No-fives subgroup temperaments: Difference between revisions

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Io is a very low-complexity temperament which tempers out the undecimal quartertone [[33/32]]. This equates very different intervals (for example, the generator itself represents both [[3/2]] and [[16/11]]), and as such some consider it to be an [[exotemperament]]. It has an extremely wide generator range, but the most accurate tunings are generally inside the range of [[flattone]] temperament.

The name ''io'' was coined by ~~[[User:CompactStar~~|CompactStar]] in 2024 based on the [[Kite's color notation|color name]] ''ilo'', prior to which it could only be termed as "undecimal temperament" with 33/32 being known as the undecimal comma.

The name ''io'' was coined by {{u|CompactStar}} in 2024 based on the [[Kite's color notation|color name]] ''ilo'', prior to which it could only be termed as "undecimal temperament" with 33/32 being known as the undecimal comma.

[[Subgroup]]: 2.3.11

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=== Alphaxenean ===

Alphaxenean tempers out the [[Alpharabian comma]] and equates a stack of four undecimal quartertones with the [[9/8|Pythagorean whole tone]]. It ~~also divides~~ the [[~~octave~~]] ~~into~~ two.

Alphaxenean tempers out the [[Alpharabian comma]], and equates a stack of four undecimal quartertones with the [[9/8|Pythagorean whole tone]]. It has a [[semi-octave]] period, and the generator can be taken to be the aforementioned undecimal quartertone or that plus a period, which represents ~[[16/11]]. The interval class of a perfect fifth is found by two generator steps and a period.

[[Subgroup]]: 2.3.11

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=== Octatonic ===

[[12/11]] is very close to 1 step of 8edo, and hence this temperament tempers out the [[undecimal octatonic comma]], the difference between a stack of eight 12/11's and the octave.

[[12/11]] is very close to one step of 8edo, and hence this temperament tempers out the [[undecimal octatonic comma]], the difference between a stack of eight 12/11's and the octave.

[[Subgroup]]: 2.3.11

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[[Badness]] (Sintel): 0.515

=== Betaxenean ===

Betaxenean tempers out the [[Betarabian comma]]. It splits a [[3/1|perfect twelfth]] into two halves of [[512/297]].

[[Subgroup]]: 2.3.11

[[Comma list]]: 264627/262144

: mapping generators: ~2, ~512/297

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.5931{{c}}, ~512/297 = 950.6872{{c}}

: [[error map]]: {{val| +0.593 -0.581 -0.791 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~512/297 = 950.1972{{c}}

: error map: {{val| 0.000 -1.561 -2.699 }}

{{Optimal ET sequence|legend=1| 5, 14e, 19, 24, 125, 149, 173, 197e, 221e, 466bee, 687beee }}

[[Badness]] (Sintel): 0.709

=== Infraug ===

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Line 1,318:

[[Badness]] (Sintel): 0.317

=== Jackpot ===

Named by {{u|CompactStar}} in 2024, jackpot tempers out the [[7-29-comma]] in 2.3.29, setting 32/29 to one step of [[7edo]] with an independent generator for the [[3/1|3rd harmonic]]. The name refers to the [[Temperament merging|EDO join]] of 7 & 77.

[[Subgroup]]: 2.3.29

[[Comma list]]: 17249876309/17179869184

: mapping generators: ~32/29, ~3

[[Optimal tuning]]s:

* [[WE]]: ~32/29 = 171.4434{{c}}, ~3/2 = 701.8515{{c}}

: [[error map]]: {{val| +0.103 -0.000 -0.503 }}

* [[CWE]]: ~32/29 = 171.4286{{c}}, ~3/2 = 701.8456{{c}}

: error map: {{val| 0.000 -0.109 -1.006 }}

{{Optimal ET sequence|legend=1| 7, 42, 49, 56, 63, 70, 77, 147, 224 }}

[[Badness]] (Sintel): 1.02

=== Sematology ===

@@ Line 861: / Line 861: @@
 Io is a very low-complexity temperament which tempers out the undecimal quartertone [[33/32]]. This equates very different intervals (for example, the generator itself represents both [[3/2]] and [[16/11]]), and as such some consider it to be an [[exotemperament]]. It has an extremely wide generator range, but the most accurate tunings are generally inside the range of [[flattone]] temperament.
-The name ''io'' was coined by [[User:CompactStar|CompactStar]] in 2024 based on the [[Kite's color notation|color name]] ''ilo'', prior to which it could only be termed as "undecimal temperament" with 33/32 being known as the undecimal comma.
+The name ''io'' was coined by {{u|CompactStar}} in 2024 based on the [[Kite's color notation|color name]] ''ilo'', prior to which it could only be termed as "undecimal temperament" with 33/32 being known as the undecimal comma.
 [[Subgroup]]: 2.3.11
@@ Line 881: / Line 881: @@
 === Alphaxenean ===
-Alphaxenean tempers out the [[Alpharabian comma]] and equates a stack of four undecimal quartertones with the [[9/8|Pythagorean whole tone]]. It also divides the [[octave]] into two.
+Alphaxenean tempers out the [[Alpharabian comma]], and equates a stack of four undecimal quartertones with the [[9/8|Pythagorean whole tone]]. It has a [[semi-octave]] period, and the generator can be taken to be the aforementioned undecimal quartertone or that plus a period, which represents ~[[16/11]]. The interval class of a perfect fifth is found by two generator steps and a period.
 [[Subgroup]]: 2.3.11
@@ Line 901: / Line 901: @@
 === Octatonic ===
-[[12/11]] is very close to 1 step of 8edo, and hence this temperament tempers out the [[undecimal octatonic comma]], the difference between a stack of eight 12/11's and the octave.
+[[12/11]] is very close to one step of 8edo, and hence this temperament tempers out the [[undecimal octatonic comma]], the difference between a stack of eight 12/11's and the octave.
 [[Subgroup]]: 2.3.11
@@ Line 919: / Line 919: @@
 [[Badness]] (Sintel): 0.515
+=== Betaxenean ===
+Betaxenean tempers out the [[Betarabian comma]]. It splits a [[3/1|perfect twelfth]] into two halves of [[512/297]].
+[[Subgroup]]: 2.3.11
+[[Comma list]]: 264627/262144
+{{Mapping|legend=2| 1 0 9 | 0 2 -7 }}
+: mapping generators: ~2, ~512/297
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.5931{{c}}, ~512/297 = 950.6872{{c}}
+: [[error map]]: {{val| +0.593 -0.581 -0.791 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~512/297 = 950.1972{{c}}
+: error map: {{val| 0.000 -1.561 -2.699 }}
+{{Optimal ET sequence|legend=1| 5, 14e, 19, 24, 125, 149, 173, 197e, 221e, 466bee, 687beee }}
+[[Badness]] (Sintel): 0.709
 === Infraug ===
@@ Line 1,298: / Line 1,318: @@
 [[Badness]] (Sintel): 0.317
+=== Jackpot ===
+Named by {{u|CompactStar}} in 2024, jackpot tempers out the [[7-29-comma]] in 2.3.29, setting 32/29 to one step of [[7edo]] with an independent generator for the [[3/1|3rd harmonic]]. The name refers to the [[Temperament merging|EDO join]] of 7 & 77.
+[[Subgroup]]: 2.3.29
+[[Comma list]]: 17249876309/17179869184
+{{Mapping|legend=2| 7 0 34 | 0 1 0 }}
+: mapping generators: ~32/29, ~3
+[[Optimal tuning]]s:
+* [[WE]]: ~32/29 = 171.4434{{c}}, ~3/2 = 701.8515{{c}}
+: [[error map]]: {{val| +0.103 -0.000 -0.503 }}
+* [[CWE]]: ~32/29 = 171.4286{{c}}, ~3/2 = 701.8456{{c}}
+: error map: {{val| 0.000 -0.109 -1.006 }}
+{{Optimal ET sequence|legend=1| 7, 42, 49, 56, 63, 70, 77, 147, 224 }}
+[[Badness]] (Sintel): 1.02
 === Sematology ===