Pentacircle clan: Difference between revisions
→Pentafrost: review |
Sorting |
||
| Line 29: | Line 29: | ||
The second comma in the comma list determines how we extend parapyth to include the harmonic 5. | The second comma in the comma list determines how we extend parapyth to include the harmonic 5. | ||
Pele adds [[441/440]] and finds the harmonic 5 by equating the [[81/80|syntonic comma (81/80)]] with the [[64/63|septimal comma (64/63)]]. Together with the slightly sharp fifth this extension makes for one of the most natural interpretations. Sensamagic adds [[245/243]] or [[385/384]], a traditional RTT favorite. Apollo adds [[100/99]] or [[225/224]], and is even simpler than sensamagic. Uni adds [[540/539]]. Melpomene adds [[56/55]] or [[81/80]]. Terrapyth adds 585640/583443, a complex entry that finds the harmonic 5 at the triple augmented unison (AAA1). These all have the same lattice structure as parapyth. | Pele adds [[441/440]] and finds the harmonic 5 by equating the [[81/80|syntonic comma (81/80)]] with the [[64/63|septimal comma (64/63)]]. Together with the slightly sharp fifth this extension makes for one of the most natural interpretations. Sensamagic adds [[245/243]] or [[385/384]], a traditional RTT favorite. Apollo adds [[100/99]] or [[225/224]], and is even simpler than sensamagic. Pentafrost adds [[245/242]]. Uni adds [[540/539]]. Melpomene adds [[56/55]] or [[81/80]]. Terrapyth adds 585640/583443, a complex entry that finds the harmonic 5 at the triple augmented unison (AAA1). These all have the same lattice structure as parapyth. | ||
Julius aka varda adds [[176/175]], splitting the octave into two. Parahemif adds [[243/242]], splitting the perfect fifth into two. Kujuku adds 14700/14641, splitting the perfect twelfth into two. Tolerant adds 2200/2187, splitting the ~33/32 into two. Finally, canta adds 472392/471625, splitting the ~14/9 into three. | Julius aka varda adds [[176/175]], splitting the octave into two. Parahemif adds [[243/242]], splitting the perfect fifth into two. Kujuku adds 14700/14641, splitting the perfect twelfth into two. Tolerant adds 2200/2187, splitting the ~33/32 into two. Finally, canta adds 472392/471625, splitting the ~14/9 into three. | ||
| Line 37: | Line 37: | ||
* ''[[Apollo]]'' → [[Marvel family #Apollo|Marvel family]] | * ''[[Apollo]]'' → [[Marvel family #Apollo|Marvel family]] | ||
* [[Sensamagic]] → [[Sensamagic family #Undecimal sensamagic|Sensamagic family]] | * [[Sensamagic]] → [[Sensamagic family #Undecimal sensamagic|Sensamagic family]] | ||
* | * [[Pele]] → [[Hemifamity family #Pele|Hemifamity family]] | ||
* ''[[Uni]]'' → [[Hemimage family #Uni|Hemimage family]] | * ''[[Uni]]'' → [[Hemimage family #Uni|Hemimage family]] | ||
* ''[[Julius]]'' or ''[[varda]]'' → [[Diaschismic rank three family #Julius aka varda|Diaschismic rank-3 family]] | * ''[[Julius]]'' or ''[[varda]]'' → [[Diaschismic rank three family #Julius aka varda|Diaschismic rank-3 family]] | ||
| Line 115: | Line 115: | ||
Badness: 1.52 × 10<sup>-3</sup> | Badness: 1.52 × 10<sup>-3</sup> | ||
== Pentafrost == | |||
Pentafrost tempers out the [[245/242|frostma]] in addition to 896/891 which also means that the [[schisma]] is tempered out, mapping prime 5 to 8 [[4/3|perfect fourths]] and -1 octaves. | |||
[[Subgroup]]: 2.3.5.7.11 | |||
[[Comma list]]: [[245/242]], [[896/891]] | |||
{{Mapping|legend=1| 1 0 15 0 7 | 0 1 -8 0 -4 | 0 0 0 1 1 }} | |||
: mapping generators: ~2, ~3, ~7 | |||
[[Optimal tuning]]s: | |||
* [[CTE]]: ~2 = 1\1, ~3/2 = 701.883, ~7/4 = 964.864 | |||
* [[CWE]]: ~2 = 1\1, ~3/2 = 701.903, ~7/4 = 964.614 | |||
* [[CE]]: ~2 = 1\1, ~3/2 = 702.006, ~7/4 = 964.085 | |||
{{Optimal ET sequence|legend=1| 12, 24, 29, 36, 41, 106d }} | |||
[[Badness]]: 1.58 × 10<sup>-3</sup> | |||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: [[105/104]], [[245/242]], [[352/351]] | |||
Mapping: {{mapping| 1 0 15 0 7 12 | 0 1 -8 0 -4 -7 | 0 0 0 1 1 1 }} | |||
: mapping generators: ~2, ~3, ~7 | |||
Optimal tunings: | |||
* CTE: ~2 = 1\1, ~3/2 = 702.106, ~7/4 = 962.655 | |||
* CWE: ~2 = 1\1, ~3/2 = 702.145, ~7/4 = 962.175 | |||
* CE: ~2 = 1\1, ~3/2 = 702.360, ~7/4 = 962.210 | |||
Optimal ET sequence: {{Optimal ET sequence| 12f, 24, 29, 41 }} | |||
Badness: 1.59 × 10<sup>-3</sup> | |||
=== Permafrost === | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: [[144/143]], [[245/242]], [[896/891]] | |||
Mapping: {{mapping| 1 0 15 0 7 -3 | 0 1 -8 0 -4 6 | 0 0 0 1 1 -1 }} | |||
: mapping generators: ~2, ~3, ~7 | |||
Optimal tunings: | |||
* CTE: 2 = 1\1, ~3/2 = 701.783, ~7/4 = 966.113 | |||
* CWE: 2 = 1\1, ~3/2 = 701.753, ~7/4 = 966.445 | |||
* CE: 2 = 1\1, ~3/2 = 701.770, ~7/4 = 965.771 | |||
Optimal ET sequence: {{Optimal ET sequence| 12, 17, 24, 36, 41, 77e }} | |||
Badness: 2.62 × 10<sup>-3</sup> | |||
== Tolerant == | == Tolerant == | ||
| Line 296: | Line 352: | ||
Badness: 1.04 × 10<sup>-3</sup> | Badness: 1.04 × 10<sup>-3</sup> | ||