User:Overthink/Dicot–kleismic equivalence continuum

This user page is editable by any wiki editor. As a general rule, most users expect their user space to be edited only by themselves, except for minor edits (e.g. maintenance), undoing obviously harmful edits such as vandalism or disruptive editing, and user talk pages. However, by including this message box, the author of this user page has indicated that this page is open to contributions from other users (e.g. content-related edits).

The dicot–kleismic equivalence continuum is a continuum of 5-limit temperaments which equate a number of classical chromatic semitones (25/24) with the Pythagorean major second (9/8). As such, it can also be called the dicot–antitonic equivalence continuum.

All temperaments in the continuum satisfy (25/24)ⁿ ~ 9/8. Varying n results in different temperaments listed in the table below. It converges to dicot as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 5-limit temperaments supported by 4edo 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 2.8853…, and temperaments having n near this value tend to be the most accurate ones.

Temperaments with integer n

n	Temperament	Comma
		Ratio	Monzo
0	Antitonic	9/8	[-3 2⟩
1	Bug	27/25	[0 3 -2⟩
2	Diminished	648/625	[3 4 -4⟩
3	Kleismic	15625/15552	[-6 -5 6⟩
4	Doublewide	390625/373248	[-9 -6 8⟩
…	…	…	…
∞	Dicot	25/24	[-3 -1 2⟩

Temperaments with fractional n

n	Temperament	Comma
		Ratio	Monzo
5/2	Mynic	10077696/9765625	[9 9 -10⟩
7/2	Quasitemp	6103515625/5804752896	[-15 -11 14⟩

Category:4edo Category:Equivalence continua