User:Godtone/diaschismic-tetracot equivalence continuum - Revision history

FloraC: Update linking

2025-04-26T12:38:06Z

Update linking

← Older revision		Revision as of 12:38, 26 April 2025
Line 5:		Line 5:
	: A reasonable way of defining this continuum equates a number of [[2048/2025\|diaschismas (2048/2025)]] with the [[393216/390625\|Würschmidt comma (393216/390625)]], so that (2048/2025)<sup>''n''</sup> ~ 393216/390625. As a result, this may also be called the ''wurschmidt-diaschismic equivalence continuum'', or the ''diaschismic-gothic equivalence continuum'', which is more or less the same thing. The just value of ''n'' is 0.5853…, and temperaments near this tend to be the most accurate. The [[17-comma\|gothic comma]] (134217728/129140163) is the characteristic [[3-limit]] comma tempered out in 34edo, and it has a value of ''n'' = 4. Therefore, one can additionally define ''k'' = 4 - ''n'', which has notable advantages - in particular, due to being determined in terms of the 3-limit comma and the comma with the next lowest power of 5, (twice the numerator of) the value of ''k'' represents the number of generator steps required to reach the 3rd harmonic.		: A reasonable way of defining this continuum equates a number of [[2048/2025\|diaschismas (2048/2025)]] with the [[393216/390625\|Würschmidt comma (393216/390625)]], so that (2048/2025)<sup>''n''</sup> ~ 393216/390625. As a result, this may also be called the ''wurschmidt-diaschismic equivalence continuum'', or the ''diaschismic-gothic equivalence continuum'', which is more or less the same thing. The just value of ''n'' is 0.5853…, and temperaments near this tend to be the most accurate. The [[17-comma\|gothic comma]] (134217728/129140163) is the characteristic [[3-limit]] comma tempered out in 34edo, and it has a value of ''n'' = 4. Therefore, one can additionally define ''k'' = 4 - ''n'', which has notable advantages - in particular, due to being determined in terms of the 3-limit comma and the comma with the next lowest power of 5, (twice the numerator of) the value of ''k'' represents the number of generator steps required to reach the 3rd harmonic.

	: All temperaments in the continuum also satisfy (15625/15552)<sup>''m''</sup> ~ 393216/390625, for a value of ''m'' defined such that 1/''n'' - 1/''m'' = 1; equivalently, we can offset ''m'' by 1, and equate a number of [[15625/15552\|kleismas (15625/15552)]] with the diaschisma, hence the name. Varying ''m'' results in different temperaments listed in the second table below. It converges to [[~~Hanson_and_cata\|~~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.		: All temperaments in the continuum also satisfy (15625/15552)<sup>''m''</sup> ~ 393216/390625, for a value of ''m'' defined such that 1/''n'' - 1/''m'' = 1; equivalently, we can offset ''m'' by 1, and equate a number of [[15625/15552\|kleismas (15625/15552)]] with the diaschisma, hence the name. Varying ''m'' results in different temperaments listed in the second 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"
Line 38:		Line 38:
	\|-		\|-
	\| 1		\| 1
	\| [[~~Hanson_and_cata~~\|Hanson/~~Kleismic~~]]		\| [[Kleismic\|Hanson/kleismic]]
	\| [[15625/15552]]		\| [[15625/15552]]
	\| {{monzo\| -6 -5 6}}		\| {{monzo\| -6 -5 6}}

Godtone at 01:24, 26 August 2024

2024-08-26T01:24:05Z

← Older revision		Revision as of 01:24, 26 August 2024
Line 168:		Line 168:
	\| {{monzo\| -6 -5 6}}		\| {{monzo\| -6 -5 6}}
	\|}		\|}

	~~[[Category:34edo]]~~
	~~[[Category:Equivalence continua]]~~

Godtone: motivation & inverted continuum

2024-07-21T21:18:17Z

motivation & inverted continuum

← Older revision		Revision as of 21:18, 21 July 2024
Line 94:		Line 94:
	: <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 3.41173… = log<sub>2</sub>~~([[~~20000~~/~~19683~~]])~~/log~~<~~sub~~>2</~~sub~~>([[~~15625~~/~~15552~~]]), ~~and~~ temperaments ~~near this tend to be the most accurate ones~~.		We may invert the continuum by setting ''k'' such that 1/''n'' - 1/''k'' = 1, resulting in a continuum for ([[15625/15552]])<sup>k</sup> ~ [[20000/19683]], motivated by many important temperaments of 34edo being defined by commas connected by kleismas.

