Porcupine family: Difference between revisions

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~~{{interwiki~~

~~| de = Porcupine~~

~~| en = Porcupine family~~

~~| es =~~

~~| ja =~~

}}

The '''porcupine family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[porcupine comma]], [[250/243]], also called the maximal diesis.

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: mapping generators: ~2, ~10/9

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 164.0621{{c}}

: error map: {{val| 0.000 +5.859 -6.624 }}

~~<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~10/9 = 164.166{{c}}~~

~~: [[error map]]: {{val| 0.000 +5.547 -7.143 }}~~

* [[POTE]]: ~2 = 1200.000{{c}}, ~10/9 = 163.950{{c}}

~~: error map: {{val| 0.000 +6.194 -6.065 }} -->~~

[[Tuning ranges]]:

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Temperaments discussed elsewhere include:

* [[Oxygen]] → [[Very low accuracy temperaments #Oxygen|Very low accuracy temperaments]].

* ''[[Oxygen]]'' → [[Very low accuracy temperaments #Oxygen|Very low accuracy temperaments]]

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

* ''[[Jamesbond]]'' → [[Whitewood family #Jamesbond|Whitewood family]]

==== Subgroup extensions ====

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Optimal tunings:

* WE: ~2 = 1200.3290{{c}}, ~10/9 = 164.1227{{c}}

* WE: ~2 = 1200.3290{{c}}, ~11/10 = 164.1227{{c}}

* CWE~~: ~2 = 1200.0000{{c}}, ~10/9 = 163.9951{{c}}~~

* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 163.9951{{c}}

~~<!-- * CTE~~: ~2 = 1200.000{{c}}, ~11/10 = 163.~~887{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~11/10 = 164.078{{c}} ~~-->~~

Badness (Sintel): 0.303

~~==== Undecimation ====~~

~~Subgroup: 2.3.5.11.13~~

~~Comma list: 55/54, 100/99, 512/507~~

~~Subgroup-val mapping: {{mapping| 1 -1 -2 0 5 | 0 6 10 8 -3 }}~~

~~: mapping generators: ~2, ~88/65~~

~~Optimal tunings:~~

* WE: ~2 = 1199.4791{{c}}, ~88/65 = 517.9845{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~88/65 = 518.1740{{c}}

~~<!-- * CTE: ~2 = 1200.000{{c}}, ~88/65 = 518.086{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~88/65 = 518.209{{c}} -->

~~{{Optimal ET sequence|legend=0| 7, 23bc, 30, 37, 44 }}~~

~~Badness (Sintel): 1.21~~

== Septimal porcupine ==

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 162.9493{{c}}

: error map: {{val| 0.000 +9.197 -1.060 +8.870 }}

~~<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~10/9 = 163.203{{c}}~~

~~: [[error map]]: {{val| 0.000 +8.435 -2.330 +10.394 }}~~

* [[POTE]]: ~2 = 1200.000{{c}}, ~10/9 = 162.880{{c}}

~~: error map: {{val| 0.000 +9.405 -0.714 +8.455 }} -->~~

[[Minimax tuning]]:

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* WE: ~2 = 1198.3250{{c}}, ~11/10 = 162.5202{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.8156{{c}}

~~<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 163.105{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~11/10 = 162.747{{c}} -->

Minimax tuning:

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* WE: ~2 = 1197.0054{{c}}, ~11/10 = 162.3022{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.8314{{c}}

~~<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 163.442{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~11/10 = 162.708{{c}} -->

Minimax tuning:

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* WE: ~2 = 1198.3206{{c}}, ~11/10 = 162.0502{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.3458{{c}}

~~<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 162.636{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~11/10 = 162.277{{c}} -->

Minimax tuning:

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Tuning ranges:

* 13-odd-limit diamond monotone: ~10/9 = [160.000, 162.162] (2\15 to 5\37)

* 13-odd-limit diamond monotone: ~11/10 = [160.000, 162.162] (2\15 to 5\37)

* 15-odd-limit diamond monotone: ~10/9 = 162.162 (5\37)

* 15-odd-limit diamond monotone: ~11/10 = 162.162 (5\37)

