166edo: Difference between revisions

166edo is [[consistent]] through the [[13-odd-limit]], yet its principle interest lies in the usefulness of its approximations. In addition to the 5-limit [[amity comma]], it [[tempering out|tempers out]] [[225/224]], [[325/324]], [[385/384]], [[540/539]], and [[729/728]], hence being an excellent tuning for the [[rank-3 temperament]] [[marvel]], in both the [[11-limit]] and in the 13-limit extension [[hecate]], the [[rank-2 temperament]] [[wizard]], which also tempers out [[4000/3993]], and [[houborizic]], which also tempers out [[2200/2197]], giving the [[optimal patent val]] for all of these. In the [[13-limit]] it tempers out 325/324, leading to hecate, and [[1573/1568]], leading to marvell, and tempering out both gives [[gizzard]], the {{nowrap|72 & 94}} temperament, for which 166 is an excellent tuning through the [[19-limit]].  

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

| {{monzo| -263 166 }}

| {{mapping| 166 263 }}

| {{mapping| 166 263 385 }}

| {{mapping| 166 263 385 466 }}

| {{mapping| 166 263 385 466 574 }}

| {{mapping| 166 263 385 466 574 614 }}

! Periods<br />per 8ve

! Associated<br />ratio*

| [[Merman]] (7-limit)

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