730edo
← 729edo | 730edo | 731edo → |
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 2^{1/730}, 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
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
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.
Sequence | Mode (suggested name) | I1 | I2 | I3 | I4 | I5 | I6 | I7 |
---|---|---|---|---|---|---|---|---|
LMsLMLs | Woolhouse Ionian | P1 | M2 | M3^{5} | P4 | P5 | M6^{5} | M7^{5} |
MsLMLsL | Woolhouse Dorian | P1 | m2_{5} | m3_{5} | P4 | P5 | m6_{5} | m7_{5} |
sLMLsLM | Woolhouse Phrygian | P1 | M2^{5} | M3^{5} | P4 | P5 | M6^{5} | m7 |
LMLsLMs | Woolhouse Lydian | P1 | m2_{5} | m3_{5} | P4 | d5^{17} | m6_{5} | m7 |
MLsLMsL | Woolhouse Mixolydian | P1 | M2^{5} | m3^{19} | P4 | - | M6^{5} | m7 |
LsLMsLM | Woolhouse Aeolian | P1 | M2 | M3^{5} | A4^{5} | P5 | d7^{17}_{5} | M7^{5} |
sLMsLML | Woolhouse Locrian | P1 | M2 | m3_{5} | P4^{19}_{7} | P5 | m6_{5} | m7_{5} |
References
- ↑ A summary of W. S. B. Woolhouse's Essay on musical intervals, 1999 by Joseph Monzo
- ↑ ^{2.0} ^{2.1} Essay on musical intervals, harmonics, and the temperament of the musical scale, &c, 1835 by Wesley Stoker Barker Woolhouse