* 13- and 15-odd-limit diamond tradeoff: ~10/9 = [150.637, 182.404]

* 13- and 15-odd-limit diamond tradeoff: ~11/10 = [150.637, 182.404]

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* WE: ~2 = 1198.0537{{c}}, ~11/10 = 162.2183{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.4665{{c}}

~~<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 163.378{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~11/10 = 162.482{{c}} -->

Minimax tuning:

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* WE: ~2 = 1200.0223{{c}}, ~11/10 = 163.6908{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 163.6874{{c}}

~~<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 163.678{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~11/10 = 163.688{{c}} -->

Minimax tuning:

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Badness (Sintel): 1.08

==== Undecimation ====

This weak extension is used by most patent vals that support porcupine and tune 13 well.

Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 64/63, 100/99, 169/168

Mapping: {{mapping| 1 -1 -2 8 0 5 | 0 6 10 -12 8 -3 }}

: mapping generators: ~2, ~35/26

Optimal tunings:

* WE: ~2 = 1198.4569{{c}}, ~35/26 = 517.9375{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~35/26 = 518.5822{{c}}

Badness (Sintel): 1.68

== Opossum ==

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 160.4589{{c}}

: error map: {{val| 0.000 +16.668 +11.392 -12.956 }}

~~<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~10/9 = 161.306{{c}}~~

~~: [[error map]]: {{val| 0.000 +14.126 +7.155 -20.583 }}~~

* [[POTE]]: ~2 = 1200.000{{c}}, ~10/9 = 159.691{{c}}

~~: error map: {{val| 0.000 +18.971 +15.229 -6.048 }} -->~~

[[Minimax tuning]]:

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* WE: ~2 = 1196.2331{{c}}, ~11/10 = 159.3050{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 160.4644{{c}}

~~<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 161.365{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~11/10 = 159.807{{c}} -->

Minimax tuning:

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* WE: ~2 = 1193.5447{{c}}, ~11/10 = 157.9505{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 159.7600{{c}}

~~<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 161.631{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~11/10 = 158.805{{c}} -->

Minimax tuning:

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 164.4060{{c}}

: error map: {{val| 0.000 +4.827 -8.344 +0.678 }}

~~<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~10/9 = 164.391{{c}}~~

~~: [[error map]]: {{val| 0.000 +4.871 -8.270 +0.913 }}~~

* [[POTE]]: ~2 = 1200.000{{c}}, ~10/9 = 164.412{{c}}

~~: error map: {{val| 0.000 +4.809 -8.375 +0.580 }} -->~~

[[Minimax tuning]]:

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* WE: ~2 = 1200.8706{{c}}, ~11/10 = 164.6715{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 164.4810{{c}}

~~<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 164.321{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~11/10 = 164.552{{c}} -->

Minimax tuning:

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* WE: ~2 = 1202.1557{{c}}, ~11/10 = 165.2494{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 164.8579{{c}}

~~<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 164.478{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~11/10 = 164.953{{c}} -->

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 166.0511{{c}}

: error map: {{val| 0.000 -0.108 -16.569 -10.161 }}

~~<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~10/9 = 166.094{{c}}~~

~~: [[error map]]: {{val| 0.000 -0.236 -16.783 -9.607 }}~~

* [[POTE]]: ~2 = 1200.000{{c}}, ~10/9 = 166.041{{c}}

~~: error map: {{val| 0.000 -0.077 -16.516 -10.299 }} -->~~

[[Minimax tuning]]:

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* WE: ~2 = 1203.0245{{c}}, ~11/10 = 166.3991{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9714{{c}}

~~<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 165.925{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~11/10 = 165.981{{c}} -->

Minimax tuning:

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* WE: ~2 = 1202.9957{{c}}, ~11/10 = 166.3885{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9843{{c}}

~~<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 166.046{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~11/10 = 165.974{{c}} -->

Minimax tuning:

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 161.2833{{c}}

: error map: {{val| 0.000 +14.195 +7.270 +69.891 }}

~~<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~10/9 = 165.185{{c}}~~

~~: [[error map]]: {{val| 0.000 +2.491 -12.236 +65.990 }}~~

* [[POTE]]: ~2 = 1200.000{{c}}, ~10/9 = 158.868{{c}}

~~: error map: {{val| 0.000 +21.442 +19.348 +72.306 }} -->~~

[[Minimax tuning]]:

