User:Eliora/Schismic-parakleismic equivalence continuum: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
Newer edit →
VisualWikitext
Eliora (talk | contribs)
autopatrolled, emailconfirmed
3,487 edits
Created page with "{{Technical data page}} The '''schismic–parakleismic equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of schismas (32805/32768) with parakleisma. 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 (...) Varying ''n'' results in different temperaments listed in the table below. It conv..."
 
Eliora (talk | contribs)
autopatrolled, emailconfirmed
3,487 edits
No edit summary
Line 1: Line 1:
{{Technical data page}}
{{Technical data page}}
The '''schismic–parakleismic equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[32805/32768|schismas (32805/32768)]] with [[parakleisma]]. This continuum is theoretically interesting in that the temperaments associated with its various commas are all 5-limit microtemperaments.
The '''schismic–parakleismic equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[32805/32768|schismas (32805/32768)]] with [[parakleisma]] (. 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 (...) 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 [[118edo]] (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 …, and temperaments having ''n'' near this value tend to be the most accurate ones.
All temperaments in the continuum satisfy (32805/32768)<sup>n</sup> = {{monzo|8 14 -13}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[parakleismic]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[118edo]] (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 2.70854..., and temperaments having ''n'' near this value tend to be the most accurate ones.
 
{| class="wikitable center-1"
|+ style="font-size: 105%;" | Temperaments with integer ''n''
|-
! rowspan="2" | ''n''
! rowspan="2" | Temperament
! colspan="2" | Comma
|-
! Ratio
! Monzo
|-
| 0
| [[Mercator]]
| (52 digits)
| {{Monzo| -84 53 }}
|-
| 1
| [[Counterschismic]]
| (44 digits)
| [[Counterschisma|{{Monzo| -69 45 -1 }}]]
|-
| 2
| [[Very high accuracy temperaments #Monzismic|Monzismic]]
| (36 digits)
| [[Monzisma|{{Monzo| 54 -37 2 }}]]
|-
| 3
| [[Alphatricot]]
| (28 digits)
| [[Alphatricot comma|{{Monzo| 39 -29 3 }}]]
|-
| 4
| [[Vulture]]
| (22 digits)
| [[Vulture comma|{{Monzo| 24 -21 4 }}]]
|-
| 5
| [[Amity]]
| [[1600000/1594323]]
| {{Monzo| 9 -13 5 }}
|-
| 6
| [[Kleismic]]
| [[15625/15552]]
| {{Monzo|-6 -5 6}}
|-
| 7
| [[Orson]]
| [[Semicomma|2109375/2097152]]
| {{Monzo|-21 3 7 }}
|-
| 8
| [[Buzzardsmic clan #Demibuzzard|Demibuzzard]]
| (22 digits)
| {{Monzo| -36 11 8 }}
|-
| 9
| [[Miscellaneous 5-limit temperaments #Untriton|Untriton]]
| (32 digits)
| {{Monzo| -51 19 9 }}
|-
| …
| …
| …
| …
|-
| ∞
| [[Schismic]]
| [[32805/32768]]
| {{Monzo| -15 8 1 }}
|}

Revision as of 15:53, 14 August 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The schismic–parakleismic equivalence continuum is a continuum of 5-limit temperaments which equate a number of schismas (32805/32768) with parakleisma (. 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 = [8 14 -13⟩. Varying n results in different temperaments listed in the table below. It converges to parakleismic as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 5-limit temperaments supported by 118edo (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 2.70854..., and temperaments having n near this value tend to be the most accurate ones.

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 Alphatricot (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 Demibuzzard (22 digits) [-36 11 8⟩
9 Untriton (32 digits) [-51 19 9⟩
… … … …
∞ Schismic 32805/32768 [-15 8 1⟩
Retrieved from "https://en.xen.wiki/index.php?title=User:Eliora/Schismic-parakleismic_equivalence_continuum&oldid=235708"