Syntonic–kleismic equivalence continuum: Difference between revisions

The '''syntonic–kleismic equivalence continuum''' (or '''syntonic–enneadecal equivalence continuum''') is a [[equivalence continuum|continuum]] of [[5-limit]] [[regular temperament|temperaments]] which equate a number of [[81/80|syntonic commas (81/80)]] with the [[19-comma]] ({{monzo| -30 19 }}).

All temperaments in the continuum satisfy {{nowrap|(81/80)^''n'' ~ {{monzo| -30 19 }}}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[meantone]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments [[support]]ed by [[19edo]] (due to it being the unique equal temperament that tempers out both commas and thus tempers out all combinations of them). The just value of ''n'' is approximately 6.376…, and temperaments having ''n'' near this value tend to be the most accurate ones.

This continuum can also be expressed as the relationship between 81/80 and the [[enneadeca]] ({{monzo| -14 -19 19 }}). That is, {{nowrap|(81/80)^''k'' ~ {{monzo| -14 -19 19 }}}}. In this case, {{nowrap| ''k'' {{=}} 3''n'' − 19 }}.

|+ style="font-size: 105%;" | Temperaments with integer ''n''

| [[Graywood]]

| {{Monzo| -30 19 }}

| {{Monzo| -26 15 1 }}

| [[Hogzilla]]

| [[Stump]]

| {{Monzo| -18 7 3 }}

| {{Monzo| -14 3 4 }}

| {{Monzo| -10 -1 5 }}

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

| [[Sensipent]]

| {{Monzo| 2 9 -7 }}

| {{Monzo| -2 13 -8 }}

| [[Xenial]]

| {{Monzo| -6 17 -9 }}

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

== Graywood ==

 

 

 

 

 

 

[[Badness]] (Sintel): 32.4

 

 

 

 

 

{{Mapping|legend=1| 1 0 11 | 0 2 -11 }}

 

 

 

 

 

 

 

 

 

 

 

[[Badness]] (Sintel): 4.71

 

== Negri (5-limit) ==

 

The 5-limit version of negri tempers out the [[negri comma]], spliting a perfect fourth into four ~16/15 generators. It corresponds to {{nowrap| ''n'' {{=}} 4 }}. The only 7-limit extension that make any sense to use is to map the hemifourth to 7/6~8/7.

 

[[Comma list]]: 16875/16384

{{Mapping|legend=1| 1 2 2 | 0 -4 3 }}

[[Optimal tuning]]s:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

* [[WE]]: ~2 = 1202.5641{{c}}, ~675/512 = 443.2124{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~675/512 = 442.2692{{c}}

[[Badness]] (Sintel): 24.9

== Unsmate ==

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

: [[error map]]: {{val| -0.001 -1.100 +1.606 }}

 

 

 

: ''For extensions, see [[Ragismic microtemperaments #Enneadecal]].''

 

The 5-limit version of enneadecal tempers out the [[enneadeca]], which simply equates a stack of nineteen [[6/5]] minor thirds with five [[2/1|octaves]]. It corresponds to {{nowrap| ''n'' {{=}} 19/3 }}, with a 19th-octave period and a generator of a [[3/2|perfect fifth]].

[[Comma list]]: 19073486328125/19042491875328

{{Mapping|legend=1| 19 0 14 | 0 1 1 }}

[[Optimal tuning]]s:

* [[CWE]]: ~648/625 = 63.1579{{c}}, ~3/2 = 701.9900{{c}}

: error map: {{val| 0.000 -0.035 -0.113 }}

{{Optimal ET sequence|legend=1| 19, 95, 114, 133, 152, 171, 323, 494, 665, 1159, 1824, 2983, 7125c }}

[[Badness]] (Sintel): 1.12

[[Comma list]]: {{monzo| 10 23 -20 }}

 

{{Mapping|legend=1| 1 -10 -11 | 0 20 23 }}

: mapping generators: ~2, ~78125/52488

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1199.9478{{c}}, ~78125/52488 = 695.0566{{c}}

: [[error map]]: {{val| -0.052 -0.301 +0.562 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~78125/52488 = 695.0846{{c}}

: error map: {{val| 0.000 -0.264 +0.631 }}

{{Optimal ET sequence|legend=1| 19, …, 126, 145, 164, 183, 713, 896c, 1079c, 1262c, 1445c }}

: ''For extensions, see [[Ragismic microtemperaments #Counterkleismic]].''

[[Comma list]]: {{monzo| -20 -24 25 }}

{{Mapping|legend=1| 1 -5 -4 | 0 25 24 }}

: mapping generators: ~2, ~6/5

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0419{{c}}, ~6/5 = 316.0916{{c}}

: [[error map]]: {{val| +0.042 +0.126 -0.282 }}

: error map: {{val| 0.000 +0.097 -0.344 }}

{{Optimal ET sequence|legend=1| 19, …, 148, 167, 186, 205, 224, 429, 653, 1082, 1735c }}

[[Badness]] (Sintel): 7.45

{{Mapping|legend=1| 1 -134 -134 | 0 184 185 }}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0094{{c}}, ~5/3 = 884.2568{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.2499{{c}}

: error map: {{val| 0.000 +0.026 -0.083 }}

[[Badness]] (Sintel): 751