730edo

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← 729edo730edo731edo →
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. The equal temperament 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 Kwazy
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 it is 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