Archytas–limmic equivalence continuum: Difference between revisions

The '''Archytas–diatonic equivalence continuum''', or '''septimal–diatonic equivalence continuum''', is a [[equivalence continuum|continuum]] of [[2.3.7 subgroup]] temperaments which equate a number of [[64/63|Archytas' commas (64/63)]] with the [[256/243|Pythagorean limma (256/243)]]. This continuum is theoretically interesting in that these are all [[2.3.7 subgroup|2.3.7-subgroup]] temperaments [[support]]ed by [[5edo]].  

All temperaments in the continuum satisfy {{nowrap|(64/63)^''n'' ~ 256/243}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[archy]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 2.3.7 subgroup temperaments supported by 5edo 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 3.3093…, and temperaments near this tend to be the most accurate ones.

256/243 is the characteristic [[3-limit]] comma tempered out in 5edo, and has many advantages as a target. In each case, ''n'' equals the order of harmonic 7 in the corresponding comma, and equals the number of generators to obtain a harmonic 3 in the generator chain.  

{| class="wikitable center-1"

! S. monzo

| {{Monzo| 8 -5 }}

| [[Trienstonian]]

| {{Monzo| 2 -3 1 }}

| {{Monzo| -4 -1 2 }}

| [[Cloudy]] retraction

| {{Monzo| -14 0 5 }}

| {{Monzo| -10 1 3 }}

| (72 digits)

| {{Monzo| 118 -16 -33 }}

| [[Slendroschismic]]

| {{Monzo| 36 -5 -10 }}

| [[Septiness]] restriction

| {{Monzo| 26 -4 -7 }}

| {{Monzo| 16 -3 -4 }}

| [[Obscenity]]

| [[4194304/4084101]]

| {{Monzo| 22 -5 -5 }}

| [[Archy]]

| {{Monzo| 6 -2 -1 }}