Breedsmic temperaments: Difference between revisions

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* [[7-odd-limit|7-]] and [[9-odd-limit]] minimax: ~49/40 = {{monzo| 1/5 0 1/25 }}

: {{monzo list| 1 0 0 0 | 7/5 0 2/25 0 | 0 0 1 0 | 8/5 0 13/25 0 }}

: [[Eigenmonzo basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.5

: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5

[[Algebraic generator]]: (2 + sqrt(2))/2

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== Unthirds ==

~~The generator for~~ unthirds temperament is ~~undecimal~~ major third, 14/11.

Despite the complexity of its mapping, unthirds is an important temperament to the structure of the [[11-limit]]; this is hinted at by unthirds' representation as the [[72edo|72]] & [[311edo|311]] temperament, the [[Temperament merging|join]] of two tuning systems well-known for their high accuracy in the 11-limit and [[41-limit]] respectively. It is generated by the interval of [[14/11]] ('''un'''decimal major '''third''', hence the name) tuned less than a cent flat, and the 23-note [[MOS]] this interval generates serves as a well temperament of, of all things, [[23edo]]. The 49-note MOS is needed to access the 3rd, 5th, 7th, and 11th harmonics, however.

The commas it tempers out include the [[breedsma]] (2401/2400), the [[lehmerisma]] (3025/3024), the [[pine comma]] (4000/3993), the [[unisquary comma]] (12005/11979), the [[argyria]] (41503/41472), and 42875/42768, all of which appear individually in various 11-limit systems. It is also notable that there is a [[restriction]] of the temperament to the 2.5/3.7/3.11/3 [[fractional subgroup]] that tempers out 3025/3024 and 12005/11979, which is of considerably less complexity, and which is shared with [[sqrtphi]] (whose generator is tuned flat of 72edo's).

[[Subgroup]]: 2.3.5.7

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Comma list: 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224

Mapping: {{mapping|1 -25 -16 -13 -26 -6 -11 | 0 74 51 44 82 27 42 }}

Mapping: {{mapping| 1 -25 -16 -13 -26 -6 -11 | 0 74 51 44 82 27 42 }}

Optimal tunings:

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Comma list: 616/615, 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224

Mapping: {{mapping|1 -25 -16 -13 -26 -6 -11 5 | 0 74 51 44 82 27 42 1}}

Mapping: {{mapping| 1 -25 -16 -13 -26 -6 -11 5 | 0 74 51 44 82 27 42 1 }}

Optimal tunings:

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== Hemigoldis ==

: ''For the 5-limit version, see [[Diaschismic–gothmic equivalence continuum #Goldis]].''

Though fairly complex in the [[7-limit]], hemigoldis does a lot better in badness metrics than pure 5-limit goldis, and yet again has many possible extensions to other primes. For example, two periods minus six generators yields a "tetracot second" which can be interpreted as ~[[21/19]] to add prime 19 or perhaps more accurately ~[[31/28]] to add prime 7, or even simply as ~[[32/29]] to add prime 29, though the other two have the benefit of clearly connecting to the 7-limit representation. Note that again [[89edo]] is a possible tuning for combining it with flat nestoria and not appearing in the optimal ET sequence.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2401/2400, 549755813888/533935546875

: mapping generators: ~2, ~7/4

[[Optimal tuning]] ([[CWE]]): ~2 = 1200.000, ~7/4 = 970.690

{{Optimal ET sequence|legend=1| 21, 47b, 68, 157, 382bccd, 529bccd }}

[[Badness]] (Sintel): 4.40

== Surmarvelpyth ==

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[[Category:Temperament collections]]

[[Category:Pages with mostly numerical content]]

[[Category:Breedsmic temperaments| ]]

[[Category:Breed| ]]

[[Category:Rank 2]]

