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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 [[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.
{{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 ==
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.


{{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 ==
== 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:
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>
<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>
<pre>
       Major-tone, t = 124  
       Major-tone, t = 124  
Line 17: Line 24:
       Comma,      c =  13
       Comma,      c =  13
</pre>
</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.  
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>
</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" />.
Woolhouse defined the following diatonic/heptonic scale for 730edo<ref name="essay" />.


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{| class="wikitable"
{| 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 />


<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 -->
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->

Latest revision as of 13:38, 13 March 2026

← 729edo 730edo 731edo →
Prime factorization 2 × 5 × 73
Step size 1.64384 ¢ 
Fifth 427\730 (701.918 ¢)
Semitones (A1:m2) 69:55 (113.4 ¢ : 90.41 ¢)
Consistency limit 15
Distinct consistency limit 15

730 equal divisions of the octave (abbreviated 730edo or 730ed2), also called 730-tone equal temperament (730tet) or 730 equal temperament (730et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 730 equal parts of about 1.64 ¢ each. Each step represents a frequency ratio of 21/730, or the 730th root of 2.

Theory

730edo is a very strong 5-limit system, but is also distinctly consistent up to the 15-odd-limit. As an equal temperament, it tempers out the [-69 45 -1⟩ (counterschisma), [-16 35 -17⟩ (minortone comma), [-53 10 16⟩ (kwazy comma), [37 25 -33⟩ (whoosh comma), and [-90 -15 49⟩ (pirate comma). In the 7-limit it tempers out 4375/4374 and [-21 0 3 5⟩, so that it supports the mitonic temperament. In the 11-limit, 3025/3024 and [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 as a logarithmic measure of interval size[1], 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

Approximation of prime harmonics in 730edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 -0.037 -0.012 -0.607 -0.633 -0.528 +0.250 +0.021 -0.329 -0.536 +0.718
Relative (%) +0.0 -2.3 -0.8 -36.9 -38.5 -32.1 +15.2 +1.3 -20.0 -32.6 +43.7
Steps
(reduced) 		730
(0) 		1157
(427) 		1695
(235) 		2049
(589) 		2525
(335) 		2701
(511) 		2984
(64) 		3101
(181) 		3302
(382) 		3546
(626) 		3617
(697)

Subsets and supersets

Since 730 factors into 2 × 5 × 73, 730edo has subset 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[2], proposed:

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

      Major-tone, t = 124 
      Minor-tone, tˌ= 111
      Limma,      θ =  68
      Comma,      c =  13

…

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.

The concordant intervals are 

      Minor-third ...... 192
      Major-third ...... 235
      Fourth ........... 303
      Fifth ............ 427
      Minor-sixth ...... 495
      Major-sixth ...... 538
      Octave ........... 730

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-1157 730⟩ [⟨730 1157]] +0.0117 0.0117 0.71
2.3.5 [-53 10 16⟩, [-16 35 -17⟩ [⟨730 1157 1695]] +0.0096 0.0100 0.61
2.3.5.7 4375/4374, 2100875/2097152, [12 -3 -14 9⟩ [⟨730 1157 1695 2049]] +0.0612 0.0899 5.47
2.3.5.7.11 3025/3024, 4375/4374, 391314/390625, 2100875/2097152 [⟨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 [⟨730 1157 1695 2049 2525 2701]] +0.0951 0.0884 5.38

Rank-2 temperaments

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 [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

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Scales

Woolhouse diatonic scale

Woolhouse defined the following diatonic/heptonic scale for 730edo[2].

According to this division of the octave into 730 degrees, which we shall here-after adopt, the diatonic scale will be — 

Key ...   0
            ... 124 ... t ... Major-tone. 
2d  ... 124
            ... 111 ... tˌ... Minor-tone. 
3d  ... 235
            ...  68 ... θ ... Limma. 
4th ... 303
            ... 124 ... t ... Major-tone. 
5th ... 427 
            ... 111 ... tˌ... Minor-tone. 
6th ... 538
            ... 124 ... t ... Major-tone.
7th ... 662
            ...  68 ... θ ... Limma. 
8th ... 730

Woolhouse's diatonic scale in Ls notation is

  • LMsLMLs - L: 124, M: 111, s: 68

Inferred modes are shown in the following table.

Woolhouse 730EDO diatonic scale
Sequence Mode (suggested name) I1 I2 I3 I4 I5 I6 I7
LMsLMLs Woolhouse Ionian P1 M2 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
LMLsLMs Woolhouse Lydian P1 m25 m35 P4 d517 m65 m7
MLsLMsL Woolhouse Mixolydian P1 M25 m319 P4 — M65 m7
LsLMsLM Woolhouse Aeolian P1 M2 M35 A45 P5 d7175 M75
sLMsLML Woolhouse Locrian P1 M2 m35 P4197 P5 m65 m75

References

  1. ↑ A summary of W. S. B. Woolhouse's Essay on musical intervals, 1999 by Joseph Monzo
  2. ↑ 2.0 2.1 Essay on musical intervals, harmonics, and the temperament of the musical scale, &c, 1835 by Wesley Stoker Barker Woolhouse

External links

Retrieved from "https://en.xen.wiki/index.php?title=730edo&oldid=225983"