Schismic–countercommatic equivalence continuum: Difference between revisions

The '''schismic–countercommatic equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[32805/32768|schismas (32805/32768)]] with the [[41-comma|Pythagorean countercomma ({{monzo| 65 -41 }})]]. This continuum is theoretically interesting in that these are all [[5-limit]] [[microtemperament]]s [[support]]ed by [[41edo]].

All temperaments in the continuum satisfy {{nowrap|(32805/32768)^''n'' ~ {{monzo| 65 -41 }}}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[schismic]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 5-limit temperaments supported by 41edo due to it being the unique equal temperament that [[tempering out|tempers out]] both commas and thus tempers out all combinations of them. The just value of ''n'' is approximately 10.1575233481…, and temperaments having ''n'' near this value tend to be the most accurate ones.  

The Pythagorean countercomma is the characteristic 3-limit comma tempered out in 41edo, and has many advantages as a target. In each case, ''n'' equals the order of [[5/1|harmonic 5]] in the corresponding comma, and equals the number of steps to obtain the interval class of [[3/1|harmonic 3]] in the generator chain. For example:

* [[Cotoneum]] ({{nowrap|''n'' {{=}} 1}}) is generated by a fifth;

* [[Newt]] ({{nowrap|''n'' {{=}} 2}}) splits its fifth in two;

* Etc.

 

{| class="wikitable center-1"

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

| −7

| [[Merman]]

| 1121008359375/1099511627776

| {{Monzo| -40 15 7 }}

| −6

| [[Ampersand]]

| 34171875/33554432

| {{Monzo| -25 7 6 }}

| −5

| [[Magic]]

| 3125/3072

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

| −4

| [[Tetracot]]

| 20000/19683

| {{monzo| 5 -9 4 }}

| −3

| [[Rodan]]

| 131072000/129140163

| {{Monzo| 20 -17 3 }}

| −2

| [[Hemififths]]

| 858993459200/847288609443

| {{monzo| 35 -25 2 }}

| −1

| [[Kwai]]

| (32 digits)

| {{Monzo| 50 -33 1 }}

| [[Countercomp]]

| (40 digits)

| {{Monzo| 65 -41 }}

| [[Cotoneum]]

| (50 digits)

| {{Monzo| 80 -49 -1 }}

| [[Newt]]

| (58 digits)

| {{Monzo| 95 -57 -2 }}

| 41 & 282

| (68 digits)

| {{Monzo| 110 -65 -3 }}

| 41 & 335

| (76 digits)

| {{Monzo| 125 -73 -4 }}

| 41 & 388

| (86 digits)

| {{Monzo| 140 -81 -5 }}

| 41 & 441

| (94 digits)

| {{Monzo| 155 -89 -6 }}

| 41 & 453

| (104 digits)

| {{Monzo| 170 -97 -7 }}

| 41 & 506

| (112 digits)

| {{Monzo| 185 -105 -8 }}

| 41 & 559

| (122 digits)

| {{Monzo| 200 -113 -9 }}

| 41 & 571

| (130 digits)

| {{Monzo| 215 -121 -10 }}

| 41 & 624

| (140 digits)

| {{Monzo| -230 129 11 }}

| 41 & 677

| (148 digits)

| {{Monzo| -245 137 12 }}

| 41 & 730

| (158 digits)

| {{Monzo| -260 145 13 }}

| {{monzo| -15 8 1 }}

* [[Septimin]] ({{nowrap|''n'' {{=}} −11/2}})

* [[Shibboleth]] ({{nowrap|''n'' {{=}} −9/2}})

* [[Pluto]] ({{nowrap|''n'' {{=}} −7/2}})

* 3737 & 5585 ({{nowrap|''n'' {{=}} 31/3 {{=}} 10.{{overline|3}}}})

* 1277 & 2513 ({{nowrap|''n'' {{=}} 21/2}})

== Kwai (5-limit) ==

: ''For extensions, see [[Hemifamity temperaments #Kwai]].''

[[Subgroup]]: 2.3.5

[[Comma list]]: {{monzo| 50 -33 1 }}

: mapping generators: ~2, ~3

* [[WE]]: ~2 = 1199.792{{c}}, ~3/2 = 702.5077{{c}}

: [[error map]]: {{val| -0.208 +0.345 -0.023 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.6243{{c}}

: error map: {{val| 0.000 +0.669 +0.288 }}

[[Badness]] (Sintel): 14.9

: ''For extensions, see [[Garischismic clan #Cotoneum]].''

[[Subgroup]]: 2.3.5

[[Comma list]]: {{monzo| 80 -49 -1 }}

{{Mapping|legend=1| 1 0 80 | 0 1 -49 }}

[[Optimal tuning]]s:  

: error map: {{val| 0.000 +0.361 +0.190 }}

{{Optimal ET sequence|legend=1| 41, 135c, 176, 217, 475, 1167, 1642, 2117b, 3759bbc }}

[[Badness]] (Sintel): 29.1

: ''For extensions, see [[Breedsmic temperaments #Hemififths]].''

[[Subgroup]]: 2.3.5

[[Comma list]]: 858993459200/847288609443

: mapping generators: ~2, ~655360/531441

* [[WE]]: ~2 = 1199.7047{{c}}, ~655360/531441 = 351.3898{{c}}

: [[error map]]: {{val| -0.295 +0.529 -0.091 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~655360/531441 = 351.4654{{c}}

{{Optimal ET sequence|legend=1| 17c, 41, 58, 99, 239, 338, 915b, 1253bc }}

[[Badness]] (Sintel): 8.75

: ''For extensions, see [[Garischismic clan #Newt]].''

[[Subgroup]]: 2.3.5

[[Comma list]]: {{monzo| 95 -57 -2 }}

{{Mapping|legend=1| 1 1 19 | 0 2 -57 }}

: mapping generators: ~2, ~{{monzo| 47 -28 -1 }}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.9120{{c}}, ~{{monzo| 47 -28 -1 }} = 351.0878{{c}}

{{Optimal ET sequence|legend=1| 41, 147c, 188, 229, 270, 1121, 1391, 1661, 1931, 3592bc, 5523bbc }}

[[Badness]] (Sintel): 35.9