			The just value of ''k'' is 3.41173… = log<sub>2</sub>([[20000/19683]])/log<sub>2</sub>([[15625/15552]]), and temperaments near this tend to be the most accurate ones.
	{\| class="wikitable center-1 center-2"		{\| class="wikitable center-1 center-2"
	\|+ ~~Temperaments with integer ''m''~~		\|+ Tetracot-kleismic continuum
	\|-		\|-
	! rowspan="2" \| ''m''		! rowspan="2" \| ''m''
	! rowspan="2" \| ''k''		! rowspan="2" \| ''n''
	! rowspan="2" \| Temperament		! rowspan="2" \| Temperament
	! colspan="2" \| Comma		! colspan="2" \| Comma
Line 107:		Line 108:
	! Monzo		! Monzo
	\|-		\|-
	\| -4		\| -2
	\| ~~8/3~~		\| ?
	\| 34 & 113		\| 34 & 113
	\| 152587890625/148769467776		\| 152587890625/148769467776
	\| {{monzo\| -7 -19 16}}		\| {{monzo\| -7 -19 16}}
	\|-		\|-
	\| -3		\| -1
	\| ~~5/2~~		\| ?
	\| [[Fifive]]		\| [[Fifive]]
	\| 9765625/9565938		\| 9765625/9565938
	\| {{monzo\| -1 -14 10}}		\| {{monzo\| -1 -14 10}}
	\|-		\|-
	\| -2		\| 0
	\| 2		\| ?
	\| [[Tetracot]]		\| [[Tetracot]]
	\| [[20000/19683]]		\| [[20000/19683]]
	\| {{monzo\| 5 -9 4}}		\| {{monzo\| 5 -9 4}}
	\|-		\|-
	\| -1		\| 1
	\| ∞		\| ?
	\| [[~~Diaschismic_family~~#~~Srutal_aka_diaschismic~~\|Srutal]]		\| [[Diaschismic family#Srutal archagall\|Srutal]]
	\| [[2048/2025]]		\| [[2048/2025]]
	\| {{monzo\| 11 -4 -2}}		\| {{monzo\| 11 -4 -2}}
	\|-		\|-
	\| 0		\| 2
	\| 4		\| ?
	\| [[~~Würschmidt_family#W.C3.BCrschmidt\|Wurschmidt~~]]		\| [[Würschmidt]]
	\| [[393216/390625]]		\| [[393216/390625]]
	\| {{monzo\| 17 1 -8}}		\| {{monzo\| 17 1 -8}}
	\|-		\|-
	\| 1		\| 3
	\| ~~7/2~~		\| ?
	\| [[~~Vishnuzmic_family#Vishnu\|~~Vishnu]]		\| [[Vishnu]]
	\| [[6115295232/6103515625]]		\| [[6115295232/6103515625]]
	\| {{monzo\| 23 6 -14}}		\| {{monzo\| 23 6 -14}}
	\|-		\|-
	\| 2		\| 4
	\| ~~10/3~~		\| ?
	\| [[Gammic_family\|Gammic]]		\| [[Gammic_family\|Gammic]]
	\| 95367431640625/95105071448064		\| 95367431640625/95105071448064
	\| {{monzo\| -29 -11 20}}		\| {{monzo\| -29 -11 20}}
	\|-		\|-
	\| 3		\| 5
	\| ~~13/4~~		\| ?
	\| [[Ragismic_microtemperaments#Quatracot\|Quatracot]]		\| [[Ragismic_microtemperaments#Quatracot\|Quatracot]]
	\| 1490116119384765625/1479074071160291328		\| 1490116119384765625/1479074071160291328
Line 162:		Line 163:
	\|-		\|-
	\| ∞		\| ∞
	\| 3		\| ?
	\| [[Hanson_and_cata\|Hanson/Kleismic]]		\| [[Hanson_and_cata\|Hanson/Kleismic]]
	\| [[15625/15552]]		\| [[15625/15552]]
	\| {{monzo\| -6 -5 6}}		\| {{monzo\| -6 -5 6}}
	\|}

