Schismic–Mercator equivalence continuum
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