Ragismic microtemperaments: Difference between revisions

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: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Enneadecal (5-limit)]].''

Enneadecal ~~temperament~~ tempers out the [[enneadeca]], {{monzo| -14 -19 19 }}, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen just minor thirds fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be ~25/24, ~27/25, ~10/9, ~5/4 or ~3/2. To this we may add possible 7-limit generators such as ~225/224, ~15/14 or ~9/7. Since enneadecal tempers out [[703125/702464]], the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)<sup>1/3</sup>. This is the interval needed to adjust the 1/3-comma meantone flat fifths and major thirds of [[19edo]] up to just ones. [[171edo]] is a good tuning for either the 5- or 7-limit, and [[494edo]] shows how to extend the temperament to the 11- or 13-limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use [[665edo]] for a tuning.

Enneadecal tempers out the [[enneadeca]], {{monzo| -14 -19 19 }}, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen [[6/5|just minor thirds]] fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be ~25/24, ~27/25, ~10/9, ~5/4 or ~3/2. To this we may add possible 7-limit generators such as ~225/224, ~15/14 or ~9/7. Since enneadecal tempers out [[703125/702464]], the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)<sup>1/3</sup>. This is the interval needed to adjust the 1/3-comma meantone flat fifths and major thirds of [[19edo]] up to just ones.

[[171edo]] is a good tuning for either the 5- or 7-limit, and [[494edo]] shows how to extend the temperament to the 11- or 13-limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use [[665edo]] for a tuning.

[[Subgroup]]: 2.3.5.7

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: ~~Mapping~~ generators: ~28/27, ~3

: mapping generators: ~28/27, ~3

[[Optimal tuning]] ([[~~CTE~~]]): ~28/27 = 63.1579{{c}}, ~3/2 = 701.~~9275~~{{c}} (~225/224 = 7.~~1907~~{{c}})

[[Optimal tuning]]s:

* [[WE]]: ~28/27 = 63.1599{{c}}, ~3/2 = 701.9027{{c}} (~225/224 = 7.1437{{c}})

: [[error map]]: {{val| +0.038 -0.014 -0.134 +0.080 }}

* [[CWE]]: ~28/27 = 63.1579{{c}}, ~3/2 = 701.9002{{c}} (~225/224 = 7.1634{{c}})

: error map: {{val| 0.000 -0.055 -0.203 +0.033 }}

{{Optimal ET sequence|legend=1| 19, …, 152, 171, 665, 836, 1007, 2185, 3192c }}

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Mapping: {{mapping| 19 0 14 -37 126 | 0 1 1 3 -2 }}

Optimal ~~tuning~~ (~~CTE~~): ~28/27 = 63.1579{{c}}, ~3/2 = 702.~~1483~~{{c}} (~225/224 = 7.~~4115~~{{c}})

Optimal tunings:

* WE: ~28/27 = 63.1431{{c}}, ~3/2 = 702.1956{{c}} (~225/224 = 7.6216{{c}})

* CWE: ~28/27 = 63.1579{{c}}, ~3/2 = 702.3164{{c}} (~225/224 = 7.5795{{c}})

{{Optimal ET sequence|legend=0| 19, 133d, 152, 323e, 475de, 627de }}

Badness (Sintel): 1.~~446~~

Badness (Sintel): 1.45

==== 13-limit ====

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Mapping: {{mapping| 19 0 14 -37 126 -20 | 0 1 1 3 -2 3 }}

Optimal ~~tuning~~ (~~CTE~~): ~28/27 = 63.1579{{c}}, ~3/2 = ~~701~~.~~9258~~{{c}} (~225/224 = 7.~~1890~~{{c}})

Optimal tunings:

* WE: ~28/27 = 63.1406{{c}}, ~3/2 = 702.0192{{c}} (~225/224 = 7.4730{{c}})

* CWE: ~28/27 = 63.1579{{c}}, ~3/2 = 702.1539{{c}} (~225/224 = 7.4171{{c}})

Badness (Sintel): 1.~~386~~

Badness (Sintel): 1.39

=== Hemienneadecal ===

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: mapping generators: ~55/54, ~3

Optimal ~~tuning~~ (~~CTE~~): ~55/54 = 31.5789{{c}}, ~3/2 = 701.~~9351~~{{c}} (~~~225~~/~~224~~ = 7.~~1983~~{{c}})

Optimal tunings:

* WE: ~55/54 = 31.5800{{c}}, ~3/2 = 701.9053{{c}} (~243/242 = 7.1448{{c}})