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* WE: ~2 = 1189.2810{{c}}, ~11/10 = 157.3322{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 160.9603{{c}}

~~<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 164.768{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~11/10 = 158.750{{c}} -->

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Hedgehog has a period 1/2 octave and a generator which can be taken to be 9/7 instead of 10/9. It also tempers out [[245/243]], the sensamagic comma. It is a strong extension of [[BPS]] (as BPS has no 2 or sqrt(2)). Its ploidacot is diploid ~~omega~~-tricot.

Hedgehog has a period 1/2 octave and a generator which can be taken to be 9/7 instead of 10/9. It also tempers out [[245/243]], the sensamagic comma. It is a strong extension of [[BPS]] (as BPS has no 2 or sqrt(2)). Its ploidacot is diploid alpha-tricot.

22edo provides an obvious tuning, which happens to be the only [[patent val|patent-val]] tuning, but if you are looking for an alternative you could try the {{val| 146 232 338 411 }} (146bccdd) val with generator 10\73, or you could try 164 cents if you are fond of round numbers. The 14-note mos gives scope for harmony while stopping well short of 22. A related temperament is [[echidna]], which offers much more accuracy. They merge on 22edo.

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* [[CWE]]: ~7/5 = 600.0000{{c}}, ~9/7 = 435.4483{{c}}

: error map: {{val| 0.000 +4.390 -9.072 +8.416 }}

~~<!-- * [[CTE]]: ~7/5 = 600.000{{c}}, ~9/7 = 435.258{{c}}~~

~~: [[error map]]: {{val| 0.000 +3.819 -10.024 +7.464 }}~~

* [[POTE]]: ~7/5 = 600.000{{c}}, ~9/7 = 435.648{{c}}

~~: error map: {{val| 0.000 +4.989 -8.074 +9.414 }} -->~~

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* WE: ~7/5 = 600.1133{{c}}, ~9/7 = 435.4680{{c}}

* CWE: ~7/5 = 600.0000{{c}}, ~9/7 = 435.4431{{c}}

~~<!-- * CTE: ~7/5 = 600.000{{c}}, ~9/7 = 435.528{{c}}~~

* POTE: ~7/5 = 600.000{{c}}, ~9/7 = 435.386{{c}} -->

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* WE: ~7/5 = 600.3651{{c}}, ~9/7 = 436.1258{{c}}

* CWE: ~7/5 = 600.0000{{c}}, ~9/7 = 436.0483{{c}}

~~<!-- * CTE: ~7/5 = 600.000{{c}}, ~9/7 = 436.309{{c}}~~

* POTE: ~7/5 = 600.000{{c}}, ~9/7 = 435.861{{c}} -->

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* WE: ~7/5 = 598.3303{{c}}, ~9/7 = 435.8617{{c}}

* CWE: ~7/5 = 600.0000{{c}}, ~9/7 = 436.3485{{c}}

~~<!-- * CTE: ~7/5 = 600.000{{c}}, ~9/7 = 435.186{{c}}~~

* POTE: ~7/5 = 600.000{{c}}, ~9/7 = 437.078{{c}} -->

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* WE: ~7/5 = 599.7917{{c}}, ~9/7 = 435.2737{{c}}

* CWE: ~7/5 = 600.0000{{c}}, ~9/7 = 435.4047{{c}}

~~<!-- * CTE: ~7/5 = 600.000{{c}}, ~9/7 = 435.329{{c}}~~

* POTE: ~7/5 = 600.000{{c}}, ~9/7 = 435.425{{c}} -->

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 82.2758{{c}}

: error map: {{val| 0.000 +4.390 -9.072 -15.653 }}

~~<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~21/20 = 81.914{{c}}~~

~~: [[error map]]: {{val| 0.000 +6.559 -5.457 -14.569 }}~~

* [[POTE]]: ~2 = 1200.000{{c}}, ~21/20 = 82.505{{c}}

~~: error map: {{val| 0.000 +3.012 -11.368 -16.342 }} -->~~

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* WE: ~2 = 1202.3781{{c}}, ~21/20 = 82.6673{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 82.2434{{c}}

