Intergan - Revision history

Ground at 05:22, 12 July 2026

2026-07-12T05:22:45Z

← Older revision		Revision as of 05:22, 12 July 2026
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	Being so close to 24edo means that 1/2, 1/3, 1/4, 1/6, and 1/8-octave temperaments are all available. 12edo's remarkable accuracy in 2.3.17.19 provides many ways to temper the subgroup and its extensions, but this requires more effort to produce a temperament that isn't virtually 12edo; Intergan's interseptimals are one option. The rank-3 temperament may be attached to one of numerous often multi-period temperaments in this subgroup to reduce it to rank-2.		Being so close to 24edo means that 1/2, 1/3, 1/4, 1/6, and 1/8-octave temperaments are all available. 12edo's remarkable accuracy in 2.3.17.19 provides many ways to temper the subgroup and its extensions, but this requires more effort to produce a temperament that isn't virtually 12edo; Intergan's interseptimals are one option. The rank-3 temperament may be attached to one of numerous often multi-period temperaments in this subgroup to reduce it to rank-2.
	* Third-octave: 121172^-11 = 2057/2048 ~~and (27/17)^3/4 = 19683/19652. The former is~~ called the Blume comma, and it is obscure despite being both accurate and low-complexity. The third-octave version of Intergan is especially useful when adding the elusive prime 29 because 29/23 is 401¢.		* Third-octave: Add 121172^-11 = 2057/2048, called the Blume comma, and it is obscure despite being both accurate and low-complexity. The third-octave version of Intergan is especially useful when adding the elusive prime 29 because 29/23 is 401¢. It is equivalent to adding the 2/(5/4)^3 = 128/125 Augmented comma. (27/17)^3/4 = 19683/19652 is a typical comma for third-octave temperaments but it has significantly more complex result.
	* Quarter-octave: ~~add~~ (19/16)^4/2=~~130321~~/131072. ~~This is one of very few~~ useful commas dividing the octave into four equal parts.		* Quarter-octave: Add the (6/5)^4/2 = 648/625 Diminished comma or the smaller 2/(19/16)^4 = 131072/130321. These are the two most useful commas dividing the octave into four equal parts besides the (25/21)^4/2 = 390625/388962 Dimcomp comma.
			2.3.5.11.13.17.19.31.37 edos:
			* Meantone: 24, 19, 43, 5, 67, 110[+5, +19], 91[+5], 115[+5]
			* Augmented: 24, 9, 15, 33[+31], 57[+5, +31]
			* Diminished: 24, 52, 76[+5]
	The different versions of Intergan may be defined by the sheer number of intervals it maps to ~19/16: 19/16, 25/21, 31/26, 32/27, 37/31, 44/37. The semiquartal temperament equates all of them besides 25/21 due to its sharp 5, instead equating it to 6/5 in the 7-limit. This includes an accurate otonal approximation of 0-1-2-3\4, 26:31:37:44.		The different versions of Intergan may be defined by the sheer number of intervals it maps to ~19/16: 19/16, 25/21, 31/26, 32/27, 37/31, 44/37. The semiquartal temperament equates all of them besides 25/21 due to its sharp 5, instead equating it to 6/5 in the 7-limit. This includes an accurate otonal approximation of 0-1-2-3\4, 26:31:37:44.

	{{Navbox regtemp}}		{{Navbox regtemp}}
	{{Cat\|temperaments}}		{{Cat\|temperaments}}

Ground at 04:53, 12 July 2026

2026-07-12T04:53:20Z

← Older revision		Revision as of 04:53, 12 July 2026
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	* Third-octave: 121172^-11 = 2057/2048 and (27/17)^3/4 = 19683/19652. The former is called the Blume comma, and it is obscure despite being both accurate and low-complexity. The third-octave version of Intergan is especially useful when adding the elusive prime 29 because 29/23 is 401¢.		* Third-octave: 121172^-11 = 2057/2048 and (27/17)^3/4 = 19683/19652. The former is called the Blume comma, and it is obscure despite being both accurate and low-complexity. The third-octave version of Intergan is especially useful when adding the elusive prime 29 because 29/23 is 401¢.
	* Quarter-octave: add (19/16)^4/2=130321/131072. This is one of very few useful commas dividing the octave into four equal parts.		* Quarter-octave: add (19/16)^4/2=130321/131072. This is one of very few useful commas dividing the octave into four equal parts.
	The different versions of Intergan may be defined by the sheer number of intervals it maps to ~19/16: 19/16, 25/21, 31/26, 32/27, 37/31, 44/37. The semiquartal temperament equates all of them besides 25/21 due to its sharp 5, instead equating it to 6/5 in the 7-limit. This includes ~~the~~ otonal approximation of ~~4edo~~, 26:31:37:44.		The different versions of Intergan may be defined by the sheer number of intervals it maps to ~19/16: 19/16, 25/21, 31/26, 32/27, 37/31, 44/37. The semiquartal temperament equates all of them besides 25/21 due to its sharp 5, instead equating it to 6/5 in the 7-limit. This includes an accurate otonal approximation of 0-1-2-3\4, 26:31:37:44.