* CWE: ~55/54 = 31.5789{{c}}, ~3/2 = 701.9034{{c}} (~243/242 = 7.1666{{c}})

{{Optimal ET sequence|legend=0| 152, 342, 836, 1178, 2014, 3192ce, 5206ce }}

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Mapping: {{mapping| 38 0 28 -74 11 -281 | 0 1 1 3 2 7 }}

Optimal ~~tuning~~ (~~CTE~~): ~55/54 = 31.5789{{c}}, ~3/2 = ~~701~~.~~9955~~{{c}} (~~~225~~/~~224~~ = 7.~~2587~~{{c}})

Optimal tunings:

* WE: ~55/54 = 31.5785{{c}}, ~3/2 = 701.9995{{c}} (~243/242 = 7.2727{{c}})

* CWE: ~55/54 = 31.5789{{c}}, ~3/2 = 702.0053{{c}} (~243/242 = 7.2685{{c}})

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Mapping: {{mapping| 38 0 28 -74 11 502 | 0 1 1 3 2 -6 }}

Optimal ~~tuning~~ (~~CTE~~): ~55/54 = 31.5789{{c}}, ~3/2 = 701.~~9812~~{{c}} (~~~225~~/~~224~~ = 7.~~2444~~{{c}})

Optimal tunings:

* WE: ~55/54 = 31.5784{{c}}, ~3/2 = 701.9736{{c}} (~243/242 = 7.2493{{c}})

* CWE: ~55/54 = 31.5789{{c}}, ~3/2 = 701.9855{{c}} (~243/242 = 7.2487{{c}})

{{Optimal ET sequence|legend=0| 152, 342, 494, 1330, 1824, 2318d }}

Badness (Sintel): 1.~~256~~

Badness (Sintel): 1.26

==== Semihemienneadecal ====

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: mapping generators: ~55/54, ~429/250

Optimal ~~tuning~~ (~~CTE~~): ~55/54 = 31.5789{{c}}, ~429/250 = 935.~~1789~~{{c}} (~144/143 = 12.~~1895~~{{c}})

Optimal tunings:

* WE: ~55/54 = 31.5799{{c}}, ~429/250 = 935.1824{{c}} (~144/143 = 12.2152{{c}})

* CWE: ~55/54 = 31.5789{{c}}, ~429/250 = 935.1617{{c}} (~144/143 = 12.2067{{c}})

{{Optimal ET sequence|legend=0| 190, 304d, 494, 684, 1178, 2850, 4028ce }}

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Mapping: {{mapping| 19 3 17 -28 82 92 159 78 | 0 10 10 30 -6 -8 -30 1 }}

Optimal ~~tuning (CTE)~~: ~28/27 = 63.1579{{c}}, ~6545/5928 = 171.~~244~~{{c}}

Optimal tunings:

* WE: ~28/27 = 63.1582{{c}}, ~6545/5928 = 171.2448{{c}}

* CWE: ~28/27 = 63.1579{{c}}, ~6545/5928 = 171.2439{{c}}

Badness (Sintel): 3.~~153~~

Badness (Sintel): 3.15

== Semidimi ==

: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Semidimi]].''

The generator of semidimi ~~temperament~~ is a semi-diminished fourth interval tuned between 162/125 and 35/27. It tempers out 5-limit {{monzo| -12 -73 55 }} and 7-limit 3955078125/3954653486, as well as 4375/4374.

The generator of semidimi is a semi-diminished fourth interval tuned between 162/125 and 35/27. It tempers out 5-limit {{monzo| -12 -73 55 }} and 7-limit 3955078125/3954653486, as well as 4375/4374.

[[Subgroup]]: 2.3.5.7

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[[Comma list]]: 4375/4374, 3955078125/3954653486

: mapping generators: ~2, ~35/27

[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1200.0000{{c}}, ~35/27 = 449.1270{{c}}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0018{{c}}, ~35/27 = 449.1277{{c}}

: [[error map]]: {{val| +0.002 +0.031 -0.040 -0.012 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/27 = 449.1270{{c}}

: error map: {{val| 0.000 +0.030 -0.043 -0.015 }}

{{Optimal ET sequence|legend=1| 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }}

{{Optimal ET sequence|legend=1| 8d, …, 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }}

