionian || P1 || [[9/8|M2]] || [[5/4|M35]] || [[4/3|P4]] || [[3/2|P5]] || [[5/3|M65]] || [[15/8|M75]]

|-

|MsLMLsL||Woolhouse Dorian||P1||m25||m35||P4||P5||m65||m75

| MsLMLsL || Woolhouse Dorian || P1 || m25 || m35 || P4 || P5 || m65 || m75

|-

|sLMLsLM||Woolhouse Phrygian||P1||M25||M35||P4||P5||M65||m7

| sLMLsLM || Woolhouse Phrygian || P1 || M25 || M35 || P4 || P5 || M65 || m7

|-

|LMLsLMs||Woolhouse Lydian||P1||m25||m35||P4||d517||m65||m7

| LMLsLMs || Woolhouse Lydian || P1 || m25 || m35 || P4 || d517 || m65 || m7

|-

|MLsLMsL||Woolhouse Mixolydian||P1||M25||m319||P4||-||M65||m7

| MLsLMsL || Woolhouse Mixolydian || P1 || M25 || m319 || P4 || — || M65 || m7

|-

|LsLMsLM||Woolhouse Aeolian||P1||M2||M35||A45||P5||d7175||M75

| LsLMsLM || Woolhouse Aeolian || P1 || M2 || M35 || A45 || P5 || d7175 || M75

|-

~~|sLMsLML||Woolhouse Locrian||P1||M2||m35||P4197||P5||m65||m75</sub~~>

|-

| sLMsLML || Woolhouse Locrian || P1 || M2 || m35 || P4197 ||P5 || m65 || m75

|}

== References ==

== External links ==

* [http://tonalsoft.com/enc/w/woolhouse-unit.aspx woolhouse-unit] on [[Tonalsoft Encyclopedia]]

