111edo: Difference between revisions

{{EDO intro|111}}

111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. With [[harmonic]]s 3 through 19 all tuned sharp, 111edo is somewhat related to [[37edo]], with which it shares the mappings for 5, 7, 11, and 13.  

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]].  

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the 31 & 80 temperament, and [[buzzard]], the 53 & 58 temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list|Comma List]]

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

! colspan="2" | Tuning Error

| {{mapping| 111 176 }}

| {{mapping| 111 176 258 }}

| {{mapping| 111 176 258 312 }}

| {{mapping| 111 176 258 312 384 }}

| {{mapping| 111 176 258 312 384 411 }}

| {{mapping| 111 176 258 312 384 411 454 }}

| {{mapping| 111 176 258 312 384 411 454 472 }}

! Generator*

! Cents*

! Associated<br>Ratio*

* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – guanyin[22] in 111edo tuning