~~<!-- * CTE: ~2 = 1200.000{{c}}, ~21/20 = 81.802{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~21/20 = 82.504{{c}} -->

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* WE: ~2 = 1202.4145{{c}}, ~21/20 = 82.6963{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 82.3130{{c}}

~~<!-- * CTE: ~2 = 1200.000{{c}}, ~21/20 = 81.912{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~21/20 = 82.530{{c}} -->

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* WE: ~2 = 1199.0072{{c}}, ~21/20 = 81.6911{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 81.8576{{c}}

~~<!-- * CTE: ~2 = 1200.000{{c}}, ~21/20 = 82.034{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~21/20 = 81.759{{c}} -->

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== Ammonite ==

Ammonite adds 686/675 to the comma list and may be described as the {{nowrap| 8d & 29 }} temperament. Its ploidacot is epsilon-enneacot. [[37edo]] provides an obvious tuning.

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 745.4994{{c}}

: error map: {{val| 0.000 +7.540 -3.823 -4.337 }}

~~<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~14/9 = 745.450{{c}}~~

~~: [[error map]]: {{val| 0.000 +7.095 -4.564 -5.276 }}~~

* [[POTE]]: ~2 = 1200.000{{c}}, ~14/9 = 745.552{{c}}

~~: error map: {{val| 0.000 +8.009 -3.040 -3.346 }} -->~~

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* WE: ~2 = 1200.0141{{c}}, ~14/9 = 745.4971{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 745.4894{{c}}

~~<!-- * CTE: ~2 = 1200.000{{c}}, ~14/9 = 745.495{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~14/9 = 745.488{{c}} -->

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* WE: ~2 = 1200.2478{{c}}, ~14/9 = 745.6252{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 745.4904{{c}}

~~<!-- * CTE: ~2 = 1200.000{{c}}, ~14/9 = 745.520{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~14/9 = 745.471{{c}} -->

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 54.5489{{c}}

: error map: {{val| 0.000 +7.105 -4.548 +12.978 }}

~~<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~36/35 = 54.804{{c}}~~

~~: [[error map]]: {{val| 0.000 +4.809 -8.374 +11.958 }}~~

* [[POTE]]: ~2 = 1200.000{{c}}, ~36/35 = 54.384{{c}}

~~: error map: {{val| 0.000 +8.585 -2.081 +13.636 }} -->~~

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* WE: ~2 = 1198.2851{{c}}, ~36/35 = 54.2986{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 54.4992{{c}}

~~<!-- * CTE: ~2 = 1200.000{{c}}, ~36/35 = 54.702{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~36/35 = 54.376{{c}} -->

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* WE: ~2 = 1200.3864{{c}}, ~36/35 = 54.6830{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 54.6396{{c}}

~~<!-- * CTE: ~2 = 1200.000{{c}}, ~36/35 = 54.575{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~36/35 = 54.665{{c}} -->

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[[Category:Temperament families]]

~~[[Category:Pages with mostly numerical content]]~~

[[Category:Porcupine family| ]]

~~[[Category:Porcupine| ]] ~~

[[Category:Rank 2]]