	{{Navbox regtemp}}		{{Navbox regtemp}}
	{{Cat\|temperaments}}		{{Cat\|temperaments}}

Ground at 04:51, 12 July 2026

2026-07-12T04:51:14Z

← Older revision		Revision as of 04:51, 12 July 2026
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	* Third-octave: 121172^-11 = 2057/2048 and (27/17)^3/4 = 19683/19652. The former is called the Blume comma, and it is obscure despite being both accurate and low-complexity. The third-octave version of Intergan is especially useful when adding the elusive prime 29 because 29/23 is 401¢.		* Third-octave: 121172^-11 = 2057/2048 and (27/17)^3/4 = 19683/19652. The former is called the Blume comma, and it is obscure despite being both accurate and low-complexity. The third-octave version of Intergan is especially useful when adding the elusive prime 29 because 29/23 is 401¢.
	* Quarter-octave: add (19/16)^4/2=130321/131072. This is one of very few useful commas dividing the octave into four equal parts.		* Quarter-octave: add (19/16)^4/2=130321/131072. This is one of very few useful commas dividing the octave into four equal parts.
	The different versions of Intergan may be defined by the sheer number of intervals it maps to ~~the~~ ~19/16 ~~interval~~: 19/16, 25/21, 31/26, 32/27, 37/31, 44/37, ~~and~~ the otonal ~~semifourth stack~~ 26:31:37:44.		The different versions of Intergan may be defined by the sheer number of intervals it maps to ~19/16: 19/16, 25/21, 31/26, 32/27, 37/31, 44/37. The semiquartal temperament equates all of them besides 25/21 due to its sharp 5, instead equating it to 6/5 in the 7-limit. This includes the otonal approximation of 4edo, 26:31:37:44.

	{{Navbox regtemp}}		{{Navbox regtemp}}
	{{Cat\|temperaments}}		{{Cat\|temperaments}}

Ground at 04:46, 12 July 2026

2026-07-12T04:46:50Z

← Older revision		Revision as of 04:46, 12 July 2026
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	* Primes 7, 23, and 29 are all more difficult to map due to their complexity, although their difficulty is in ascending order. They are more easily reached by splitting the semifourth generator in half in a manner similar to Negri.		* Primes 7, 23, and 29 are all more difficult to map due to their complexity, although their difficulty is in ascending order. They are more easily reached by splitting the semifourth generator in half in a manner similar to Negri.
	{\| class="wikitable"		{\| class="wikitable"
	\|+Semiquartal Intergan, 14\67 generator, 37-limit patent val		\|+Semiquartal Intergan, 14\67 generator, 37-limit patent val, 37-odd-limit
	!Gens		!Gens
	!Cents		!Cents

Ground at 04:43, 12 July 2026

2026-07-12T04:43:41Z

Show changes

Ground at 02:53, 12 July 2026

2026-07-12T02:53:05Z

← Older revision		Revision as of 02:53, 12 July 2026
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	'''Intergan''' is a set of temperaments related to 24edo semiquartal. The term is short for "interseptimal Ganassi," after 12edo's 2.3.17.19 Ganassi temperament. It seeks to cover all of the notable concordances generated by a meantone-range semifourth.		'''Intergan''' is a set of temperaments related to 24edo semiquartal. The term is short for "interseptimal Ganassi," after 12edo's 2.3.17.19 Ganassi temperament. It seeks to cover all of the notable concordances generated by a meantone-range semifourth.

