Dicot family: Difference between revisions

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=== Overview to extensions ===

The ~~dicot~~ comma~~, 25/24, factors into~~ the ~~7-limit as (~~[[~~49/48~~]])([[~~50/49~~]]). ~~Since [[49/48]] is the difference between [[8/7]] and~~ [[7/6]], ~~and [[50/49]] is the difference between~~ [[7/5]] ~~and [[10/7]], it makes the most sense to temper out both of them~~, ~~leading to decimal. In the 11-limit, we have the identity 25/24=([[45/44]])(~~[[55/54]]), ~~but also 50/49=([[99/98]])(~~[[~~100~~/99]]), ~~so it makes sense to temper out~~ all ~~of them. That interestingly leads to a rank-1 temperament,~~ the ~~14c val of [[14edo]]. However, since dicot is an exotemperament anyway, the other extensions should also be considered~~.

==== 7-limit extensions ====

The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which [[7-limit]] family member we are looking at. Septimal dicot adds [[36/35]], flattie adds [[21/20]], sharpie adds [[28/27]], and dichotic adds [[64/63]], all retaining the same period and generator.

The ~~second~~ comma of the [[~~normal lists|normal comma list~~]] ~~defines which~~ [[~~7-limit~~]] ~~family member we are looking at~~. ~~Septimal dicot adds~~ [[36/35]]~~, flattie adds~~ [[21/20]]~~, sharpie adds~~ [[28/27]], and ~~dichotic adds~~ [[64/63]]~~, all retaining~~ the ~~same period~~ and ~~generator.~~

The dicot comma, 25/24, factors into the 7-limit as ([[49/48]])⋅([[50/49]]). Since [[49/48]] is the difference between [[8/7]] and [[7/6]], and [[50/49]] is the difference between [[7/5]] and [[10/7]], it makes sense to extend dicot to temper them all out, leading to decimal, a weak extension where the octave and twelfth are split in halves. Other weak extensions include sidi, which adds [[245/243]], and jamesbond, which adds [[16/15]]. Here sidi uses 14/9 as a generator, with two of them making up the combined [[5/2]][[~]][[12/5]] neutral tenth. Jamesbond has a period of 1/7 octave, and uses an approximate 15/14 as generator.

~~Decimal adds~~ [[49/48]], sidi adds [[245/243]], and jamesbond adds [[16/15]]. Here ~~decimal divides the [[period]] to a [[sqrt(2)|semi-octave]], and~~ sidi uses 14/9 as a generator, with two of them making up the combined 5/2~12/5 neutral tenth. Jamesbond has a period of 1/7 octave, and uses an approximate 15/14 as generator.

Temperaments discussed elsewhere are:

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* ''[[Jamesbond]]'' → [[7th-octave temperaments #Jamesbond|7th-octave temperaments]]

The rest are considered below.

The rest are considered in each sections below.

==== Subgroup extensions ====

In the 11-limit, we have the identity 25/24 = ([[45/44]])⋅([[55/54]]), so it makes sense to temper out all of them. This leads to the very natural subgroup temperament where [[11/9]]~[[27/22]] is mapped to the neutral third. As such, this is also the path that most of the septimal extensions take to get their 11-limit versions.

=== 2.3.5.11 subgroup ===

~~The 2.3.5.11-subgroup extension maps [[11/9]]~[[27/22]] to the neutral third. As such, it is related to most of the septimal extensions.~~

Subgroup: 2.3.5.11

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Decimal tempers out 49/48 and [[50/49]], and has a semi-octave period for 7/5~10/7 and a hemitwelfth generator for 7/4~12/7. Its ploidacot is diploid dicot. [[10edo]] makes for a good tuning, from which it derives its name. [[14edo]] in the 14c val and [[24edo]] in the 24c val are also among the possibilities.

Decimal can be extended to the 11-limit by the usual path of tempering out 45/44 and 55/54. There is an alternative due to the identity 50/49 = ([[99/98]])⋅([[100/99]]), in which case it also tempers out 33/32. The two mappings meet at the 14c val of [[14edo]].

