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''

! rowspan="2" | ''n''

| [[Graywood]]

| {{Monzo| -30 19 }}

| 7c & 12c

| {{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 }}

|+ style="font-size: 105%;" | Notable temperaments of fractional ''n''

! Temperament !! ''n'' !! Comma

| [[Unsmate]] || 9/2 = 4.5 || {{monzo| -24 2 9 }}

| [[Sycamore]] || 11/2 = 5.5 || {{monzo| -16 -6 11 }}

| [[Counterhanson]] || 25/4 = 6.25 || {{monzo| -20 -24 25 }}

| [[Enneadecal]] || 19/3 = 6.{{overline|3}} || {{monzo| -14 -19 19 }}

| [[Egads]] || 51/8 = 6.375 || {{monzo| -36 -52 51 }}

| [[Acrokleismic]] || 32/5 = 6.4 || {{monzo| 22 33 -32 }}

 

Named by [[CompactStar]] in 2024, graywood tempers out the [[19-comma]], corresponding to {{nowrap| ''n'' {{=}} 0 }}. It takes [[19edo]]'s closed [[circle of fifths]], but adds an independent generator for [[prime interval|prime]] [[5/1|5]]. 19 is the only equal temperament that makes it to the optimal ET sequence as all the small edo tunings, e.g. [[38edo|38c-edo]] or [[57edo|57c-edo]], are not nearly as accurate as 19 itself.

 

 

 

: mapping generators: ~2187/2048, ~5

 

* [[WE]]: ~2187/2048 = 63.2773{{c}}, ~5/4 = 381.7568{{c}}

* [[CWE]]: ~2187/2048 = 63.1579{{c}}, ~5/4 = 382.7889{{c}}

 

{{Optimal ET sequence|legend=1| 19 }}

 

[[Badness]] (Sintel): 32.4

 

: ''For extensions, see [[Semaphoresmic clan #Helayo]].''

 

Hogzilla is similar to [[godzilla]] in that it is generated by a [[semitwelfth]]. It corresponds to {{nowrap| ''n'' {{=}} 2 }}.

 

 

 

 

* [[CWE]]: ~2 = 1200.0000{{c}}, ~2048/1215 = 947.2462{{c}}

: error map: {{val| 0.000 -7.463 -6.022 }}

 

{{Optimal ET sequence|legend=1| 14, 19 }}

 

[[Badness]] (Sintel): 9.96

 

: ''For extensions, see [[Marvel temperaments #Triton]] and [[Sensamagic clan #Pycnic]].''

 

Stump splits the [[3/1|3rd]] [[harmonic]] into three equal parts, each for [[~]][[64/45]]. It corresponds to {{nowrap| ''n'' {{=}} 3 }}.

 

 

[[Comma list]]: 273375/262144

 

{{Mapping|legend=1| 1 0 6 | 0 3 -7 }}

 

* [[WE]]: ~2 = 1202.6288{{c}}, ~64/45 = 633.1214{{c}}

: error map: {{val| 0.000 -6.921 -8.059 }}

 

{{Optimal ET sequence|legend=1| 17, 19, 207bbccc }}

 

[[Badness]] (Sintel): 4.71

 

 

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.  

[[Subgroup]]: 2.3.5

[[Comma list]]: 16875/16384

: mapping generators: ~2, ~16/15

* [[WE]]: ~2 = 1202.3403{{c}}, ~16/15 = 126.0002{{c}}

: [[error map]]: {{val| +2.340 -1.275 -3.633 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~16/15 = 125.6610{{c}}

: error map: {{val| 0.000 -4.599 -9.331 }}

[[Badness]] (Sintel): 2.04

: ''For extensions, see [[Starling temperaments #Xenial]] and [[Sensamagic clan #Xenia]].''

Named by [[User:Xenllium|Xenllium]] in 2026, xenial splits the [[8/3|perfect eleventh]] into nine equal parts, each for ~[[10/9]]. It corresponds to {{nowrap| ''n'' {{=}} 9 }}. Its [[ploidacot]] is zeta-enneacot, and from this it derives its name.

 

 

 

 

[[Badness]] (Sintel): 8.84

 

 

 

 

 

{{Optimal ET sequence|legend=1| 8c, 11, 19 }}

 

[[Badness]] (Sintel): 24.9

== Unsmate ==

 

 

 

* [[CWE]]: ~2 = 1200.0000{{c}}, ~1875/1024 = 1011.0348{{c}}

: error map: {{val| 0.000 -2.642 -8.383 }}

 

 

 

: ''For extensions, see [[Ragismic microtemperaments #Parakleismic]] and [[Starling temperaments #Paraguay]].''

 

The 5-limit version of parakleismic tempers out the [[parakleisma]]. It corresponds to {{nowrap| ''n'' {{=}} 13/2 }}, and 13 generator steps give the interval class of [[3/1|3]].

 

 

 

 

 

 

 

 

 

 

 

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1199.9478{{c}}, ~27/25 = 126.7236{{c}}

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

* [[CWE]]: ~2 = 1200.0000{{c}}, ~27/25 = 126.7237{{c}}

{{Optimal ET sequence|legend=1| 19, 85c, 104c, 123, 142, 161, 303 }}

[[Badness]] (Sintel): 15.3

== Enneadecal (5-limit) ==

: ''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]].  

[[Subgroup]]: 2.3.5

[[Comma list]]: 19073486328125/19042491875328

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

* [[WE]]: ~648/625 = 63.1579{{c}}, ~3/2 = 701.9861{{c}}

: [[error map]]: {{val| +0.013 +0.044 -0.095 }}

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

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

[[Badness]] (Sintel): 1.12

[[Subgroup]]: 2.3.5

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

: mapping generators: ~2, ~78125/52488

: [[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 }}

 

== Counterhanson ==

 

 

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

 

 

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

 

 

 

 

 

 

[[Comma list]]: {{monzo| -134 -185 184 }}

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

: mapping generators: ~2, ~5/3

[[Optimal tuning]]s:  

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

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

{{Optimal ET sequence|legend=1| 19, …, 1600, 3219, 4819 }}

[[Badness]] (Sintel): 751