612edo: Difference between revisions

612edo is a very strong [[5-limit]] system, a fact noted by {{w|Isaac Newton}}<ref>[https://emusicology.org/index.php/EMR/article/view/7647/6030 Muzzulini, Daniel. 2021. "Isaac Newton's Microtonal Approach to Just Intonation". ''Empirical Musicology Review'' 15 (3-4):223-48. https://doi.org/10.18061/emr.v15i3-4.7647.]</ref>, {{w|Robert Holford Macdowall Bosanquet|R. H. M. Bosanquet}}{{citation needed}} and {{w|James Murray Barbour}}{{citation needed}}. It [[tempers out]] the {{monzo| 485 -306 }} ([[sasktel comma]]) in the 3-limit, and in the 5-limit {{monzo| 1 -27 18 }} ([[ennealimma]]), {{monzo| -52 -17 34 }} ([[septendecima]]), {{monzo| -53 10 16 }} ([[kwazy comma]]), {{monzo| 54 -37 2 }} ([[monzisma]]), {{monzo| -107 47 14 }} (fortune comma), and {{monzo| 161 -84 -12 }} ([[atom]]). In the 7-limit it tempers out [[2401/2400]] and [[4375/4374]], so that it [[support]]s the [[ennealimmal]] temperament, and in fact provides the [[optimal patent val]] for ennealimmal. The 7-limit val for 612 can be characterized as the ennealimmal commas plus the kwazy comma. In the 11-limit, it tempers out [[3025/3024]] and [[9801/9800]], so that 612 supports the [[hemiennealimmal]] temperament. In the 13-limit, it tempers [[2200/2197]] and [[4096/4095]].

The 612edo has been proposed as the logarithmic [[interval size measure]] '''skisma''' (or '''sk'''), since one step is nearly the same size as the [[schisma]] (32805/32768), 1/12 of a [[Pythagorean comma]] or 1/11 of a [[syntonic comma]]. Since 612 is divisible by {{EDOs| 2, 3, 4, 6, 9, 12, 17, 18, 34, 36, 51, 68, 102, 153, 204 and 306 }}, it can readily express the step sizes of the 12, 17, 34, and 68 divisions. A table of intervals approximated by 612 can be found under [[Table of 612edo intervals]].

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

| {{mapping| 612 970 1421 }}

| {{mapping| 612 970 1421 1718 }}

| {{mapping| 612 970 1421 1718 2117 }}

| {{mapping| 612 970 1421 1718 2117 2265 }}

| {{mapping| 612 970 1421 1718 2117 2265 2600 }}

! Periods<br />per 8ve

! Associated<br />ratio*

| [[Kwazy]]

| 194\612<br />(41\612)

| 380.39<br />(80.39)

| 81/65<br />(22/21)

| 133\612<br />(25\612)

| 315.69<br />(49.02)

| 6/5<br />(36/35)

| 124\612<br />(22\612)

| 243.137<br />(43.14)

| 3145728/2734375<br />(?)

| 254\612<br />(1\612)

| 498.04<br />(1.96)

| 4/3<br />(32805/32768)

| 127\612<br />(17\612)

| 249.02<br />(33.33)

| {{monzo| -23 5 9 -2 }}<br />(100352/98415)

| 127\612<br />(9\612)

| 249.02<br />(17.65)

| 231/200<br />(99/98)

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