730edo

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730edo

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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

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 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 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 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, 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.

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