Gammic family - Revision history

FloraC at 05:14, 23 June 2026

2026-06-23T05:14:57Z

← Older revision		Revision as of 05:14, 23 June 2026
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	==== Subgroup extensions ====		==== Subgroup extensions ====
	Gammic also naturally extends with the [[17/1\|17th harmonic]], as is given in [[#Subgroup extensions]].		Gammic also naturally extends with the [[17/1\|17th harmonic]], as is given in [[#Subgroup extensions_2\|#Subgroup extensions]].

	== Septimal gammic ==		== Septimal gammic ==

FloraC: On the subgroup extension

2026-06-23T05:13:49Z

On the subgroup extension

← Older revision		Revision as of 05:13, 23 June 2026
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	=== Overview to extensions ===		=== Overview to extensions ===
			==== 7-limit extensions ====
	Because 171 is such a strong [[7-limit]] system, it is well motivated to extend gammic to the 7-limit. This we may do by adding [[4375/4374]] to the comma list. 96 gammic generators finally reach 7, which is a long way to go compared to the 39 generator steps of [[pontiac]]. If someone wants to make the trip, a 103-note generator chain is possible.		Because 171 is such a strong [[7-limit]] system, it is well motivated to extend gammic to the 7-limit. This we may do by adding [[4375/4374]] to the comma list. 96 gammic generators finally reach 7, which is a long way to go compared to the 39 generator steps of [[pontiac]]. If someone wants to make the trip, a 103-note generator chain is possible.

			==== Subgroup extensions ====
			Gammic also naturally extends with the [[17/1\|17th harmonic]], as is given in [[#Subgroup extensions]].

	== Septimal gammic ==		== Septimal gammic ==
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	== Neptune ==		== Neptune ==
	A more interesting extension is to neptune, which divides an octave plus a gammic generator in half, to get a 10/7 generator. Neptune adds [[2401/2400]] to the gammic comma, and may be described as the 68&~~amp;~~171 temperament. The generator chain goes merrily on, stacking one 10/7 over another, until after eighteen generator steps 6/5 (up nine octaves) is reached. Then in succession we get 12/7, the neutral third, 7/4 and 5/4. Two neutral thirds then gives a fifth, and these intervals with their inverses are the full set of [[7-odd-limit]] consonances. [[171edo]] makes a good tuning, and we can also choose to make any of the consonances besides 7/5 and 10/7 just, including the fifth, which gives a tuning extending [[Carlos Gamma]].		A more interesting extension is to neptune, which divides an octave plus a gammic generator in half, to get a 10/7 generator. Neptune adds [[2401/2400]] to the gammic comma, and may be described as the {{nowrap\| 68 & 171 }} temperament. The generator chain goes merrily on, stacking one 10/7 over another, until after eighteen generator steps [[6/5]] (up nine octaves) is reached. Then in succession we get [[12/7]], the neutral third, [[7/4]] and [[5/4]]. Two neutral thirds then gives a fifth, and these intervals with their inverses are the full set of [[7-odd-limit]] consonances. [[171edo]] makes a good tuning, and we can also choose to make any of the consonances besides 7/5 and 10/7 just, including the fifth, which gives a tuning extending [[Carlos Gamma]].

	Adding 385/384 or 1375/1372 to the list of commas allows for an extension to the [[11-limit]], where (7/5)<sup>3</sup> equates to 11/4.		Adding 385/384 or 1375/1372 to the list of commas allows for an extension to the [[11-limit]], where (7/5)<sup>3</sup> equates to 11/4.