	~~{\| class="wikitable"~~
	~~\|+ Temperaments with fractional ''m'' and ''n''~~
	\|-
	~~! Temperament !! ''m'' !! ''n'' !! ''k''~~
	\|-
	~~\| [[High_badness_temperaments#Majvam\|Majvam]] \|\| 1/2 = 0.5 \|\| 1/3 = 0.{{overline\|3}} \|\| 11/3 = 3.{{overline\|6}}~~
	\|-
	~~\| [[Chlorine]] \|\| 3/2 = 1.5 \|\| 3/5 = 0.6 \|\| 17/5 = 3.4~~
	\|}		\|}

	[[Category:34edo]]		[[Category:34edo]]
	[[Category:Equivalence continua]]		[[Category:Equivalence continua]]

Godtone: just value of m & exact forms for just values

2024-07-21T20:13:42Z

just value of m & exact forms for just values

← Older revision		Revision as of 20:13, 21 July 2024
Line 1:		Line 1:
	The '''diaschismic-tetracot 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]]. It is equivalent to the ''diaschismic-gothic equivalence continuum'', where the [[gothic comma]] is found as ([[2048/2025]])<sup>2</sup> * [[20000/19683]], at ''n'' = -2.		The '''diaschismic-tetracot 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]]. It is equivalent to the ''diaschismic-gothic equivalence continuum'', where the [[gothic comma]] is found as ([[2048/2025]])<sup>2</sup> * [[20000/19683]], at ''n'' = -2.

	Each ''n'' on the continuum is defined by equating ([[2048/2025]])<sup>''n''</sup> with [[20000/19683]]. The just value of ''n'' is 1.41464…, and temperaments near this tend to be the most accurate. However, due to this continuum being defined through two reasonably-accurate temperaments and due to the strength of 34edo as a 5-limit temperament (supporting many notable tempered equivalences), simple fractional values of ''n'' in the general proximity of the just value are also often notable.		Each ''n'' on the continuum is defined by equating ([[2048/2025]])<sup>''n''</sup> with [[20000/19683]]. The just value of ''n'' is 1.41464… = log<sub>2</sub>([[20000/19683]])/log<sub>2</sub>([[2048/2025]]), and temperaments near this tend to be the most accurate. However, due to this continuum being defined through two reasonably-accurate temperaments and due to the strength of 34edo as a 5-limit temperament (supporting many notable tempered equivalences), simple fractional values of ''n'' in the general proximity of the just value are also often notable.

	: A reasonable way of defining this continuum equates a number of [[2048/2025\|diaschismas (2048/2025)]] with the [[393216/390625\|Würschmidt comma (393216/390625)]], so that (2048/2025)<sup>''n''</sup> ~ 393216/390625. As a result, this may also be called the ''wurschmidt-diaschismic equivalence continuum'', or the ''diaschismic-gothic equivalence continuum'', which is more or less the same thing. The just value of ''n'' is 0.5853…, and temperaments near this tend to be the most accurate. The [[17-comma\|gothic comma]] (134217728/129140163) is the characteristic [[3-limit]] comma tempered out in 34edo, and it has a value of ''n'' = 4. Therefore, one can additionally define ''k'' = 4 - ''n'', which has notable advantages - in particular, due to being determined in terms of the 3-limit comma and the comma with the next lowest power of 5, (twice the numerator of) the value of ''k'' represents the number of generator steps required to reach the 3rd harmonic.		: A reasonable way of defining this continuum equates a number of [[2048/2025\|diaschismas (2048/2025)]] with the [[393216/390625\|Würschmidt comma (393216/390625)]], so that (2048/2025)<sup>''n''</sup> ~ 393216/390625. As a result, this may also be called the ''wurschmidt-diaschismic equivalence continuum'', or the ''diaschismic-gothic equivalence continuum'', which is more or less the same thing. The just value of ''n'' is 0.5853…, and temperaments near this tend to be the most accurate. The [[17-comma\|gothic comma]] (134217728/129140163) is the characteristic [[3-limit]] comma tempered out in 34edo, and it has a value of ''n'' = 4. Therefore, one can additionally define ''k'' = 4 - ''n'', which has notable advantages - in particular, due to being determined in terms of the 3-limit comma and the comma with the next lowest power of 5, (twice the numerator of) the value of ''k'' represents the number of generator steps required to reach the 3rd harmonic.
Line 94:		Line 94:
	: <nowiki>*</nowiki> in projective tuning space, ∞ = -∞.		: <nowiki>*</nowiki> in projective tuning space, ∞ = -∞.