	* The most basic version is 2.3.11.13/5.17.19 5&24. 4/3 is split into two semifourths most simply represented by 15/13, and 22/17 is equated to 13/10 and 22/19 is equated to 15/13. It ~~contains~~ the 16:19:22 +1+1 sub-diminished triad.		* The most basic version is 2.3.11.13/5.17.19 5&24. 4/3 is split into two semifourths most simply represented by 15/13, and 22/17 is equated to 13/10 and 22/19 is equated to 15/13. It features the remarkably simple 10:13:15 ultramajor triad and its retroversion found in Island temperaments, along with the 16:19:22 +1+1 sub-diminished triad and 16:19:24:27 +3+5+3 minor add6 tetrad. 11:12:17:19 is present, although this adding 13, 14, or 15 to the subgroup will significantly improve its usefulness.
	* Tempering out 81/80 is somewhat obvious, resulting in 2.3.5.11.13.17.19 5&24.		* Tempering out 81/80 is somewhat obvious, resulting in 2.3.5.11.13.17.19 5&24.
	* Primes 31 and 37 are also obvious to add, equating 37/32 to 22/19 and 37/31 to 19/16, resulting in 2.3.5.11.13.17.19.31.37 5&24. 67edo is roughly optimal due to its very mild meantone fourth being split in half. The 2.31.37 restriction is highly accurate in 43edo.		* Primes 31 and 37 are also obvious to add, equating 37/32 to 22/19 and 37/31 to 19/16, resulting in 2.3.5.11.13.17.19.31.37 5&24. 67edo is roughly optimal due to its very mild meantone fourth being split in half. The 2.31.37 restriction is highly accurate in 43edo.

Ground at 02:41, 12 July 2026

2026-07-12T02:41:08Z

← Older revision		Revision as of 02:41, 12 July 2026
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	\|}		\|}
	Being so close to 24edo means that 1/2, 1/3, 1/4, 1/6, and 1/8-octave temperaments are all available. 12edo's accuracy ~~provides many ways to temper~~ 2.3.17.19, but this requires more effort to produce a temperament that isn't virtually 12edo. Intergan's interseptimals are one option. The rank-3 temperament ~~should~~ be attached to one of numerous often multi-period temperaments in this subgroup.		Being so close to 24edo means that 1/2, 1/3, 1/4, 1/6, and 1/8-octave temperaments are all available. 12edo's remarkable accuracy in 2.3.17.19 provides many ways to temper the subgroup and its extensions, but this requires more effort to produce a temperament that isn't virtually 12edo; Intergan's interseptimals are one option. The rank-3 temperament may be attached to one of numerous often multi-period temperaments in this subgroup to reduce it to rank-2.
	* Third-octave: 121172^-11 = 2057/2048 and (27/17)^3/4 = 19683/19652. The former is called the Blume comma, and it is obscure despite being both accurate and low-complexity.		* Third-octave: 121172^-11 = 2057/2048 and (27/17)^3/4 = 19683/19652. The former is called the Blume comma, and it is obscure despite being both accurate and low-complexity. The third-octave version of Intergan is especially useful when adding the elusive prime 29 because 29/23 is 401¢.
	* Quarter-octave: add (19/16)^4/2=130321/131072. This is one of very few useful commas dividing the octave into four equal parts.		* Quarter-octave: add (19/16)^4/2=130321/131072. This is one of very few useful commas dividing the octave into four equal parts.
	The different versions of Intergan may be defined by the sheer number of intervals it maps to the ~19/16 interval: 19/16, 25/21, 31/26, 32/27, 37/31, 44/37, and the otonal semifourth stack 26:31:37:44.		The different versions of Intergan may be defined by the sheer number of intervals it maps to the ~19/16 interval: 19/16, 25/21, 31/26, 32/27, 37/31, 44/37, and the otonal semifourth stack 26:31:37:44.

Ground at 02:35, 12 July 2026

2026-07-12T02:35:42Z

← Older revision		Revision as of 02:35, 12 July 2026
Line 1:		Line 1:
	'''Intergan''' is a set of temperaments related to 24edo semiquartal. The term is short for "interseptimal Ganassi," after 12edo's 2.3.17.19 Ganassi temperament. It seeks to cover all of the notable concordances generated by a meantone-range semifourth.		'''Intergan''' is a set of temperaments related to 24edo semiquartal. The term is short for "interseptimal Ganassi," after 12edo's 2.3.17.19 Ganassi temperament. It seeks to cover all of the notable concordances generated by a meantone-range semifourth.