@@ Line 1: / Line 1: @@
-{{interwiki
-| de = Porcupine
-| en = Porcupine family
-| es =
-| ja =
-}}
 {{Technical data page}}
 The '''porcupine family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[porcupine comma]], [[250/243]], also called the maximal diesis.
@@ Line 18: / Line 12: @@
 {{Mapping|legend=1| 1 2 3 | 0 -3 -5 }}
 : mapping generators: ~2, ~10/9
@@ Line 26: / Line 19: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 164.0621{{c}}
 : error map: {{val| 0.000 +5.859 -6.624 }}
-<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~10/9 = 164.166{{c}}
-: [[error map]]: {{val| 0.000 +5.547 -7.143 }}
-* [[POTE]]: ~2 = 1200.000{{c}}, ~10/9 = 163.950{{c}}
-: error map: {{val| 0.000 +6.194 -6.065 }} -->
 [[Tuning ranges]]:
@@ Line 53: / Line 42: @@
 Temperaments discussed elsewhere include:
-* [[Oxygen]] → [[Very low accuracy temperaments #Oxygen|Very low accuracy temperaments]].
+* ''[[Oxygen]]'' → [[Very low accuracy temperaments #Oxygen|Very low accuracy temperaments]]
-* [[Jamesbond]] → [[7th-octave temperaments #Jamesbond|7th-octave temperaments]].
+* ''[[Jamesbond]]'' → [[Whitewood family #Jamesbond|Whitewood family]]
 ==== Subgroup extensions ====
@@ Line 69: / Line 58: @@
 Optimal tunings:
-* WE: ~2 = 1200.3290{{c}}, ~10/9 = 164.1227{{c}}
+* WE: ~2 = 1200.3290{{c}}, ~11/10 = 164.1227{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 163.9951{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 163.9951{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 163.887{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~11/10 = 164.078{{c}} -->
 {{Optimal ET sequence|legend=0| 7, 15, 22, 73ce, 95ce }}
 Badness (Sintel): 0.303
-==== Undecimation ====
-Subgroup: 2.3.5.11.13
-Comma list: 55/54, 100/99, 512/507
-Subgroup-val mapping: {{mapping| 1 -1 -2 0 5 | 0 6 10 8 -3 }}
-: mapping generators: ~2, ~88/65
-Optimal tunings:
-* WE: ~2 = 1199.4791{{c}}, ~88/65 = 517.9845{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~88/65 = 518.1740{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~88/65 = 518.086{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~88/65 = 518.209{{c}} -->
-{{Optimal ET sequence|legend=0| 7, 23bc, 30, 37, 44 }}
-Badness (Sintel): 1.21
 == Septimal porcupine ==
@@ Line 113: / Line 81: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 162.9493{{c}}
 : error map: {{val| 0.000 +9.197 -1.060 +8.870 }}
-<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~10/9 = 163.203{{c}}
-: [[error map]]: {{val| 0.000 +8.435 -2.330 +10.394 }}
-* [[POTE]]: ~2 = 1200.000{{c}}, ~10/9 = 162.880{{c}}
-: error map: {{val| 0.000 +9.405 -0.714 +8.455 }} -->
 [[Minimax tuning]]:
@@ Line 143: / Line 107: @@
 * WE: ~2 = 1198.3250{{c}}, ~11/10 = 162.5202{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.8156{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 163.105{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~11/10 = 162.747{{c}} -->
 Minimax tuning:
@@ Line 170: / Line 132: @@
 * WE: ~2 = 1197.0054{{c}}, ~11/10 = 162.3022{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.8314{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 163.442{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~11/10 = 162.708{{c}} -->
 Minimax tuning:
@@ Line 198: / Line 158: @@
 * WE: ~2 = 1198.3206{{c}}, ~11/10 = 162.0502{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.3458{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 162.636{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~11/10 = 162.277{{c}} -->
 Minimax tuning:
@@ Line 206: / Line 164: @@
 Tuning ranges:
-* 13-odd-limit diamond monotone: ~10/9 = [160.000, 162.162] (2\15 to 5\37)
+* 13-odd-limit diamond monotone: ~11/10 = [160.000, 162.162] (2\15 to 5\37)
-* 15-odd-limit diamond monotone: ~10/9 = 162.162 (5\37)