	We may invert the continuum by setting ''n'' such that 1/''n'' - 1/''m'' = 1. The just value of ''m'' is 1.~~4117…~~, and temperaments near this tend to be the most accurate ones.		We may invert the continuum by setting ''m'' such that 1/''n'' - 1/''m'' = 1. The just value of ''m'' is 3.41173… = log<sub>2</sub>([[20000/19683]])/log<sub>2</sub>([[15625/15552]]), and temperaments near this tend to be the most accurate ones.

	{\| class="wikitable center-1 center-2"		{\| class="wikitable center-1 center-2"

Godtone at 19:14, 21 July 2024

2024-07-21T19:14:33Z

← Older revision		Revision as of 19:14, 21 July 2024
Line 34:		Line 34:
	\| 1/2		\| 1/2
	\| [[Fifive]]		\| [[Fifive]]
	\| [[9765625/9565938]]		\| 9765625/9565938
	\| {{monzo\| -1 -14 10}}		\| {{monzo\| -1 -14 10}}
	\|-		\|-

Godtone at 19:13, 21 July 2024

2024-07-21T19:13:26Z

← Older revision		Revision as of 19:13, 21 July 2024
Line 63:		Line 63:
	\|-		\|-
	\| 2		\| 2
	\| [[~~würschmidt~~]]		\| [[Würschmidt]]
	\| [[393216/390625]]		\| [[393216/390625]]
	\| {{monzo\| 17 1 -8}}		\| {{monzo\| 17 1 -8}}

Godtone: create continuum

2024-07-21T19:13:08Z

create continuum

← Older revision		Revision as of 19:13, 21 July 2024
Line 1:		Line 1:
	The '''diaschismic-~~kleismic~~ 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-tetracot 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]]. It is equivalent to the ''diaschismic-gothic equivalence continuum'', where the [[gothic comma]] is found as ([[2048/2025]])<sup>2</sup> * [[20000/19683]], at ''n'' = -2.

	~~A reasonable way of defining this~~ continuum ~~equates a number of~~ [[2048/2025~~\|diaschismas (2048/2025)~~]] ~~with the [[393216/390625\|Würschmidt comma (393216/390625)]], so that (2048/2025~~)<sup>''n''</sup> ~~~ 393216~~/~~390625. As a result, this may also be called the ''wurschmidt-diaschismic equivalence continuum'', or the ''diaschismic-gothic equivalence continuum'', which is more or less the same thing~~. The just value of ''n'' is 0.~~5853…~~, and temperaments near this tend to be the most accurate. The [[17-comma\|gothic comma]] (134217728/129140163) is the characteristic [[3-limit]] comma tempered out in 34edo, and it has a value of ''n'' = 4. Therefore, one can additionally define ''k'' = 4 - ''n'', which has notable advantages - in particular, due to being ~~determined in terms of the 3~~-~~limit comma~~ and the ~~comma with the next lowest power~~ of 5, (~~twice the numerator of~~) ~~the value~~ of ''k'' ~~represents~~ the ~~number~~ of ~~generator steps required to reach~~ the ~~3rd harmonic~~.		Each ''n'' on the continuum is defined by equating ([[2048/2025]])<sup>''n''</sup> with [[20000/19683]]. The just value of ''n'' is 1.41464…, and temperaments near this tend to be the most accurate. However, due to this continuum being defined through two reasonably-accurate temperaments and due to the strength of 34edo as a 5-limit temperament (supporting many notable tempered equivalences), simple fractional values of ''n'' in the general proximity of the just value are also often notable.

