Xenharmonic Wiki

Diaschismic–gothmic equivalence continuum: Difference between revisions

← Older editNewer edit →
VisualWikitext
Godtone (talk | contribs)
autopatrolled, emailconfirmed
2,464 edits
m note that this also implies fractional k (as integer n <=> integer k)
Lériendil (talk | contribs)
8,046 edits
think the inversion on the gothic is pointless given no new temperaments are listed within that
Line 1: Line 1:
The '''diaschismic-gothmic equivalence continuum''' is a [[equivalence continuum|continuum]] of [[5-limit]] [[regular temperament|temperaments]] that describes the set of all [[5-limit]] temperaments supported by [[34edo]].
The '''diaschismic-gothic equivalence continuum''' is a [[equivalence continuum|continuum]] of [[5-limit]] [[regular temperament|temperaments]] that describes the set of all [[5-limit]] temperaments supported by [[34edo]].


All temperaments in the continuum satisfy (2048/2025)<sup>''n''</sup> ~ {{monzo| 27 -17 }}, equating a number of [[2048/2025|diaschismas (2048/2025)]] with the [[gothic comma]]. Varying ''n'' results in different temperaments listed in the table below. It converges to [[diaschismic]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 5-limit temperaments supported by 34edo due to it being the unique equal temperament that [[tempering out|tempers out]] both commas and thus tempers out all combinations of them. The just value of ''n'' is approximately 3.41464…, and temperaments having ''n'' near this value tend to be the most accurate ones.  
All temperaments in the continuum satisfy (2048/2025)<sup>''n''</sup> ~ {{monzo| 27 -17 }}, equating a number of [[2048/2025|diaschismas (2048/2025)]] with the [[gothic comma]]. Varying ''n'' results in different temperaments listed in the table below. It converges to [[diaschismic]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 5-limit temperaments supported by 34edo due to it being the unique equal temperament that [[tempering out|tempers out]] both commas and thus tempers out all combinations of them. The just value of ''n'' is approximately 3.41464…, and temperaments having ''n'' near this value tend to be the most accurate ones.  
Line 73: Line 73:
|}
|}
: <nowiki>*</nowiki> in projective tuning space, ∞ = -∞.
: <nowiki>*</nowiki> in projective tuning space, ∞ = -∞.
We may invert the continuum by setting ''m'' such that 1/''n'' + 1/''m'' = 1. The just value of ''m'' is 1.41414…, and temperaments near this tend to be the most accurate ones.
{| class="wikitable center-1"
|+ Temperaments with integer ''m''
|-
! rowspan="2" | ''m''
! rowspan="2" | Temperament
! colspan="2" | Comma
|-
! Ratio
! Monzo
|-
| 0
| [[Gothic]]
| [[134217728/129140163]]
| {{monzo| 27 -17 }}
|-
| 1
| [[Srutal]]
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
|-
| 2
| [[Tetracot]]
| [[20000/19683]]
| {{monzo| 5 -9 4 }}
|-
| …
| …
| …
| …
|-
| ∞
| [[Immunity]]
| [[1638400/1594323]]
| {{monzo| 16 -13 2 }}
|}


