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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
| Prime factorization = 2 × 5 × 73
| Step size = 1.64384¢
| Fifth = 427\730 (701.92¢)
| Semitones = 69:55 (113.42¢ : 90.41¢)
| Consistency = 15
}}
{{EDO intro|730}}


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 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.


{{Primes in edo|730}}
[[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.
 
=== Prime harmonics ===
{{Harmonics in equal|730|columns=11}}


== 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 27:
       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 43:
</blockquote>
</blockquote>


== Woolhouse diatonic scale ==
== 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" />.


Line 85: Line 95:


== References ==
== References ==
<references />
<references />


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

Revision as of 15:48, 16 August 2022

← 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

Template:EDO intro

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, [-69 45 -1⟩, the minortone comma, [-16 35 -17⟩, the kwazy comma, [-53 10 16⟩, the whoosh comma, [37 25 -33⟩, and the pirate comma, [-90 -15 49⟩. 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[1] as a 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.

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)

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

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 Joe 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
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