730edo: Difference between revisions

(20 intermediate revisions by 8 users not shown)

Line 1:

The '''730 equal divisions of the octave''' ('''730edo'''), or the '''730(-tone) equal temperament''' ('''730tet''', '''730et''') divides the [[octave]] into 730 [[equal]] parts of 1.644 [[cent]]s each. It is a very strong five-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 supports 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">~~[~~http://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~~, ~~which can be regarded as either a good thing or a bad one~~.

== Theory ==

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|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 ==

W. S. B. Woolhouse, in his 1835 essay<ref name="essay">[https://archive.org/details/essayonmusicali00woolgoog/page/n34/mode/2up Essay on musical intervals, harmonics, and the temperament of the musical scale, &c], 1835 by Wesley Stoker Barker Woolhouse</ref>, proposed:

~~...~~ dividing the octave into 730 equal intervals, which we shall call ''degrees'', the elemental intervals will be:

… dividing the octave into 730 equal intervals, which we shall call ''degrees'', the elemental intervals will be:

<pre>

Major-tone, t = 124

Line 17:

Line 24:

Comma, c = 13

</pre>

~~...~~

…

These numbers present a more accurate measurement of the musical scale than any other, unless we go to very high numbers. The greatest error which can arise from their natural or melodious combinations is that of the fifth, and does not amount to one half of the error of the major-tone above mentioned.

Line 33:

Line 40:

</blockquote>

== ~~Woolhouse diatonic scale~~ ==

== 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 (¢)

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

! [[TE simple badness|Relative]] (%)

|-

| 2.3

| {{Monzo| -1157 730 }}

| {{Mapping| 730 1157 }}

| +0.0117

| 0.0117

| 0.71

|-

| 2.3.5

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

| {{Mapping| 730 1157 1695 }}

| +0.0096

| 0.0100

| 0.61

|-

| 2.3.5.7

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

| {{Mapping| 730 1157 1695 2049 }}

| +0.0612

| 0.0899

| 5.47

|-

| 2.3.5.7.11

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

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

| +0.0856

| 0.0940

| 5.72

|-

| 2.3.5.7.11.13

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

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

| +0.0951

| 0.0884

| 5.38

|}

=== Rank-2 temperaments ===

{| class="wikitable center-all left-5"

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ratio*

! Temperaments

|-

| 1

| 99\730

| 162.74

| 1125/1024

| [[Crazy]]

|-

| 1

| 111\730

| 182.47

| 10/9

| [[Mitonic]]

|-

| 1

| 113\730

| 185.75

| {{Monzo| 24 4 -13 }}

| [[Pirate]]

|-

| 1

| 303\730

| 498.08

| 4/3

| [[Counterschismic]]

|-

| 1

| 341\730

| 560.55

| 864/625

| [[Whoosh]]

|-

| 2

| 111\730

| 182.47

| 10/9

| [[Seminar]]

|-

| 10

| 192\730 (27\730)

| 315.62 (44.38)

| 6/5 (40/39)

| [[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 ==

=== Woolhouse diatonic scale ===

Woolhouse defined the following diatonic/heptonic scale for 730edo<ref name="essay" />.

Line 64:

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~~

|-

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

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

|-

|~~MsLMLsL~~||Woolhouse ~~Dorian~~||P1||~~m2~~5</~~sub>|~~|m3<~~sub~~>5</~~sub~~>||P4||P5||m6<~~sub~~>5</~~sub~~>||m7<~~sub~~>5</~~sub~~>

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

|-

|~~sLMLsLM~~||Woolhouse ~~Phrygian~~||P1||M2<~~sup~~>5</~~sup~~>||M3<~~sup~~>5</~~sup~~>||P4||P5||M6<~~sup~~>5</~~sup~~>||m7

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

|-

|~~LMLsLMs~~||Woolhouse ~~Lydian~~||P1||m2<~~sub~~>5</~~sub~~>||m3<~~sub~~>5</~~sub~~>||P4||~~d517~~||m6<~~sub~~>5</~~sub~~>||m7

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

|-

|~~MLsLMsL~~||Woolhouse ~~Mixolydian~~||P1||M2<~~sup~~>5</~~sup~~>||m3<~~sup~~>19</~~sup~~>||P4||-||M6<~~sup~~>5</~~sup~~>||m7

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

|-

|~~LsLMsLM~~||Woolhouse ~~Aeolian~~||P1||M2~~||M3~~5||A45||P5||~~d7175~~||M75

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

|-

|~~sLMsLML~~||Woolhouse ~~Locrian~~||P1||M2||m3<~~sub~~>5</~~sub~~>||P419<~~sub~~>7</~~sub~~>~~||P5||m6~~5||m7<~~sub~~>5</~~sub~~>

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

|-

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

|}

== References ==

~~<references~~ />

== External links ==

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

[[Category:Equal divisions of the octave]]