+* 15-odd-limit diamond monotone: ~11/10 = 162.162 (5\37)
-* 13- and 15-odd-limit diamond tradeoff: ~10/9 = [150.637, 182.404]
+* 13- and 15-odd-limit diamond tradeoff: ~11/10 = [150.637, 182.404]
 {{Optimal ET sequence|legend=0| 15, 22, 37 }}
@@ Line 224: / Line 182: @@
 * WE: ~2 = 1198.0537{{c}}, ~11/10 = 162.2183{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.4665{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 163.378{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~11/10 = 162.482{{c}} -->
 Minimax tuning:
@@ Line 245: / Line 201: @@
 * WE: ~2 = 1200.0223{{c}}, ~11/10 = 163.6908{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~11/10 = 163.6874{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 163.678{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~11/10 = 163.688{{c}} -->
 Minimax tuning:
@@ Line 255: / Line 209: @@
 Badness (Sintel): 1.08
+==== Undecimation ====
+This weak extension is used by most patent vals that support porcupine and tune 13 well.
+Subgroup: 2.3.5.7.11.13
+Comma list: 55/54, 64/63, 100/99, 169/168
+Mapping: {{mapping| 1 -1 -2 8 0 5 | 0 6 10 -12 8 -3 }}
+: mapping generators: ~2, ~35/26
+Optimal tunings:
+* WE: ~2 = 1198.4569{{c}}, ~35/26 = 517.9375{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~35/26 = 518.5822{{c}}
+{{Optimal ET sequence|legend=0| 7, 30, 37, 81bd }}
+Badness (Sintel): 1.68
 == Opossum ==
@@ Line 272: / Line 244: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 160.4589{{c}}
 : error map: {{val| 0.000 +16.668 +11.392 -12.956 }}
-<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~10/9 = 161.306{{c}}
-: [[error map]]: {{val| 0.000 +14.126 +7.155 -20.583 }}
-* [[POTE]]: ~2 = 1200.000{{c}}, ~10/9 = 159.691{{c}}
-: error map: {{val| 0.000 +18.971 +15.229 -6.048 }} -->
 [[Minimax tuning]]:
@@ Line 294: / Line 262: @@
 * WE: ~2 = 1196.2331{{c}}, ~11/10 = 159.3050{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~11/10 = 160.4644{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 161.365{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~11/10 = 159.807{{c}} -->
 Minimax tuning:
@@ Line 314: / Line 280: @@
 * WE: ~2 = 1193.5447{{c}}, ~11/10 = 157.9505{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~11/10 = 159.7600{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 161.631{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~11/10 = 158.805{{c}} -->
 Minimax tuning:
@@ Line 338: / Line 302: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 164.4060{{c}}
 : error map: {{val| 0.000 +4.827 -8.344 +0.678 }}
-<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~10/9 = 164.391{{c}}
-: [[error map]]: {{val| 0.000 +4.871 -8.270 +0.913 }}
-* [[POTE]]: ~2 = 1200.000{{c}}, ~10/9 = 164.412{{c}}
-: error map: {{val| 0.000 +4.809 -8.375 +0.580 }} -->
 [[Minimax tuning]]:
@@ Line 361: / Line 321: @@
 * WE: ~2 = 1200.8706{{c}}, ~11/10 = 164.6715{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~11/10 = 164.4810{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 164.321{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~11/10 = 164.552{{c}} -->
 Minimax tuning:
@@ Line 382: / Line 340: @@
 * WE: ~2 = 1202.1557{{c}}, ~11/10 = 165.2494{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~11/10 = 164.8579{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 164.478{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~11/10 = 164.953{{c}} -->
 {{Optimal ET sequence|legend=0| 7d, 22, 29, 51f, 80cdeff }}
@@ Line 406: / Line 362: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 166.0511{{c}}
 : error map: {{val| 0.000 -0.108 -16.569 -10.161 }}
-<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~10/9 = 166.094{{c}}
-: [[error map]]: {{val| 0.000 -0.236 -16.783 -9.607 }}
-* [[POTE]]: ~2 = 1200.000{{c}}, ~10/9 = 166.041{{c}}