{| class="wikitable"
{| class="wikitable"
|+ Temperaments with fractional ''m'', ''n'' and ''k''
|+ Temperaments with fractional ''n''
|-
|-
! ''n'' !! ''m'' !! Temperament !! Comma
! ''n'' !! ''k'' !! Temperament !! Comma
|-
|-
| 5/2 = 2.5 || 5/3 = 1.{{overline|6}} || [[Fifive]] || {{monzo| -1 -14 10 }}
| 5/2 = 2.5 || 3/2 = 1.5 || [[Fifive]] || {{monzo| -1 -14 10 }}
|-
|-
| 13/4 = 3.25 || 13/9 = 1.{{overline|4}} || [[Quatracot]] || {{monzo| -33 -16 26 }}
| 13/4 = 3.25 || 3/4 = 0.75 || [[Quatracot]] || {{monzo| -33 -16 26 }}
|-
|-
| 10/3 = 3.{{overline|3}} || 10/7 = 1.{{overline|428571}} || [[Gammic]] || {{monzo| -29 -11 20 }}
| 10/3 = 3.{{overline|3}} || 2/3 = 0.{{overline|6}} || [[Gammic]] || {{monzo| -29 -11 20 }}
|-
|-
| 17/5 = 3.{{overline|4}} || 17/12 = 1.41{{overline|6}} || [[Chlorine]] || {{monzo| -52 -17 34 }}
| 17/5 = 3.4 || 3/5 = 0.6 || [[Chlorine]] || {{monzo| -52 -17 34 }}
|-
|-
| 7/2 = 3.5 || 7/5 = 1.4 || [[Vishnu]] || {{monzo| 23 6 -14 }}
| 7/2 = 3.5 || 1/2 = 0.5 || [[Vishnu]] || {{monzo| 23 6 -14 }}
|-
|-
| 11/3 = 3.{{overline|6}} || 11/8 = 1.375 || [[Majvam]] || {{monzo| 40 7 -22 }}
| 11/3 = 3.{{overline|6}} || 1/3 = 0.{{overline|3}} || [[Majvam]] || {{monzo| 40 7 -22 }}
|-
|-
| 9/2 = 4.5 || 9/7 = 1.{{overline|285714}} || [https://sintel.pythonanywhere.com/result?subgroup=2.3.5.13.17&reduce=on&weights=tenney&target=&edos=&commas=35184372088832%2F34332275390625%2C+289%2F288%2C+2197%2F2187&submit_comma=submit 34 & 142] || {{monzo| 45 -2 -18 }}
| 9/2 = 4.5 || -1/2 = -0.5 || [https://sintel.pythonanywhere.com/result?subgroup=2.3.5.13.17&reduce=on&weights=tenney&target=&edos=&commas=35184372088832%2F34332275390625%2C+289%2F288%2C+2197%2F2187&submit_comma=submit 34 & 142] || {{monzo| 45 -2 -18 }}
|}
|}


All temperaments in the continuum also satisfy (15625/15552)<sup>''l''</sup> ~ 393216/390625, for a value of ''l'' defined such that 1/''k'' - 1/''l'' = 1; equivalently, we can offset ''l'' by 1, and equate a number of [[15625/15552|kleismas (15625/15552)]] with the diaschisma, giving rise to the name ''diaschismic-kleismic equivalence continuum''. Varying ''l'' results in different temperaments listed in the table below. It converges to [[hanson]] as ''l'' approaches infinity, and is motivated by the fact that many important temperaments of 34edo follow a chain of commas connected by kleismas.  
All temperaments in the continuum also satisfy (15625/15552)<sup>''m''</sup> ~ 393216/390625, for a value of ''m'' defined such that 1/''k'' - 1/''m'' = 1; equivalently, we can offset ''m'' by 1, and equate a number of [[15625/15552|kleismas (15625/15552)]] with the diaschisma, giving rise to the name ''diaschismic-kleismic equivalence continuum''. Varying ''m'' results in different temperaments listed in the table below. It converges to [[hanson]] as ''m'' approaches infinity, and is motivated by the fact that many important temperaments of 34edo follow a chain of commas connected by kleismas.  


{| class="wikitable center-1"
{| class="wikitable center-1"
|+ Temperaments with integer ''l'' in the kleismic-würschmidt continuum
|+ Temperaments with integer ''m'' in the kleismic-würschmidt continuum
|-
|-
! rowspan="2" | ''l''
! rowspan="2" | ''m''
! rowspan="2" | Temperament
! rowspan="2" | Temperament
! colspan="2" | Comma
! colspan="2" | Comma
Line 196: Line 158:


{| class="wikitable"
{| class="wikitable"
|+ Temperaments with fractional ''k'' and ''l''
|+ Temperaments with fractional ''k'' and ''m''
|-
|-
! ''l'' !! ''k'' !! Temperament !! Comma
! ''m'' !! ''k'' !! Temperament !! Comma
|-
|-
| 1/2 = 0.5 || 1/3 = 0.{{overline|3}} || [[Majvam]] || {{monzo| 40 7 -22 }}
| 1/2 = 0.5 || 1/3 = 0.{{overline|3}} || [[Majvam]] || {{monzo| 40 7 -22 }}
Retrieved from "https://en.xen.wiki/w/Diaschismic–gothmic_equivalence_continuum"