[[Badness]] (Sintel): 0.382

@@ Line 70: / Line 70: @@
 : ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Enneadecal (5-limit)]].''
-Enneadecal temperament tempers out the [[enneadeca]], {{monzo| -14 -19 19 }}, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen just minor thirds fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be ~25/24, ~27/25, ~10/9, ~5/4 or ~3/2. To this we may add possible 7-limit generators such as ~225/224, ~15/14 or ~9/7. Since enneadecal tempers out [[703125/702464]], the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)<sup>1/3</sup>. This is the interval needed to adjust the 1/3-comma meantone flat fifths and major thirds of [[19edo]] up to just ones. [[171edo]] is a good tuning for either the 5- or 7-limit, and [[494edo]] shows how to extend the temperament to the 11- or 13-limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use [[665edo]] for a tuning.
+Enneadecal tempers out the [[enneadeca]], {{monzo| -14 -19 19 }}, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen [[6/5|just minor thirds]] fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be ~25/24, ~27/25, ~10/9, ~5/4 or ~3/2. To this we may add possible 7-limit generators such as ~225/224, ~15/14 or ~9/7. Since enneadecal tempers out [[703125/702464]], the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)<sup>1/3</sup>. This is the interval needed to adjust the 1/3-comma meantone flat fifths and major thirds of [[19edo]] up to just ones.
+[[171edo]] is a good tuning for either the 5- or 7-limit, and [[494edo]] shows how to extend the temperament to the 11- or 13-limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use [[665edo]] for a tuning.
 [[Subgroup]]: 2.3.5.7
@@ Line 77: / Line 79: @@
 {{Mapping|legend=1| 19 0 14 -37 | 0 1 1 3 }}
-: Mapping generators: ~28/27, ~3
+: mapping generators: ~28/27, ~3
-[[Optimal tuning]] ([[CTE]]): ~28/27 = 63.1579{{c}}, ~3/2 = 701.9275{{c}} (~225/224 = 7.1907{{c}})
+[[Optimal tuning]]s:
+* [[WE]]: ~28/27 = 63.1599{{c}}, ~3/2 = 701.9027{{c}} (~225/224 = 7.1437{{c}})
+: [[error map]]: {{val| +0.038 -0.014 -0.134 +0.080 }}
+* [[CWE]]: ~28/27 = 63.1579{{c}}, ~3/2 = 701.9002{{c}} (~225/224 = 7.1634{{c}})
+: error map: {{val| 0.000 -0.055 -0.203 +0.033 }}
 {{Optimal ET sequence|legend=1| 19, …, 152, 171, 665, 836, 1007, 2185, 3192c }}
@@ Line 92: / Line 98: @@
 Mapping: {{mapping| 19 0 14 -37 126 | 0 1 1 3 -2 }}
-Optimal tuning (CTE): ~28/27 = 63.1579{{c}}, ~3/2 = 702.1483{{c}} (~225/224 = 7.4115{{c}})
+Optimal tunings:
+* WE: ~28/27 = 63.1431{{c}}, ~3/2 = 702.1956{{c}} (~225/224 = 7.6216{{c}})
+* CWE: ~28/27 = 63.1579{{c}}, ~3/2 = 702.3164{{c}} (~225/224 = 7.5795{{c}})
 {{Optimal ET sequence|legend=0| 19, 133d, 152, 323e, 475de, 627de }}
-Badness (Sintel): 1.446
+Badness (Sintel): 1.45
 ==== 13-limit ====
@@ Line 105: / Line 113: @@
 Mapping: {{mapping| 19 0 14 -37 126 -20 | 0 1 1 3 -2 3 }}
-Optimal tuning (CTE): ~28/27 = 63.1579{{c}}, ~3/2 = 701.9258{{c}} (~225/224 = 7.1890{{c}})
+Optimal tunings:
+* WE: ~28/27 = 63.1406{{c}}, ~3/2 = 702.0192{{c}} (~225/224 = 7.4730{{c}})
+* CWE: ~28/27 = 63.1579{{c}}, ~3/2 = 702.1539{{c}} (~225/224 = 7.4171{{c}})
 {{Optimal ET sequence|legend=0| 19, 133df, 152f, 323ef }}
-Badness (Sintel): 1.386
+Badness (Sintel): 1.39
 === Hemienneadecal ===
@@ Line 119: / Line 129: @@
 : mapping generators: ~55/54, ~3