	All temperaments in the continuum also satisfy (15625/15552)<sup>''m''</sup> ~ 393216/390625, for a value of ''m'' defined such that 1/''n'' - 1/''m'' = 1; equivalently, we can offset ''m'' by 1, and equate a number of [[15625/15552\|kleismas (15625/15552)]] with the diaschisma, hence the name. Varying ''m'' results in different temperaments listed in the second table below. It converges to [[Hanson_and_cata\|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.		: A reasonable way of defining this continuum equates a number of [[2048/2025\|diaschismas (2048/2025)]] with the [[393216/390625\|Würschmidt comma (393216/390625)]], so that (2048/2025)<sup>''n''</sup> ~ 393216/390625. As a result, this may also be called the ''wurschmidt-diaschismic equivalence continuum'', or the ''diaschismic-gothic equivalence continuum'', which is more or less the same thing. The just value of ''n'' is 0.5853…, and temperaments near this tend to be the most accurate. The [[17-comma\|gothic comma]] (134217728/129140163) is the characteristic [[3-limit]] comma tempered out in 34edo, and it has a value of ''n'' = 4. Therefore, one can additionally define ''k'' = 4 - ''n'', which has notable advantages - in particular, due to being determined in terms of the 3-limit comma and the comma with the next lowest power of 5, (twice the numerator of) the value of ''k'' represents the number of generator steps required to reach the 3rd harmonic.