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 * [[7-odd-limit|7-]] and [[9-odd-limit]] minimax: ~49/40 = {{monzo| 1/5 0 1/25 }}
 : {{monzo list| 1 0 0 0 | 7/5 0 2/25 0 | 0 0 1 0 | 8/5 0 13/25 0 }}
-: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
 [[Algebraic generator]]: (2 + sqrt(2))/2
@@ Line 586: / Line 586: @@
 == Unthirds ==
-The generator for unthirds temperament is undecimal major third, 14/11.
+Despite the complexity of its mapping, unthirds is an important temperament to the structure of the [[11-limit]]; this is hinted at by unthirds' representation as the [[72edo|72]] & [[311edo|311]] temperament, the [[Temperament merging|join]] of two tuning systems well-known for their high accuracy in the 11-limit and [[41-limit]] respectively. It is generated by the interval of [[14/11]] ('''un'''decimal major '''third''', hence the name) tuned less than a cent flat, and the 23-note [[MOS]] this interval generates serves as a well temperament of, of all things, [[23edo]]. The 49-note MOS is needed to access the 3rd, 5th, 7th, and 11th harmonics, however.
+The commas it tempers out include the [[breedsma]] (2401/2400), the [[lehmerisma]] (3025/3024), the [[pine comma]] (4000/3993), the [[unisquary comma]] (12005/11979), the [[argyria]] (41503/41472), and 42875/42768, all of which appear individually in various 11-limit systems. It is also notable that there is a [[restriction]] of the temperament to the 2.5/3.7/3.11/3 [[fractional subgroup]] that tempers out 3025/3024 and 12005/11979, which is of considerably less complexity, and which is shared with [[sqrtphi]] (whose generator is tuned flat of 72edo's).
 [[Subgroup]]: 2.3.5.7
@@ Line 1,106: / Line 1,108: @@
 Comma list: 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224
-Mapping: {{mapping|1 -25 -16 -13 -26 -6 -11 | 0 74 51 44 82 27 42 }}
+Mapping: {{mapping| 1 -25 -16 -13 -26 -6 -11 | 0 74 51 44 82 27 42 }}
 Optimal tunings:
@@ Line 1,121: / Line 1,123: @@
 Comma list: 616/615, 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224
-Mapping: {{mapping|1 -25 -16 -13 -26 -6 -11 5 | 0 74 51 44 82 27 42 1}}
+Mapping: {{mapping| 1 -25 -16 -13 -26 -6 -11 5 | 0 74 51 44 82 27 42 1 }}
 Optimal tunings:
@@ Line 1,128: / Line 1,130: @@
 {{Optimal ET sequence|legend=0| 103, 167, 270 }}
+== Hemigoldis ==
+: ''For the 5-limit version, see [[Diaschismic–gothmic equivalence continuum #Goldis]].''
+Though fairly complex in the [[7-limit]], hemigoldis does a lot better in badness metrics than pure 5-limit goldis, and yet again has many possible extensions to other primes. For example, two periods minus six generators yields a "tetracot second" which can be interpreted as ~[[21/19]] to add prime 19 or perhaps more accurately ~[[31/28]] to add prime 7, or even simply as ~[[32/29]] to add prime 29, though the other two have the benefit of clearly connecting to the 7-limit representation. Note that again [[89edo]] is a possible tuning for combining it with flat nestoria and not appearing in the optimal ET sequence.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 2401/2400, 549755813888/533935546875
+{{Mapping|legend=1| 1 21 -9 2 | 0 -24 14 1 }}
+: mapping generators: ~2, ~7/4
+[[Optimal tuning]] ([[CWE]]): ~2 = 1200.000, ~7/4 = 970.690
+{{Optimal ET sequence|legend=1| 21, 47b, 68, 157, 382bccd, 529bccd }}
+[[Badness]] (Sintel): 4.40
 == Surmarvelpyth ==
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 [[Category:Temperament collections]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Breedsmic temperaments| ]] <!-- main article -->
 [[Category:Breed| ]] <!-- key article -->
 [[Category:Rank 2]]