730edo: Difference between revisions

{{EDO intro|730}}

730edo is a very strong 5-limit system, but is also [[consistency|distinctly consistent]] up to the [[15-odd-limit]]. The equal temperament [[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.

{{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.

{{Harmonics in equal|730|columns=11}}

Since 730 factors into {{factorization|730}}, 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.  

! rowspan="2" | Optimal 8ve<br>stretch (¢)

| {{monzo| -1157 730 }}

| {{mapping| 730 1157 }}

| {{monzo| -53 10 16 }}, {{monzo| -16 35 -17 }}

| {{mapping| 730 1157 1695 }}

| {{mapping| 730 1157 1695 2049 }}

| {{mapping| 730 1157 1695 2049 2525 }}

| {{mapping| 730 1157 1695 2049 2525 2701 }}

|+Table of rank-2 temperaments by generator

! Associated<br>Ratio*

| [[Kwazy]]

| {{monzo| 24 4 -13 }}

<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

|+Woolhouse 730EDO diatonic scale

|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