-Optimal tuning (CTE): ~55/54 = 31.5789{{c}}, ~3/2 = 701.9351{{c}} (~225/224 = 7.1983{{c}})
+Optimal tunings:
+* WE: ~55/54 = 31.5800{{c}}, ~3/2 = 701.9053{{c}} (~243/242 = 7.1448{{c}})
+* CWE: ~55/54 = 31.5789{{c}}, ~3/2 = 701.9034{{c}} (~243/242 = 7.1666{{c}})
 {{Optimal ET sequence|legend=0| 152, 342, 836, 1178, 2014, 3192ce, 5206ce }}
@@ Line 132: / Line 144: @@
 Mapping: {{mapping| 38 0 28 -74 11 -281 | 0 1 1 3 2 7 }}
-Optimal tuning (CTE): ~55/54 = 31.5789{{c}}, ~3/2 = 701.9955{{c}} (~225/224 = 7.2587{{c}})
+Optimal tunings:
+* WE: ~55/54 = 31.5785{{c}}, ~3/2 = 701.9995{{c}} (~243/242 = 7.2727{{c}})
+* CWE: ~55/54 = 31.5789{{c}}, ~3/2 = 702.0053{{c}} (~243/242 = 7.2685{{c}})
 {{Optimal ET sequence|legend=0| 152f, 342f, 494 }}
@@ Line 145: / Line 159: @@
 Mapping: {{mapping| 38 0 28 -74 11 502 | 0 1 1 3 2 -6 }}
-Optimal tuning (CTE): ~55/54 = 31.5789{{c}}, ~3/2 = 701.9812{{c}} (~225/224 = 7.2444{{c}})
+Optimal tunings:
+* WE: ~55/54 = 31.5784{{c}}, ~3/2 = 701.9736{{c}} (~243/242 = 7.2493{{c}})
+* CWE: ~55/54 = 31.5789{{c}}, ~3/2 = 701.9855{{c}} (~243/242 = 7.2487{{c}})
 {{Optimal ET sequence|legend=0| 152, 342, 494, 1330, 1824, 2318d }}
-Badness (Sintel): 1.256
+Badness (Sintel): 1.26
 ==== Semihemienneadecal ====
@@ Line 159: / Line 175: @@
 : mapping generators: ~55/54, ~429/250
-Optimal tuning (CTE): ~55/54 = 31.5789{{c}}, ~429/250 = 935.1789{{c}} (~144/143 = 12.1895{{c}})
+Optimal tunings:
+* WE: ~55/54 = 31.5799{{c}}, ~429/250 = 935.1824{{c}} (~144/143 = 12.2152{{c}})
+* CWE: ~55/54 = 31.5789{{c}}, ~429/250 = 935.1617{{c}} (~144/143 = 12.2067{{c}})
 {{Optimal ET sequence|legend=0| 190, 304d, 494, 684, 1178, 2850, 4028ce }}
@@ Line 174: / Line 192: @@
 Mapping: {{mapping| 19 3 17 -28 82 92 159 78 | 0 10 10 30 -6 -8 -30 1 }}
-Optimal tuning (CTE): ~28/27 = 63.1579{{c}}, ~6545/5928 = 171.244{{c}}
+Optimal tunings:
+* WE: ~28/27 = 63.1582{{c}}, ~6545/5928 = 171.2448{{c}}
+* CWE: ~28/27 = 63.1579{{c}}, ~6545/5928 = 171.2439{{c}}
 {{Optimal ET sequence|legend=0| 855, 988, 1843 }}
-Badness (Sintel): 3.153
+Badness (Sintel): 3.15
 == Semidimi ==
 : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Semidimi]].''
-The generator of semidimi temperament is a semi-diminished fourth interval tuned between 162/125 and 35/27. It tempers out 5-limit {{monzo| -12 -73 55 }} and 7-limit 3955078125/3954653486, as well as 4375/4374.
+The generator of semidimi is a semi-diminished fourth interval tuned between 162/125 and 35/27. It tempers out 5-limit {{monzo| -12 -73 55 }} and 7-limit 3955078125/3954653486, as well as 4375/4374.
 [[Subgroup]]: 2.3.5.7
@@ Line 189: / Line 209: @@
 [[Comma list]]: 4375/4374, 3955078125/3954653486
-{{Mapping|legend=1| 1 36 48 61 | 0 -55 -73 -93 }}
+{{Mapping|legend=1| 1 -19 -25 -32 | 0 55 73 93 }}
+: mapping generators: ~2, ~35/27
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~35/27 = 449.1270{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0018{{c}}, ~35/27 = 449.1277{{c}}
+: [[error map]]: {{val| +0.002 +0.031 -0.040 -0.012 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/27 = 449.1270{{c}}
+: error map: {{val| 0.000 +0.030 -0.043 -0.015 }}
-{{Optimal ET sequence|legend=1| 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }}
+{{Optimal ET sequence|legend=1| 8d, …, 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }}
 [[Badness]] (Sintel): 0.382