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

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|730}}
+{{ED intro}}
 == Theory ==
-edo is a very strong 5-limit system, but is also distinctly consistent up to the [[15-odd-limit]]. It tempers out the [[counterschisma]], {{monzo| -69 45 -1 }}, the minortone comma, {{monzo| -16 35 -17 }}, the kwazy comma, {{monzo| -53 10 16 }}, the whoosh comma, {{monzo| 37 25 -33 }}, and the pirate comma, {{monzo| -90 -15 49 }}. In the 7-limit it tempers out [[4375/4374]] and {{monzo| -21 0 3 5 }}, so that it [[support]]s the [[mitonic]] temperament. In the 11-limit, [[3025/3024]] and {{monzo| 4 -3 -6 4 1 }}, so that it supports the [[deca]] temperament. In the 13-limit, [[1001/1000]] and [[4225/4224]], supporting 13-limit deca.
+edo is a very strong 5-limit system, but is also [[consistency|distinctly consistent]] up to the [[15-odd-limit]]. As an equal temperament, it [[tempering out|tempers out]] the {{monzo| -69 45 -1 }} ([[counterschisma]]), {{monzo| -16 35 -17 }} (minortone comma), {{monzo| -53 10 16 }} ([[kwazy comma]]), {{monzo| 37 25 -33 }} (whoosh comma), and {{monzo| -90 -15 49 }} (pirate comma). In the 7-limit it tempers out [[4375/4374]] and {{monzo| -21 0 3 5 }}, so that it [[support]]s the [[mitonic]] temperament. In the 11-limit, [[3025/3024]] and {{monzo| 4 -3 -6 4 1 }}, so that it supports the [[deca]] temperament. In the 13-limit, [[1001/1000]] and [[4225/4224]], supporting 13-limit deca.
-[[W. S. B. Woolhouse]] proposed 730edo<ref name="summary">[https://www.webcitation.org/5zxZzQ3eS A summary of W. S. B. Woolhouse's Essay on musical intervals], 1999 by [[Joe Monzo]]</ref> as a [[Interval size measure|logarithmic measure of interval size]], sometimes called the '''Woolhouse unit'''. While 730 is divisible by 2, 5, 10, 73, 146 and 365, it is not divisible by 12 and it is also deficient, with [[abundancy index]] of 0.82, which limits its application as an interval size measure.
+{{W|W. S. B. Woolhouse}} proposed 730edo as a [[interval size measure|logarithmic measure of interval size]]<ref name="summary">[https://www.webcitation.org/5zxZzQ3eS A summary of W. S. B. Woolhouse's Essay on musical intervals], 1999 by [[Joseph Monzo]]</ref>, sometimes called the '''Woolhouse unit'''. While 730 is divisible by 2, 5, 10, 73, 146, and 365, it is not divisible by 12 and it is also deficient, with [[abundancy index]] of 0.82, which limits its application as an interval size measure.
 === Prime harmonics ===
-{{Harmonics in equal|730|columns=11}}
+{{Harmonics in equal|730}}
+=== Subsets and supersets ===
+Since 730 factors into 2 × 5 × 73, 730edo has subset edos {{EDOs| 2, 5, 10, 73, 146, and 365 }}. 1460edo, which doubles it, gives alternative approximations to harmonics 7, 11, and 13. [[2190edo]], which triples it, corrects these harmonics to near-just levels of accuracy. [[4380edo]] gives a possible full 31-limit system.
 == Intervals ==
@@ Line 39: / Line 42: @@
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
 ! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal 8ve<br>stretch (¢)
+! rowspan="2" | Optimal<br>8ve stretch (¢)
 ! colspan="2" | Tuning error
 |-
@@ Line 49: / Line 53: @@
 |-
 | 2.3
-| {{monzo| -1157 730 }}
+| {{Monzo| -1157 730 }}
-| [{{val| 730 1157 }}]
+| {{Mapping| 730 1157 }}
 | +0.0117
 | 0.0117
@@ Line 56: / Line 60: @@
 |-
 | 2.3.5
-| {{monzo| -53 10 16 }}, {{monzo| -16 35 -17 }}