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

@@ Line 1: / Line 1: @@
-The '''730 equal divisions of the octave''' ('''730edo'''), or the '''730(-tone) equal temperament''' ('''730tet''', '''730et''') divides the [[octave]] into 730 [[equal]] parts of 1.644 [[cent]]s each. It is a very strong five-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 supports 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.
+{{Infobox ET}}
+{{ED intro}}
-W. S. B. Woolhouse proposed 730edo<ref name="summary">[http://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, which can be regarded as either a good thing or a bad one.
+== Theory ==
+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.
-{{Primes in edo|730}}
+{{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}}
+=== 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 ==
 W. S. B. Woolhouse, in his 1835 essay<ref name="essay">[https://archive.org/details/essayonmusicali00woolgoog/page/n34/mode/2up Essay on musical intervals, harmonics, and the temperament of the musical scale, &c], 1835 by Wesley Stoker Barker Woolhouse</ref>, proposed:
 <blockquote>
-... dividing the octave into 730 equal intervals, which we shall call ''degrees'', the elemental intervals will be:
+… dividing the octave into 730 equal intervals, which we shall call ''degrees'', the elemental intervals will be:
 <pre>
        Major-tone, t = 124
@@ Line 17: / Line 24: @@
        Comma,      c =  13
 </pre>
-...
+…
 These numbers present a more accurate measurement of the musical scale than any other, unless we go to very high numbers. The greatest error which can arise from their natural or melodious combinations is that of the fifth, and does not amount to one half of the error of the major-tone above mentioned.
@@ Line 33: / Line 40: @@
 </blockquote>
-== Woolhouse diatonic scale ==
+== 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| -1157 730 }}
+| {{Mapping| 730 1157 }}
+| +0.0117
+| 0.0117
+| 0.71
+|-
+| 2.3.5
+| {{Monzo| -53 10 16 }}, {{monzo| -16 35 -17 }}
+| {{Mapping| 730 1157 1695 }}
+| +0.0096
+| 0.0100
+| 0.61
+|-
+| 2.3.5.7
+| 4375/4374, 2100875/2097152, {{monzo| 12 -3 -14 9 }}
+| {{Mapping| 730 1157 1695 2049 }}
+| +0.0612
+| 0.0899
+| 5.47
+|-
+| 2.3.5.7.11
+| 3025/3024, 4375/4374, 391314/390625, 2100875/2097152
+| {{Mapping| 730 1157 1695 2049 2525 }}
+| +0.0856
+| 0.0940
+| 5.72
+|-
+| 2.3.5.7.11.13
+| 1001/1000, 3025/3024, 4225/4224, 4375/4374, 2100875/2097152
+| {{Mapping| 730 1157 1695 2049 2525 2701 }}
+| +0.0951
+| 0.0884
+| 5.38
+|}
+=== Rank-2 temperaments ===
+{| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br>per 8ve
+! Generator*
+! Cents*
+! Associated<br>ratio*
+! Temperaments
+|-
+| 1
+| 99\730
+| 162.74
+| 1125/1024
+| [[Crazy]]
+|-
+| 1
+| 111\730
+| 182.47
+| 10/9
+| [[Mitonic]]
+|-
+| 1
+| 113\730
+| 185.75
+| {{Monzo| 24 4 -13 }}
+| [[Pirate]]
+|-
+| 1
+| 303\730
+| 498.08
+| 4/3
+| [[Counterschismic]]
+|-
+| 1
+| 341\730
+| 560.55
+| 864/625
+| [[Whoosh]]
+|-
+| 2
+| 111\730
+| 182.47
+| 10/9
+| [[Seminar]]
+|-
+| 10
+| 192\730<br>(27\730)
+| 315.62<br>(44.38)
+| 6/5<br>(40/39)
+| [[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 ==
+=== Woolhouse diatonic scale ===
 Woolhouse defined the following diatonic/heptonic scale for 730edo<ref name="essay" />.
@@ Line 64: / 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
 |-
-|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>]]
+! Sequence !! Mode (suggested name) !! I1 !! I2 !! I3 !! I4 !! I5 !! I6 !! I7
 |-
-|MsLMLsL||Woolhouse Dorian||P1||m2<sub>5</sub>||m3<sub>5</sub>||P4||P5||m6<sub>5</sub>||m7<sub>5</sub>
+| 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>]]
 |-
-|sLMLsLM||Woolhouse Phrygian||P1||M2<sup>5</sup>||M3<sup>5</sup>||P4||P5||M6<sup>5</sup>||m7
+| MsLMLsL || Woolhouse Dorian || P1 || m2<sub>5</sub> || m3<sub>5</sub> || P4 || P5 || m6<sub>5</sub> || m7<sub>5</sub>
 |-
-|LMLsLMs||Woolhouse Lydian||P1||m2<sub>5</sub>||m3<sub>5</sub>||P4||d5<sup>17</sup>||m6<sub>5</sub>||m7
+| sLMLsLM || Woolhouse Phrygian || P1 || M2<sup>5</sup> || M3<sup>5</sup> || P4 || P5 || M6<sup>5</sup> || m7
 |-
-|MLsLMsL||Woolhouse Mixolydian||P1||M2<sup>5</sup>||m3<sup>19</sup>||P4||-||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
 |-
-|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>
+| MLsLMsL || Woolhouse Mixolydian || P1 || M2<sup>5</sup> || m3<sup>19</sup> || P4 || &mdash; || M6<sup>5</sup> || m7
 |-
-|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>
+| 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>
 |}
 == References ==
+<references />
-<references />
+== External links ==
+* [http://tonalsoft.com/enc/w/woolhouse-unit.aspx woolhouse-unit] on [[Tonalsoft Encyclopedia]]
-[[Category:Equal divisions of the octave]]
+[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->