Schismic–commatic equivalence continuum: Difference between revisions

The '''Schismic-Pythagorean equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[32805/32768|schismas (32805/32768)]] with [[Pythagorean comma|Pythagorean comma ({{monzo|-19 12}})]]. This continuum is theoretically interesting in that these are all 5-limit temperaments supported by [[12edo]].

All temperaments in the continuum satisfy (32805/32768)^''n'' ~ {{monzo|-19 12}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[schismic]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[12edo]] (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 12.0078623975..., and temperaments having ''n'' near this value tend to be the most accurate ones – indeed, the fact that this number is so close to 12 reflects how small [[Kirnberger's atom]] (the difference between 12 schismas and the Pythagorean comma) is.

| [[Hemifamity temperaments|Undim]]

Vals: {{Val list| 12, 79, 91, 103 }}

Vals: {{Val list| 12, 109, 121, 133 }}

{{see also| Hemifamity temperaments #Undim }}

Vals: {{Val list| 12, 140, 152, 164, 1076bc, 1228bc }}

Vals: {{Val list| 12, 193, 205, 217, 422 }}

Vals: {{Val list| 12, 258, 270, 1878 }}