+| {{Monzo| -53 10 16 }}, {{monzo| -16 35 -17 }}
-| [{{val| 730 1157 1695 }}]
+| {{Mapping| 730 1157 1695 }}
 | +0.0096
 | 0.0100
@@ Line 64: / Line 68: @@
 | 2.3.5.7
 | 4375/4374, 2100875/2097152, {{monzo| 12 -3 -14 9 }}
-| [{{val| 730 1157 1695 2049 }}]
+| {{Mapping| 730 1157 1695 2049 }}
 | +0.0612
 | 0.0899
@@ Line 71: / Line 75: @@
 | 2.3.5.7.11
 | 3025/3024, 4375/4374, 391314/390625, 2100875/2097152
-| [{{val| 730 1157 1695 2049 2525 }}]
+| {{Mapping| 730 1157 1695 2049 2525 }}
 | +0.0856
 | 0.0940
@@ Line 78: / Line 82: @@
 | 2.3.5.7.11.13
 | 1001/1000, 3025/3024, 4225/4224, 4375/4374, 2100875/2097152
-| [{{val| 730 1157 1695 2049 2525 2701 }}]
+| {{Mapping| 730 1157 1695 2049 2525 2701 }}
 | +0.0951
 | 0.0884
@@ Line 86: / Line 90: @@
 === 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)
+! Generator*
-! Cents<br>(Reduced)
+! Cents*
-! Associated<br>Ratio
+! Associated<br>ratio*
 ! Temperaments
 |-
@@ Line 97: / Line 102: @@
 | 162.74
 | 1125/1024
-| [[Kwazy]]
+| [[Crazy]]
 |-
 | 1
@@ Line 108: / Line 113: @@
 | 113\730
 | 185.75
-| {{monzo| 24 4 -13 }}
+| {{Monzo| 24 4 -13 }}
 | [[Pirate]]
 |-
@@ Line 135: / Line 140: @@
 | [[Deca]]
 |}
+<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
 == Scales ==
@@ Line 167: / Line 173: @@
 {| class="wikitable"
-|+Woolhouse 730EDO diatonic scale
+|+ style="font-size: 105%;" | Woolhouse 730EDO diatonic scale
 |-
-!Sequence!!Mode (suggested name)!!I1!!I2!!I3!!I4!!I5!!I6!!I7
+! Sequence !! Mode (suggested name) !! I1 !! I2 !! I3 !! I4 !! I5 !! I6 !! I7
 |-
-|LMsLMLs||Woolhouse Ionian||P1||[[9/8|M2]]||[[5/4|M3<sup>5</sup>]]||[[4/3|P4]]||[[3/2|P5]]||[[5/3|M6<sup>5</sup>]]||[[15/8|M7<sup>5</sup>]]
+| LMsLMLs || Woolhouse Ionian || P1 || [[9/8|M2]] || [[5/4|M3<sup>5</sup>]] || [[4/3|P4]] || [[3/2|P5]] || [[5/3|M6<sup>5</sup>]] || [[15/8|M7<sup>5</sup>]]
 |-
-|MsLMLsL||Woolhouse Dorian||P1||m2<sub>5</sub>||m3<sub>5</sub>||P4||P5||m6<sub>5</sub>||m7<sub>5</sub>
+| MsLMLsL || Woolhouse Dorian || P1 || m2<sub>5</sub> || m3<sub>5</sub> || P4 || P5 || m6<sub>5</sub> || m7<sub>5</sub>
 |-
-|sLMLsLM||Woolhouse Phrygian||P1||M2<sup>5</sup>||M3<sup>5</sup>||P4||P5||M6<sup>5</sup>||m7
+| sLMLsLM || Woolhouse Phrygian || P1 || M2<sup>5</sup> || M3<sup>5</sup> || P4 || P5 || M6<sup>5</sup> || m7
 |-
-|LMLsLMs||Woolhouse Lydian||P1||m2<sub>5</sub>||m3<sub>5</sub>||P4||d5<sup>17</sup>||m6<sub>5</sub>||m7
+| LMLsLMs || Woolhouse Lydian || P1 || m2<sub>5</sub> || m3<sub>5</sub> || P4 || d5<sup>17</sup> || m6<sub>5</sub> || m7
 |-
-|MLsLMsL||Woolhouse Mixolydian||P1||M2<sup>5</sup>||m3<sup>19</sup>||P4||-||M6<sup>5</sup>||m7
+| MLsLMsL || Woolhouse Mixolydian || P1 || M2<sup>5</sup> || m3<sup>19</sup> || P4 || &mdash; || M6<sup>5</sup> || m7
 |-
-|LsLMsLM||Woolhouse Aeolian||P1||M2||M3<sup>5</sup>||A4<sup>5</sup>||P5||d7<sup>17</sup><sub>5</sub>||M7<sup>5</sup>
+| LsLMsLM || Woolhouse Aeolian || P1 || M2 || M3<sup>5</sup> || A4<sup>5</sup> || P5 || d7<sup>17</sup><sub>5</sub> || M7<sup>5</sup>
-|-
-|sLMsLML||Woolhouse Locrian||P1||M2||m3<sub>5</sub>||P4<sup>19</sup><sub>7</sub>||P5||m6<sub>5</sub>||m7<sub>5</sub>
 |-
+| sLMsLML || Woolhouse Locrian || P1 || M2 || m3<sub>5</sub> || P4<sup>19</sup><sub>7</sub> ||P5 || m6<sub>5</sub> || m7<sub>5</sub>
 |}
 == References ==
 <references />
+== External links ==
+* [http://tonalsoft.com/enc/w/woolhouse-unit.aspx woolhouse-unit] on [[Tonalsoft Encyclopedia]]
 [[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->