			: All temperaments in the continuum also satisfy (15625/15552)<sup>''m''</sup> ~ 393216/390625, for a value of ''m'' defined such that 1/''n'' - 1/''m'' = 1; equivalently, we can offset ''m'' by 1, and equate a number of [[15625/15552\|kleismas (15625/15552)]] with the diaschisma, hence the name. Varying ''m'' results in different temperaments listed in the second table below. It converges to [[Hanson_and_cata\|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 ''n''~~		\|+ Diaschismic-tetracot equivalence continuum
	\|-		\|-
	! rowspan="2" \| ''n''		! rowspan="2" \| ''n''
	~~! rowspan="2" \| ''k''~~
	! rowspan="2" \| Temperament		! rowspan="2" \| Temperament
	! colspan="2" \| Comma		! colspan="2" \| Comma
Line 16:		Line 17:
	! Monzo		! Monzo
	\|-		\|-
	\| 4		\| -2
	~~\| 0~~
	\| [[Gothic]]		\| [[Gothic]]
	\| [[134217728/129140163]]		\| [[134217728/129140163]]
	\| {{monzo\| 27 -17}}		\| {{monzo\| 27 -17}}
	\|-		\|-
	\| 3		\| -1
	\| 1
	\| [[Immunity]]		\| [[Immunity]]
	\| [[1638400/1594323]]		\| [[1638400/1594323]]
	\| {{monzo\| 16 -13 2}}		\| {{monzo\| 16 -13 2}}
	\|-		\|-
	\| 2		\| 0
	~~\| 2~~
	\| [[Tetracot]]		\| [[Tetracot]]
	\| [[20000/19683]]		\| [[20000/19683]]
	\| {{monzo\| 5 -9 4}}		\| {{monzo\| 5 -9 4}}
			\|-
			\| 1/2
			\| [[Fifive]]
			\| [[9765625/9565938]]
			\| {{monzo\| -1 -14 10}}
	\|-		\|-
	\| 1		\| 1
	~~\| 3~~
	\| [[Hanson_and_cata\|Hanson/Kleismic]]		\| [[Hanson_and_cata\|Hanson/Kleismic]]
	\| [[15625/15552]]		\| [[15625/15552]]
	\| {{monzo\| -6 -5 6}}		\| {{monzo\| -6 -5 6}}
	\|-		\|-
	\| 0		\| 4/3
	\| 4		\| [[Gammic]]
	\| [[~~Würschmidt_family#W.C3.BCrschmidt~~\|~~Wurschmidt~~]]		\| 95367431640625/95105071448064
			\| {{monzo\| -29 -11 20 }}
			\|-
			\| 7/5
			\| [[Chlorine]]
			\| [very long but equal to ([[25/24]])<sup>17</sup> / [[2/1\|2]]]
			\| {{monzo\| -52 -17 34 }}
			\|-
			\| 3/2
			\| [[Vishnu]]
			\| [[6115295232/6103515625]]
			\| {{monzo\| 23 6 -14}}
			\|-
			\| 5/3
			\| [[Majvam]]
			\| 2404631929946112/2384185791015625
			\| {{monzo\| 40 7 -22 }}
			\|-
			\| 2
			\| [[würschmidt]]
	\| [[393216/390625]]		\| [[393216/390625]]
	\| {{monzo\| 17 1 -8}}		\| {{monzo\| 17 1 -8}}
	\|-		\|-
	\| -1		\| 5/2
	\| 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]
			\| 35184372088832/34332275390625
			\| {{monzo\| 45 -2 18 }}
			\|-
			\| 3
	\| [[Mabila_family#Mabila\|Mabila]]		\| [[Mabila_family#Mabila\|Mabila]]
	\| 268435456/263671875		\| 268435456/263671875
	\| {{monzo\| 28 -3 -10}}		\| {{monzo\| 28 -3 -10}}
	\|-		\|-
	\| -2		\| 4
	~~\| 6~~
	\| [[User:Userminusone/Goldis_comma\|Goldis]]		\| [[User:Userminusone/Goldis_comma\|Goldis]]
	\| 549755813888/533935546875		\| 549755813888/533935546875
	\| {{monzo\| 39 -7 -12}}		\| {{monzo\| 39 -7 -12}}
	\|-		\|-
	~~\| …~~
	\| …		\| …
	\| …		\| …
Line 64:		Line 87:
	\| …		\| …
	\|-		\|-
	\| ∞*
	\| ∞		\| ∞
	\| [[Diaschismic_family#Srutal_aka_diaschismic\|Srutal]]		\| [[Diaschismic_family#Srutal_aka_diaschismic\|Srutal]]

Godtone: draft page for testing my idea to centre this on tetracot and to correct miscellaneous errors/inconsistencies

2024-07-21T18:40:12Z

draft page for testing my idea to centre this on tetracot and to correct miscellaneous errors/inconsistencies

New page

The '''diaschismic-kleismic 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]].

A reasonable way of defining this continuum equates a number of [[2048/2025|diaschismas (2048/2025)]] with the [[393216/390625|Würschmidt comma (393216/390625)]], so that (2048/2025)<sup>''n''</sup> ~ 393216/390625. As a result, this may also be called the ''wurschmidt-diaschismic equivalence continuum'', or the ''diaschismic-gothic equivalence continuum'', which is more or less the same thing. The just value of ''n'' is 0.5853…, and temperaments near this tend to be the most accurate. The [[17-comma|gothic comma]] (134217728/129140163) is the characteristic [[3-limit]] comma tempered out in 34edo, and it has a value of ''n'' = 4. Therefore, one can additionally define ''k'' = 4 - ''n'', which has notable advantages - in particular, due to being determined in terms of the 3-limit comma and the comma with the next lowest power of 5, (twice the numerator of) the value of ''k'' represents the number of generator steps required to reach the 3rd harmonic.