FloraC: Re-organize

2026-06-23T05:08:26Z

Re-organize

@@ Line 1: / Line 1: @@
 {{Technical data page}}
-The [[Carlos Gamma]] rank-1 temperament divides 3/2 into 20 equal parts, 11 of which give a 5/4. This is closely related to the rank-2 microtemperament tempering out {{monzo| -29 -11 20 }}, the [[gammic comma]]. This temperament, '''gammic''', takes 11 [[generator]] steps to reach 5/4, and 20 to reach 3/2. The generator in question is 1990656/1953125 = {{monzo| 13 5 -9 }}, which when suitably tempered is very close to 5/171 octaves, which makes for an ideal gammic tuning. As a 5-limit temperament supported by [[171edo]], [[Schismatic family|schismatic]] temperament makes for a natural comparison. Schismic, tempering out {{monzo| -15 8 1 }}, the [[schisma]], is plainly much less complex than gammic, but people seeking the exotic might prefer gammic even so. The 34-note mos is interesting, being a 1L 33s refinement of the [[34edo]] tuning. Of course gammic can be tuned to 34, which makes the two equivalent, and would rather remove the point of Carlos Gamma if used for it.
+The '''gammic family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[gammic comma]] ({{monzo|legend=1| -29 -11 20 }}), a [[5-limit]] comma of about 4.77 cents in size.
-Because 171 is such a strong [[7-limit]] system, it is natural to extend gammic to the 7-limit. This we may do by adding [[4375/4374]] to the comma list. 96 gammic generators finally reach 7, which is a long way to go compared to the 39 generator steps of pontiac. If someone wants to make the trip, a 103-note mos is possible.
+== Gammic ==
 [[Subgroup]]: 2.3.5
@@ Line 22: / Line 26: @@
 [[Badness]] (Sintel): 2.06
-=== 2.3.5.17 subgroup ===
+=== Overview to extensions ===
-The interval of 3 generators represents 1/3 of [[6/5]], which is very close to [[17/16]], with the comma between 6/5 and (17/16)<sup>3</sup> being [[24576/24565]] ([[S-expression|S16/S17]]). This then naturally interprets the generator as [[51/50]] with two generators representing [[25/24]], tempering out [[15625/15606]] (S49⋅S50<sup>2</sup>).
+Because 171 is such a strong [[7-limit]] system, it is well motivated to extend gammic to the 7-limit. This we may do by adding [[4375/4374]] to the comma list. 96 gammic generators finally reach 7, which is a long way to go compared to the 39 generator steps of [[pontiac]]. If someone wants to make the trip, a 103-note generator chain is possible.
 == Septimal gammic ==
@@ Line 328: / Line 317: @@
 Badness (Sintel): 1.38
 [[Category:Temperament families]]
 [[Category:Gammic family| ]] <!-- main article -->
 [[Category:Rank 2]]

FloraC: Switch to Sintel's badness, WE & CWE tunings

2026-06-23T05:00:02Z

Switch to Sintel's badness, WE & CWE tunings

Show changes

FloraC: Units, misc. cleanup, - redundant category

2026-06-22T09:30:17Z

Units, misc. cleanup, - redundant category

Show changes

Overthink: /* Neptune */ depends on whether you consider the 7- or 9-odd-limit as consonances

2026-05-24T17:55:39Z

Neptune: depends on whether you consider the 7- or 9-odd-limit as consonances

← Older revision		Revision as of 17:55, 24 May 2026
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	== Neptune ==		== Neptune ==
	A more interesting extension is to neptune, which divides an octave plus a gammic generator in half, to get a 10/7 generator. Neptune adds [[2401/2400]] to the gammic comma, and may be described as the 68&171 temperament. The generator chain goes merrily on, stacking one 10/7 over another, until after eighteen generator steps 6/5 (up nine octaves) is reached. Then in succession we get 12/7, the neutral third, 7/4 and 5/4. Two neutral thirds then gives a fifth, and these intervals with their inverses are the full set of ~~septimal~~ consonances. [[171edo]] makes a good tuning, and we can also choose to make any of the consonances besides 7/5 and 10/7 just, including the fifth, which gives a tuning extending [[Carlos Gamma]].		A more interesting extension is to neptune, which divides an octave plus a gammic generator in half, to get a 10/7 generator. Neptune adds [[2401/2400]] to the gammic comma, and may be described as the 68&171 temperament. The generator chain goes merrily on, stacking one 10/7 over another, until after eighteen generator steps 6/5 (up nine octaves) is reached. Then in succession we get 12/7, the neutral third, 7/4 and 5/4. Two neutral thirds then gives a fifth, and these intervals with their inverses are the full set of [[7-odd-limit]] consonances. [[171edo]] makes a good tuning, and we can also choose to make any of the consonances besides 7/5 and 10/7 just, including the fifth, which gives a tuning extending [[Carlos Gamma]].

