Schismic–Mercator equivalence continuum: Difference between revisions

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For a similar but perhaps more intuitive and practical concept, see [[Syntonic-chromatic equivalence continuum]].
For a similar but perhaps more intuitive and practical concept, see [[Syntonic-chromatic equivalence continuum]].


{| class="wikitable center-1 center-2"
{| class="wikitable center-1"
|+ Temperaments in the continuum
|+ Temperaments with integer ''n''
|-
|-
! rowspan="2" | ''n''
! rowspan="2" | ''n''
Line 16: Line 16:
|-
|-
| 0
| 0
| [[Mercator's comma|Mercator]]
| [[Mercator]]
|  
| (52 digits)
| {{monzo|-84 53}}
| {{monzo| -84 53 }}
|-
|-
| 1
| 1
| Counterschismic
| Counterschismic
|  
| (44 digits)
| {{monzo|-69 45 -1}}
| [[Counterschisma|{{monzo| -69 45 -1 }}]]
|-
|-
| 2
| 2
| [[Very high accuracy temperaments #Monzismic|Monzismic]]
| [[Very high accuracy temperaments #Monzismic|Monzismic]]
|  
| (36 digits)
| [[Monzisma|{{monzo|54 -37 2}}]]
| [[Monzisma|{{monzo| 54 -37 2 }}]]
|-
|-
| 3
| 3
| [[Tricot]]
| [[Tricot]]
|  
| (28 digits)
| {{monzo| 39 -29 3}}
| [[Tricot comma|{{monzo| 39 -29 3 }}]]
|-
|-
| 4
| 4
| [[Vulture]]
| [[Vulture]]
|  
| (22 digits)
| {{monzo| 24 -21 4 }}
| [[Vulture comma|{{monzo| 24 -21 4 }}]]
|-
|-
| 5
| 5
Line 56: Line 56:
|-
|-
| 8
| 8
| [[Marvel temperaments|Submajor]]
| [[Submajor (temperament)|Submajor]]
|  
| (22 digits)
| {{monzo|-36 11 8 }}
| {{monzo| -36 11 8 }}
|-
|-
| 9
| 9
| [[High badness temperaments|Untriton]]
| [[High badness temperaments #Untriton|Untriton]]
|  
| (32 digits)
| {{monzo|-51 19 9 }}
| {{monzo| -51 19 9 }}
|-
|-
| …
| …
Line 73: Line 73:
| [[Schismic]]
| [[Schismic]]
| [[32805/32768]]
| [[32805/32768]]
| {{monzo| -15 8 1}}
| {{monzo| -15 8 1 }}
|}
|}


Examples of temperaments with fractional values of ''n'':
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…
* [[Countritonic]] (''n'' = 4.5)
 
* [[Quartonic]] (''n'' = 5.5)
{| class="wikitable center-1"
* [[Ditonic]] (''n'' = 6.5)
|+ Temperaments with integer ''m''
* 53 & 3684 (''n'' = 11/6 = 1.8{{overline|3}})
|-
* 53 & 4190 (''n'' = 13/7 = 1.{{overline|857142}})
! rowspan="2" | ''m''
! rowspan="2" | Temperament
! colspan="2" | Comma
|-
! Ratio
! Monzo
|-
| 0
| [[Mercator]]
| (52 digits)
| {{monzo| -84 53 }}
|-
| 1
| [[Schismic]]
| [[32805/32768]]
| {{monzo| -15 8 1 }}
|-
| 2
| [[Very high accuracy temperaments #Monzismic|Monzismic]]
| (36 digits)
| [[Monzisma|{{monzo| 54 -37 2 }}]]
|-
| …
| …
| …
| …
|-
| ∞
| Counterschismic
| (44 digits)
| [[Counterschisma|{{monzo| -69 45 -1 }}]]
|}
 
{| class="wikitable"
|+ Temperaments with fractional ''n'' and ''m''
|-
! Temperament !! ''n'' !! ''m''
|-
| 53 & 3684 || 11/6 = 1.8{{overline|3}} || 11/5 = 2.2
|-
| 53 & 4190 || 13/7 = 1.{{overline|857142}} || 13/6 = 2.1{{overline|6}}
|-
| [[Countritonic]] || 9/2 = 4.5 || 9/7 = 1.{{overline|285714}}
|-
| [[Quartonic]] || 11/2 = 5.5 || 11/9 = 1.{{overline|2}}
|-
| [[Ditonic]] || 13/2 = 6.5 || 13/11 = 1.{{overline|18}}
|}


== Mercator ==
== Mercator ==

Revision as of 14:32, 16 July 2024

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.

Temperaments with integer n
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…

Temperaments with integer m
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⟩
Temperaments with fractional n and m
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

See also: Mercator's comma
 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

See also: Counterschisma

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

See also: High badness temperaments § 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

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