All temperaments in the continuum also satisfy (15625/15552)<sup>''m''</sup> ~ 393216/390625, for a value of ''m'' defined such that 1/''n'' - 1/''m'' = 1; equivalently, we can offset ''m'' by 1, and equate a number of [[15625/15552|kleismas (15625/15552)]] with the diaschisma, hence the name. Varying ''m'' results in different temperaments listed in the second table below. It converges to [[Hanson_and_cata|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"
|+ Temperaments with integer ''n''
|-
! rowspan="2" | ''n''
! rowspan="2" | ''k''
! rowspan="2" | Temperament
! colspan="2" | Comma
|-
! Ratio
! Monzo
|-
| 4
| 0
| [[Gothic]]
| [[134217728/129140163]]
| {{monzo| 27 -17}}
|-
| 3
| 1
| [[Immunity]]
| [[1638400/1594323]]
| {{monzo| 16 -13 2}}
|-
| 2
| 2
| [[Tetracot]]
| [[20000/19683]]
| {{monzo| 5 -9 4}}
|-
| 1
| 3
| [[Hanson_and_cata|Hanson/Kleismic]]
| [[15625/15552]]
| {{monzo| -6 -5 6}}
|-
| 0
| 4
| [[Würschmidt_family#W.C3.BCrschmidt|Wurschmidt]]
| [[393216/390625]]
| {{monzo| 17 1 -8}}
|-
| -1
| 5
| [[Mabila_family#Mabila|Mabila]]
| 268435456/263671875
| {{monzo| 28 -3 -10}}
|-
| -2
| 6
| [[User:Userminusone/Goldis_comma|Goldis]]
| 549755813888/533935546875
| {{monzo| 39 -7 -12}}
|-
| …
| …
| …
| …
| …
|-
| ∞*
| ∞
| [[Diaschismic_family#Srutal_aka_diaschismic|Srutal]]
| [[2048/2025]]
| {{monzo| 11 -4 -2}}
|}
: <nowiki>*</nowiki> in projective tuning space, ∞ = -∞.

We may invert the continuum by setting ''n'' such that 1/''n'' - 1/''m'' = 1. The just value of ''m'' is 1.4117…, and temperaments near this tend to be the most accurate ones.

{| class="wikitable center-1 center-2"
|+ Temperaments with integer ''m''
|-
! rowspan="2" | ''m''
! rowspan="2" | ''k''
! rowspan="2" | Temperament
! colspan="2" | Comma
|-
! Ratio
! Monzo
|-
| -4
| 8/3
| 34 & 113
| 152587890625/148769467776
| {{monzo| -7 -19 16}}
|-
| -3
| 5/2
| [[Fifive]]
| 9765625/9565938
| {{monzo| -1 -14 10}}
|-
| -2
| 2
| [[Tetracot]]
| [[20000/19683]]
| {{monzo| 5 -9 4}}
|-
| -1
| ∞
| [[Diaschismic_family#Srutal_aka_diaschismic|Srutal]]
| [[2048/2025]]
| {{monzo| 11 -4 -2}}
|-
| 0
| 4
| [[Würschmidt_family#W.C3.BCrschmidt|Wurschmidt]]
| [[393216/390625]]
| {{monzo| 17 1 -8}}
|-
| 1
| 7/2
| [[Vishnuzmic_family#Vishnu|Vishnu]]
| [[6115295232/6103515625]]
| {{monzo| 23 6 -14}}
|-
| 2
| 10/3
| [[Gammic_family|Gammic]]
| 95367431640625/95105071448064
| {{monzo| -29 -11 20}}
|-
| 3
| 13/4
| [[Ragismic_microtemperaments#Quatracot|Quatracot]]
| 1490116119384765625/1479074071160291328
| {{monzo| -35 -16 26}}
|-
| …
| …
| …
| …
| …
|-
| ∞
| 3
| [[Hanson_and_cata|Hanson/Kleismic]]
| [[15625/15552]]
| {{monzo| -6 -5 6}}
|}

{| class="wikitable"
|+ Temperaments with fractional ''m'' and ''n''
|-
! Temperament !! ''m'' !! ''n'' !! ''k''
|-
| [[High_badness_temperaments#Majvam|Majvam]] || 1/2 = 0.5 || 1/3 = 0.{{overline|3}} || 11/3 = 3.{{overline|6}}
|-
| [[Chlorine]] || 3/2 = 1.5 || 3/5 = 0.6 || 17/5 = 3.4
|}

[[Category:34edo]]
[[Category:Equivalence continua]]