Schismic–Mercator equivalence continuum: Difference between revisions
Cmloegcmluin (talk | contribs) "optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence |
+countritonic |
||
| Line 77: | Line 77: | ||
Examples of temperaments with fractional values of ''n'': | Examples of temperaments with fractional values of ''n'': | ||
* [[Countritonic]] (''n'' = 4.5) | |||
* [[Quartonic]] (''n'' = 5.5) | * [[Quartonic]] (''n'' = 5.5) | ||
* [[Ditonic]] (''n'' = 6.5) | * [[Ditonic]] (''n'' = 6.5) | ||
| Line 87: | Line 88: | ||
Comma list: {{monzo| -84 53 }} | Comma list: {{monzo| -84 53 }} | ||
Mapping: {{mapping| 53 84 123 | 0 0 1 }} | |||
{{Multival|legend=1| 0 53 84 }} | |||
Optimal tuning (CTE): ~531441/524288 = 1\53, ~5/4 = 386.264 | |||
{{Optimal ET sequence|legend=1| 53, 477, 530, 583, 636, 689, 742, 795, 848, 901, 1749, 2650 }} | {{Optimal ET sequence|legend=1| 53, 477, 530, 583, 636, 689, 742, 795, 848, 901, 1749, 2650 }} | ||
| Line 101: | Line 102: | ||
Counterschismic is much like [[schismic]], but the harmonic 5 is located at +45 fifths instead of schismic's -8. They unite in [[53edo]], of course. | Counterschismic is much like [[schismic]], but the harmonic 5 is located at +45 fifths instead of schismic's -8. They unite in [[53edo]], of course. | ||
Subgroup: 2.3.5 | |||
Comma list: {{monzo| -69 45 -1 }} | Comma list: {{monzo| -69 45 -1 }} | ||
Mapping: {{mapping| 1 2 21 | 0 -1 -45 }} | |||
{{Multival|legend=1| 1 45 69 }} | |||
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.9175 | |||
{{Optimal ET sequence|legend=1| 53, 412, 465, 518, 571, 624, 677, 730, 2973, 3703, 4433, 5163, 11056}} | {{Optimal ET sequence|legend=1| 53, 412, 465, 518, 571, 624, 677, 730, 2973, 3703, 4433, 5163, 11056}} | ||
Badness: 0.09123 | Badness: 0.09123 | ||
== Countritonic == | |||
{{See also| Ragismic microtemperaments #Countritonic }} | |||
Subgroup: 2.3.5 | |||
Comma list: {{monzo| 33 -34 9 }} | |||
Mapping: {{mapping| 1 6 19 | 0 -9 -34 }} | |||
Optimal tuning (CTE): ~2 = 1\1, ~14348907/10240000 = 588.636 | |||
{{Optimal ET sequence|legend=1| 53, 263, 316, 369, 422, 475, 528, 2587b, 3115b, 3643b }} | |||
Badness: 0.256 | |||
== 53 & 3684 == | == 53 & 3684 == | ||
Subgroup: 2.3.5 | |||
Comma list: {{monzo| -339 230 -11 }} | Comma list: {{monzo| -339 230 -11 }} | ||
Mapping: {{mapping| 1 2 11 | 0 -11 -230 }} | |||
{{Multival|legend=1| 11 230 339 }} | |||
Optimal tuning (CTE): ~2 = 1\1, ~10737418240/10460353203 = 45.2769 | |||
{{Optimal ET sequence|legend=1| 53, 3684, 11105 }} | {{Optimal ET sequence|legend=1| 53, 3684, 11105 }} | ||
| Line 128: | Line 148: | ||
== 53 & 4190 == | == 53 & 4190 == | ||
Subgroup: 2.3.5 | |||
Comma list: {{monzo| 393 -267 13 }} | Comma list: {{monzo| 393 -267 13 }} | ||
Mapping: {{mapping| 1 6 93 | 0 -13 -267 }} | |||
{{Multival|legend=1| 13 267 393 }} | |||
Optimal tuning (CTE): ~2 = 1\1, ~{{monzo| -60 41 -2 }} = 407.5419 | |||
{{Optimal ET sequence|legend=1| 53, 4190, 4243, 4296 }} | {{Optimal ET sequence|legend=1| 53, 4190, 4243, 4296 }} | ||
Revision as of 09:19, 6 September 2023
The schismic-Mercator equivalence continuum is a continuum of 5-limit temperaments which equate a number of schismas (32805/32768) with Mercator's comma ([-84 53⟩). This continuum is theoretically interesting in that these are all 5-limit microtemperaments.