	Adding 385/384 or 1375/1372 to the list of commas allows for an extension to the [[11-limit]], where (7/5)<sup>3</sup> equates to 11/4.		Adding 385/384 or 1375/1372 to the list of commas allows for an extension to the [[11-limit]], where (7/5)<sup>3</sup> equates to 11/4.

Lériendil: the 7-limit header should still be there because it has subgroup temps under it

2026-05-24T06:23:50Z

the 7-limit header should still be there because it has subgroup temps under it

← Older revision		Revision as of 06:23, 24 May 2026
Line 145:		Line 145:
	Adding 385/384 or 1375/1372 to the list of commas allows for an extension to the [[11-limit]], where (7/5)<sup>3</sup> equates to 11/4.		Adding 385/384 or 1375/1372 to the list of commas allows for an extension to the [[11-limit]], where (7/5)<sup>3</sup> equates to 11/4.

	[[Gene Ward Smith]] once described [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_6001.html neptune as an analog of miracle].		[[Gene Ward Smith]] once described [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_6001.html neptune as an analog of miracle].

			=== 7-limit ===
	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7

Overthink: /* Neptune */ remove extraneous header

2026-05-24T00:57:03Z

Neptune: remove extraneous header

← Older revision		Revision as of 00:57, 24 May 2026
Line 145:		Line 145:
	Adding 385/384 or 1375/1372 to the list of commas allows for an extension to the [[11-limit]], where (7/5)<sup>3</sup> equates to 11/4.		Adding 385/384 or 1375/1372 to the list of commas allows for an extension to the [[11-limit]], where (7/5)<sup>3</sup> equates to 11/4.

	[[Gene Ward Smith]] once described [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_6001.html neptune as an analog of miracle].		[[Gene Ward Smith]] once described [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_6001.html neptune as an analog of miracle].

	~~=== 7-limit ===~~
	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7

FloraC: Text replacement - "Category:Temperament families↵Category:Pages with mostly numerical content" to "Category:Temperament families"

2025-11-27T12:23:48Z

Text replacement - "Category:Temperament families↵Category:Pages with mostly numerical content" to "Category:Temperament families"

← Older revision		Revision as of 12:23, 27 November 2025
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	[[Category:Temperament families]]		[[Category:Temperament families]]
	~~[[Category:Pages with mostly numerical content]]~~
	[[Category:Gammic family\| ]] <!-- main article -->		[[Category:Gammic family\| ]] <!-- main article -->
	[[Category:Gammic\| ]] <!-- key article -->		[[Category:Gammic\| ]] <!-- key article -->
	[[Category:Rank 2]]		[[Category:Rank 2]]

Lériendil: /* 2.3.5.7.17 subgroup */

2025-09-18T18:45:48Z

2.3.5.7.17 subgroup

← Older revision		Revision as of 18:45, 18 September 2025
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	==== 2.3.5.7.17 subgroup ====		==== 2.3.5.7.17 subgroup ====
	Extending 2.3.5.17 gammic via neptune, we find that both 2401/2400 ({S\|49}) and 2500/2499 (S50) are tempered out; their product, 1225/1224 (S35) is therefore also tempered out.		Extending 2.3.5.17 gammic via neptune, we find that both 2401/2400 ({{S\|49}}) and 2500/2499 (S50) are tempered out; their product, 1225/1224 (S35) is therefore also tempered out.

	[[Subgroup]]: 2.3.5.7.17		[[Subgroup]]: 2.3.5.7.17