	* The most basic version is 2.3.11.13/5.17.19 5&24. 22/17 is equated to 13/10 and 22/19 is equated to 15/13. It contains the 16:19:22 sub-diminished triad.		* The most basic version is 2.3.11.13/5.17.19 5&24. 4/3 is split into two semifourths most simply represented by 15/13, and 22/17 is equated to 13/10 and 22/19 is equated to 15/13. It contains the 16:19:22 +1+1 sub-diminished triad.
	* Tempering out 81/80 is somewhat obvious, resulting in 2.3.5.11.13.17.19 5&24.		* Tempering out 81/80 is somewhat obvious, resulting in 2.3.5.11.13.17.19 5&24.
	* Primes 31 and 37 are also obvious to add, equating 37/32 to 22/19 and 37/31 to 19/16, resulting in 2.3.5.11.13.17.19.31.37 5&24. 67edo is roughly optimal due to its very mild meantone fourth being split in half. The 2.31.37 restriction is highly accurate in 43edo.		* Primes 31 and 37 are also obvious to add, equating 37/32 to 22/19 and 37/31 to 19/16, resulting in 2.3.5.11.13.17.19.31.37 5&24. 67edo is roughly optimal due to its very mild meantone fourth being split in half. The 2.31.37 restriction is highly accurate in 43edo.
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	Being so close to 24edo means that 1/2, 1/3, 1/4, 1/6, and 1/8-octave temperaments are all available. 12edo's accuracy provides many ways to temper 2.3.17.19, but this requires more effort to produce a temperament that isn't virtually 12edo. Intergan's interseptimals are one option. The rank-3 temperament should be attached to one of numerous often multi-period temperaments in this subgroup.		Being so close to 24edo means that 1/2, 1/3, 1/4, 1/6, and 1/8-octave temperaments are all available. 12edo's accuracy provides many ways to temper 2.3.17.19, but this requires more effort to produce a temperament that isn't virtually 12edo. Intergan's interseptimals are one option. The rank-3 temperament should be attached to one of numerous often multi-period temperaments in this subgroup.
	* Third-octave: 121172^-11 = 2057/2048 and (27/17)^3/4 = 19683/19652. The former is called the Blume comma, and it is obscure despite being both accurate and low-complexity.		* Third-octave: 121172^-11 = 2057/2048 and (27/17)^3/4 = 19683/19652. The former is called the Blume comma, and it is obscure despite being both accurate and low-complexity.
	* Quarter-octave: add (19/16)^4/2=130321/131072.		* Quarter-octave: add (19/16)^4/2=130321/131072. This is one of very few useful commas dividing the octave into four equal parts.
	The different versions of Intergan may be defined by the sheer number of intervals it maps to the ~19/16 interval: 19/16, 25/21, 31/26, 32/27, 37/31, 44/37, and the otonal semifourth stack 26:31:37:44.		The different versions of Intergan may be defined by the sheer number of intervals it maps to the ~19/16 interval: 19/16, 25/21, 31/26, 32/27, 37/31, 44/37, and the otonal semifourth stack 26:31:37:44.

	{{Navbox regtemp}}		{{Navbox regtemp}}
	{{Cat\|temperaments}}		{{Cat\|temperaments}}

Ground at 02:32, 12 July 2026

2026-07-12T02:32:22Z

← Older revision		Revision as of 02:32, 12 July 2026
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	* The most basic version is 2.3.11.13/5.17.19 5&24. 22/17 is equated to 13/10 and 22/19 is equated to 15/13. It contains the 16:19:22 sub-diminished triad.		* The most basic version is 2.3.11.13/5.17.19 5&24. 22/17 is equated to 13/10 and 22/19 is equated to 15/13. It contains the 16:19:22 sub-diminished triad.
	* Tempering out 81/80 is somewhat obvious, resulting in 2.3.5.11.13.17.19 5&24.		* Tempering out 81/80 is somewhat obvious, resulting in 2.3.5.11.13.17.19 5&24.
	* Primes 31 and 37 are also obvious to add, equating 37/32 to 22/19 and 37/31 to 19/16, resulting in 2.3.5.11.13.17.19.31.37 5&24. The 2.31.37 restriction is highly accurate in 43edo.		* Primes 31 and 37 are also obvious to add, equating 37/32 to 22/19 and 37/31 to 19/16, resulting in 2.3.5.11.13.17.19.31.37 5&24. 67edo is roughly optimal due to its very mild meantone fourth being split in half. The 2.31.37 restriction is highly accurate in 43edo.
	* Primes 7, 23, and 29 are all more difficult to map due to their complexity.		* Primes 7, 23, and 29 are all more difficult to map due to their complexity, although their difficulty is in ascending order. They are more easily reached by splitting the semifourth generator in half in a manner similar to Negri.