[[Subgroup]]: 2.3.5.7

@@ Line 32: / Line 32: @@
 === Overview to extensions ===
-The dicot comma, 25/24, factors into the 7-limit as ([[49/48]])([[50/49]]). Since [[49/48]] is the difference between [[8/7]] and [[7/6]], and [[50/49]] is the difference between [[7/5]] and [[10/7]], it makes the most sense to temper out both of them, leading to decimal. In the 11-limit, we have the identity 25/24=([[45/44]])([[55/54]]), but also 50/49=([[99/98]])([[100/99]]), so it makes sense to temper out all of them. That interestingly leads to a rank-1 temperament, the 14c val of [[14edo]]. However, since dicot is an exotemperament anyway, the other extensions should also be considered.
+==== 7-limit extensions ====
+The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which [[7-limit]] family member we are looking at. Septimal dicot adds [[36/35]], flattie adds [[21/20]], sharpie adds [[28/27]], and dichotic adds [[64/63]], all retaining the same period and generator.
-The second comma of the [[normal lists|normal comma list]] defines which [[7-limit]] family member we are looking at. Septimal dicot adds [[36/35]], flattie adds [[21/20]], sharpie adds [[28/27]], and dichotic adds [[64/63]], all retaining the same period and generator.
+The dicot comma, 25/24, factors into the 7-limit as ([[49/48]])⋅([[50/49]]). Since [[49/48]] is the difference between [[8/7]] and [[7/6]], and [[50/49]] is the difference between [[7/5]] and [[10/7]], it makes sense to extend dicot to temper them all out, leading to decimal, a weak extension where the octave and twelfth are split in halves. Other weak extensions include sidi, which adds [[245/243]], and jamesbond, which adds [[16/15]]. Here sidi uses 14/9 as a generator, with two of them making up the combined [[5/2]][[~]][[12/5]] neutral tenth. Jamesbond has a period of 1/7 octave, and uses an approximate 15/14 as generator.
-Decimal adds [[49/48]], sidi adds [[245/243]], and jamesbond adds [[16/15]]. Here decimal divides the [[period]] to a [[sqrt(2)|semi-octave]], and sidi uses 14/9 as a generator, with two of them making up the combined 5/2~12/5 neutral tenth. Jamesbond has a period of 1/7 octave, and uses an approximate 15/14 as generator.
 Temperaments discussed elsewhere are:
@@ Line 42: / Line 41: @@
 * ''[[Jamesbond]]'' → [[7th-octave temperaments #Jamesbond|7th-octave temperaments]]
-The rest are considered below.
+The rest are considered in each sections below.
+==== Subgroup extensions ====
+In the 11-limit, we have the identity 25/24 = ([[45/44]])⋅([[55/54]]), so it makes sense to temper out all of them. This leads to the very natural subgroup temperament where [[11/9]]~[[27/22]] is mapped to the neutral third. As such, this is also the path that most of the septimal extensions take to get their 11-limit versions.
 === 2.3.5.11 subgroup ===
-The 2.3.5.11-subgroup extension maps [[11/9]]~[[27/22]] to the neutral third. As such, it is related to most of the septimal extensions.
 Subgroup: 2.3.5.11
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 Decimal tempers out 49/48 and [[50/49]], and has a semi-octave period for 7/5~10/7 and a hemitwelfth generator for 7/4~12/7. Its ploidacot is diploid dicot. [[10edo]] makes for a good tuning, from which it derives its name. [[14edo]] in the 14c val and [[24edo]] in the 24c val are also among the possibilities.
+Decimal can be extended to the 11-limit by the usual path of tempering out 45/44 and 55/54. There is an alternative due to the identity 50/49 = ([[99/98]])⋅([[100/99]]), in which case it also tempers out 33/32. The two mappings meet at the 14c val of [[14edo]].
 [[Subgroup]]: 2.3.5.7