Augmented–cloudy equivalence continuum: Difference between revisions

The '''augmented-cloudy equivalence continuum''' is a continuum of 2.5.7 subgroup temperaments which equate a number of [[128/125|dieses (128/125)]] with the [[16807/16384|cloudy comma (16807/16384)]].

All temperaments in the continuum satisfy (128/125)^''n'' ~ 16807/16384. Varying ''n'' results in different temperaments listed in the table below. It converges to [[augmented]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[Just intonation subgroup|2.5.7 subgroup]] temperaments supported by [[15edo]] (due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them). The just value of ''n'' is approximately 1.0747..., and temperaments having ''n'' near this value tend to be the most accurate ones.

{| class="wikitable center-1 center-2 center-4 center-5"

|+ Temperaments in the continuum

! rowspan="2" | 1 / ''n''

| -2

| {{Monzo|0 0 -6 5}}

| -2

| {{Monzo|-21 0 -3 10}}

| -1

| {{Monzo|-7 0 -3 5}}

| -1

| {{Monzo|-7 0 -3 5}}

| {{Monzo|-14 0 0 5}}

| [[Augmented family|Augmented]]

| {{Monzo|7 0 -3}}

| [[Subgroup temperaments#Rainy|Rainy]]

| {{Monzo|-21 0 3 5}}

| [[Subgroup temperaments#Rainy|Rainy]]

| {{Monzo|-21 0 3 5}}

| {{Monzo|-28 0 6 5}}

| {{Monzo|-35 0 3 10}}

| {{Monzo|-35 0 9 5}}

| {{Monzo|-49 0 3 15}}

| [[Augmented family|Augmented]]

| {{Monzo|7 0 -3}}

| {{Monzo|-14 0 0 5}}

* 15 & 72 (''n'' = 2/3)

* 379 & 4184 (''n'' = 13/12)

* 410 & 3675 (''n'' = 14/13)

* 851 & 1687 (''n'' = 29/27)

* 441 & 1308 (''n'' = 15/14)

* 68 & 15 (''n'' = 3/2)