All temperaments in the continuum satisfy (32805/32768)n ~ [-84 53⟩. 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 53edo (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 approximately 1.8503390493…, and temperaments having n near this value tend to be the most accurate ones.
For a similar but perhaps more intuitive and practical concept, see Syntonic-chromatic equivalence continuum.
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| 0 | Mercator | [-84 53⟩ | |
| 1 | Counterschismic | [-69 45 -1⟩ | |
| 2 | Monzismic | [54 -37 2⟩ | |
| 3 | Tricot | [39 -29 3⟩ | |
| 4 | Vulture | [24 -21 4⟩ | |
| 5 | Amity | 1600000/1594323 | [9 -13 5⟩ |
| 6 | Kleismic | 15625/15552 | [-6 -5 6⟩ |
| 7 | Orson | 2109375/2097152 | [-21 3 7⟩ |
| 8 | Submajor | [-36 11 8⟩ | |
| 9 | Untriton | [-51 19 9⟩ | |
| … | … | … | … |
| ∞ | Schismic | 32805/32768 | [-15 8 1⟩ |
Examples of temperaments with fractional values of n:
- Countritonic (n = 4.5)
- Quartonic (n = 5.5)
- Ditonic (n = 6.5)
- 53 & 3684 (n = 11/6 = 1.83)
- 53 & 4190 (n = 13/7 = 1.857142)
Mercator
- and Mercator family
Comma list: [-84 53⟩
Mapping: [⟨53 84 123], ⟨0 0 1]]
Wedgie: ⟨⟨ 0 53 84 ]]
Optimal tuning (CTE): ~531441/524288 = 1\53, ~5/4 = 386.264
Optimal ET sequence: 53, 477, 530, 583, 636, 689, 742, 795, 848, 901, 1749, 2650
Badness: 0.2843
Counterschismic
Counterschismic is much like schismic, but the harmonic 5 is located at +45 fifths instead of schismic's -8. They unite in 53edo, of course.
Subgroup: 2.3.5
Comma list: [-69 45 -1⟩
Mapping: [⟨1 2 21], ⟨0 -1 -45]]
Wedgie: ⟨⟨ 1 45 69 ]]
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.9175
Optimal ET sequence: 53, 412, 465, 518, 571, 624, 677, 730, 2973, 3703, 4433, 5163, 11056
Badness: 0.09123
Countritonic
Subgroup: 2.3.5
Comma list: [33 -34 9⟩
Mapping: [⟨1 6 19], ⟨0 -9 -34]]
Optimal tuning (CTE): ~2 = 1\1, ~14348907/10240000 = 588.636
Optimal ET sequence: 53, 263, 316, 369, 422, 475, 528, 2587b, 3115b, 3643b
Badness: 0.256
53 & 3684
Subgroup: 2.3.5
Comma list: [-339 230 -11⟩
Mapping: [⟨1 2 11], ⟨0 -11 -230]]
Wedgie: ⟨⟨ 11 230 339 ]]
Optimal tuning (CTE): ~2 = 1\1, ~10737418240/10460353203 = 45.2769
Optimal ET sequence: 53, 3684, 11105
Badness: 0.276036
53 & 4190
Subgroup: 2.3.5
Comma list: [393 -267 13⟩
Mapping: [⟨1 6 93], ⟨0 -13 -267]]
Wedgie: ⟨⟨ 13 267 393 ]]
Optimal tuning (CTE): ~2 = 1\1, ~[-60 41 -2⟩ = 407.5419
Optimal ET sequence: 53, 4190, 4243, 4296
Badness: 0.173433