Schismic–Mercator equivalence continuum: Difference between revisions
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We may invert the continuum by setting ''m'' such that {{nowrap|1/''m'' + 1/''n'' {{=}} 1}}. This may be called the ''counterschismic–Mercator equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 2.17600… | We may invert the continuum by setting ''m'' such that {{nowrap|1/''m'' + 1/''n'' {{=}} 1}}. This may be called the ''counterschismic–Mercator equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 2.17600… While the counterschisma is of comparable size as the schisma, it is way more complex, so this continuum does not contain as many useful temperaments at integer points. | ||
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== Mercator == | == Mercator == | ||
{{See also| Mercator's comma }} | {{See also| Mercator's comma | Mercator family }} | ||
Comma list: {{monzo| -84 53 }} | Comma list: {{monzo| -84 53 }} | ||
Revision as of 13:35, 5 February 2025
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 the temperaments associated with its various commas 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 | (52 digits) | [-84 53⟩ |
| 1 | Counterschismic | (44 digits) | [-69 45 -1⟩ |
| 2 | Monzismic | (36 digits) | [54 -37 2⟩ |
| 3 | Tricot | (28 digits) | [39 -29 3⟩ |
| 4 | Vulture | (22 digits) | [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 | (22 digits) | [-36 11 8⟩ |
| 9 | Untriton | (32 digits) | [-51 19 9⟩ |
| … | … | … | … |
| ∞ | Schismic | 32805/32768 | [-15 8 1⟩ |
We may invert the continuum by setting m such that 1/m + 1/n = 1. This may be called the counterschismic–Mercator equivalence continuum, which is essentially the same thing. The just value of m is 2.17600… While the counterschisma is of comparable size as the schisma, it is way more complex, so this continuum does not contain as many useful temperaments at integer points.
| m | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| 0 | Mercator | (52 digits) | [-84 53⟩ |
| 1 | Schismic | 32805/32768 | [-15 8 1⟩ |
| 2 | Monzismic | (36 digits) | [54 -37 2⟩ |
| … | … | … | … |
| ∞ | Counterschismic | (44 digits) | [-69 45 -1⟩ |
| Temperament | n | m |
|---|---|---|
| 53 & 3684 | 11/6 = 1.83 | 11/5 = 2.2 |
| 53 & 4190 | 13/7 = 1.857142 | 13/6 = 2.16 |
| Countritonic | 9/2 = 4.5 | 9/7 = 1.285714 |
| Quartonic | 11/2 = 5.5 | 11/9 = 1.2 |
| Ditonic | 13/2 = 6.5 | 13/11 = 1.18 |
Mercator
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
- For extensions, see Ragismic microtemperaments #Countritonic and Hemifamity temperaments #Countriton.
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