	Like other important temperaments by [[User:Ground]], it may be understood as a relatively low-complexity rank-3 for greater flexibility. 81/80 is an obvious comma to temper out as seen in the standard rank-2 version, but leaving it in creates a significant amount of generator complexity. With this in mind, rank-3 Intergan is constructed as 2.3.5.11.13.17.19.31.37 5&9&24 with the following comma list:		Like other important temperaments by [[User:Ground]], it may be understood as a relatively low-complexity rank-3 for greater flexibility. 81/80 is an obvious comma to temper out as seen in the standard rank-2 version, but leaving it in creates a significant amount of generator complexity. With this in mind, rank-3 Intergan is constructed as 2.3.5.11.13.17.19.31.37 5&9&24 with the following comma list:
	* 4/3/(15/13)^2 = 676/675		* 4/3 / (15/13)^2 = 676/675
	* 22/19/(15/13) = 286/285		* 22/19 / (15/13) = 286/285
	* 13/10/(22/17) = 221/220		* 13/10 / (22/17) = 221/220
	* 16/15/(17/16) = 256/255		* 16/15 / (17/16) = 256/255
	* 37/32/(15/13) = 481/480		* 37/32 / (15/13) = 481/480
	* 37/31/(19/16) = 592/589		* 37/31 / (19/16) = 592/589
	It seems obvious to add (22/17)^2/(5/3) = 1452/1445, but that's equivalent to adding 81/80, as is 5/4/(19/17)^2 = 1445/1444.		It seems obvious to add (22/17)^2/(5/3) = 1452/1445, but that's equivalent to adding 81/80, as is 5/4/(19/17)^2 = 1445/1444.
	{\| class="wikitable"		{\| class="wikitable"
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	\|}		\|}
	Being so close to 24edo means that 1/2, 1/3, 1/4, 1/6, and 1/8-octave temperaments are all available. 12edo's accuracy provides many ways to temper 2.3.17.19, but this requires more effort to produce a temperament that isn't virtually 12edo. Intergan's interseptimals are one option. The rank-3 temperament should be attached to one of numerous often multi-period temperaments in this subgroup.		Being so close to 24edo means that 1/2, 1/3, 1/4, 1/6, and 1/8-octave temperaments are all available. 12edo's accuracy provides many ways to temper 2.3.17.19, but this requires more effort to produce a temperament that isn't virtually 12edo. Intergan's interseptimals are one option. The rank-3 temperament should be attached to one of numerous often multi-period temperaments in this subgroup.
	* Third-octave: 121172^-11 = 2057/2048 and (27/17)^3/4 = 19683/19652.		* Third-octave: 121172^-11 = 2057/2048 and (27/17)^3/4 = 19683/19652. The former is called the Blume comma, and it is obscure despite being both accurate and low-complexity.
	* Quarter-octave: add (19/16)^4/2=130321/131072		* Quarter-octave: add (19/16)^4/2=130321/131072.
	The different versions of Intergan may be defined by the sheer number of intervals it maps to the ~19/16 interval: 19/16, 25/21, 31/26, 32/27, 37/31, 44/37, and the otonal semifourth stack 26:31:37:44.		The different versions of Intergan may be defined by the sheer number of intervals it maps to the ~19/16 interval: 19/16, 25/21, 31/26, 32/27, 37/31, 44/37, and the otonal semifourth stack 26:31:37:44.

	{{Navbox regtemp}}		{{Navbox regtemp}}
	{{Cat\|temperaments}}		{{Cat\|temperaments}}

Hkm at 00:35, 12 July 2026

2026-07-12T00:35:33Z

← Older revision		Revision as of 00:35, 12 July 2026
Line 1:		Line 1:
	'''Intergan''' is a set of temperaments related to 24edo semiquartal. The term is short for "interseptimal Ganassi, after 12edo's 2.3.17.19 Ganassi temperament. It seeks to cover all of the notable concordances generated by a meantone-range semifourth.		'''Intergan''' is a set of temperaments related to 24edo semiquartal. The term is short for "interseptimal Ganassi," after 12edo's 2.3.17.19 Ganassi temperament. It seeks to cover all of the notable concordances generated by a meantone-range semifourth.

	* The most basic version is 2.3.11.13/5.17.19 5&24. 22/17 is equated to 13/10 and 22/19 is equated to 15/13. It contains the 16:19:22 sub-diminished triad.		* The most basic version is 2.3.11.13/5.17.19 5&24. 22/17 is equated to 13/10 and 22/19 is equated to 15/13. It contains the 16:19:22 sub-diminished triad.