Ennealimmal–enneadecal equivalence continuum: Difference between revisions

The '''ennealimmal-enneadecal equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[ennealimma|ennealimmas ({{monzo|1 -27 18}})]] with [[Enneadeca|enneadeca comma ({{monzo|-14 -19 19}})]]. This continuum is theoretically interesting in that these are all 5-limit microtemperaments.

All temperaments in the continuum satisfy (2･3^-27･5¹⁸)^''n'' ~ (2^-14･3^-19･5¹⁹). Varying ''n'' results in different temperaments listed in the table below. It converges to [[ennealimmal]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[171edo]] (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 3.2669545024..., and temperaments having ''n'' near this value tend to be the most accurate ones.  

|+ Temperaments in the continuum

| -4

| -3

| -2

| -1

* 171 & 3193 (''n'' = -5/2 = -2.5)

* 171 & 2140 (''n'' = -3/2 = -1.5)

* 171 & 1087 (''n'' = -1/2 = -0.5)

* [[Landscape microtemperaments #Pnict|Pnict]] (''n'' = 1/3 = 0.{{overline|3}})

* [[Gammic family|Gammic]] (''n'' = 1/2 = 0.5)

* [[Horwell temperaments #Mutt|Mutt]] (''n'' = 2/3 = 0.{{overline|6}})

* [[Landscape microtemperaments #Septichrome|Septichrome]] (''n'' = 4/3 = 1.{{overline|3}})

* [[Metric microtemperaments #Geb|Geb]] (''n'' = 3/2 = 1.5)

* 171 & 1901 (''n'' = 5/2 = 2.5)

* 171 & 4125 (''n'' = 10/3 = 3.{{overline|3}})

* 171 & 3125 (''n'' = 7/2 = 3.5)