-: error map: {{val| 0.000 -0.077 -16.516 -10.299 }} -->
 [[Minimax tuning]]:
@@ Line 429: / Line 381: @@
 * WE: ~2 = 1203.0245{{c}}, ~11/10 = 166.3991{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9714{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 165.925{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~11/10 = 165.981{{c}} -->
 Minimax tuning:
@@ Line 450: / Line 400: @@
 * WE: ~2 = 1202.9957{{c}}, ~11/10 = 166.3885{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9843{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 166.046{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~11/10 = 165.974{{c}} -->
 Minimax tuning:
@@ Line 475: / Line 423: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 161.2833{{c}}
 : error map: {{val| 0.000 +14.195 +7.270 +69.891 }}
-<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~10/9 = 165.185{{c}}
-: [[error map]]: {{val| 0.000 +2.491 -12.236 +65.990 }}
-* [[POTE]]: ~2 = 1200.000{{c}}, ~10/9 = 158.868{{c}}
-: error map: {{val| 0.000 +21.442 +19.348 +72.306 }} -->
 [[Minimax tuning]]:
@@ Line 498: / Line 442: @@
 * WE: ~2 = 1189.2810{{c}}, ~11/10 = 157.3322{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~11/10 = 160.9603{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 164.768{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~11/10 = 158.750{{c}} -->
 {{Optimal ET sequence|legend=0| 7, 8d, 15d }}
@@ Line 508: / Line 450: @@
 {{See also| Sensamagic clan | Stearnsmic clan }}
-Hedgehog has a period 1/2 octave and a generator which can be taken to be 9/7 instead of 10/9. It also tempers out [[245/243]], the sensamagic comma. It is a strong extension of [[BPS]] (as BPS has no 2 or sqrt(2)). Its ploidacot is diploid omega-tricot.
+Hedgehog has a period 1/2 octave and a generator which can be taken to be 9/7 instead of 10/9. It also tempers out [[245/243]], the sensamagic comma. It is a strong extension of [[BPS]] (as BPS has no 2 or sqrt(2)). Its ploidacot is diploid alpha-tricot.
 edo provides an obvious tuning, which happens to be the only [[patent val|patent-val]] tuning, but if you are looking for an alternative you could try the {{val| 146 232 338 411 }} (146bccdd) val with generator 10\73, or you could try 164 cents if you are fond of round numbers. The 14-note mos gives scope for harmony while stopping well short of 22. A related temperament is [[echidna]], which offers much more accuracy. They merge on 22edo.
@@ Line 525: / Line 467: @@
 * [[CWE]]: ~7/5 = 600.0000{{c}}, ~9/7 = 435.4483{{c}}
 : error map: {{val| 0.000 +4.390 -9.072 +8.416 }}
-<!-- * [[CTE]]: ~7/5 = 600.000{{c}}, ~9/7 = 435.258{{c}}
-: [[error map]]: {{val| 0.000 +3.819 -10.024 +7.464 }}
-* [[POTE]]: ~7/5 = 600.000{{c}}, ~9/7 = 435.648{{c}}
-: error map: {{val| 0.000 +4.989 -8.074 +9.414 }} -->
 {{Optimal ET sequence|legend=1| 8d, 14c, 22 }}
@@ Line 544: / Line 482: @@
 * WE: ~7/5 = 600.1133{{c}}, ~9/7 = 435.4680{{c}}
 * CWE: ~7/5 = 600.0000{{c}}, ~9/7 = 435.4431{{c}}
-<!-- * CTE: ~7/5 = 600.000{{c}}, ~9/7 = 435.528{{c}}
-* POTE: ~7/5 = 600.000{{c}}, ~9/7 = 435.386{{c}} -->
 {{Optimal ET sequence|legend=0| 8d, 14c, 22, 58ce }}
@@ Line 561: / Line 497: @@
 * WE: ~7/5 = 600.3651{{c}}, ~9/7 = 436.1258{{c}}
 * CWE: ~7/5 = 600.0000{{c}}, ~9/7 = 436.0483{{c}}
-<!-- * CTE: ~7/5 = 600.000{{c}}, ~9/7 = 436.309{{c}}
-* POTE: ~7/5 = 600.000{{c}}, ~9/7 = 435.861{{c}} -->
 {{Optimal ET sequence|legend=0| 8d, 14cf, 22 }}
@@ Line 578: / Line 512: @@
 * WE: ~7/5 = 598.3303{{c}}, ~9/7 = 435.8617{{c}}
 * CWE: ~7/5 = 600.0000{{c}}, ~9/7 = 436.3485{{c}}
-<!-- * CTE: ~7/5 = 600.000{{c}}, ~9/7 = 435.186{{c}}
-* POTE: ~7/5 = 600.000{{c}}, ~9/7 = 437.078{{c}} -->
 {{Optimal ET sequence|legend=0| 14c, 22f }}
@@ Line 595: / Line 527: @@
 * WE: ~7/5 = 599.7917{{c}}, ~9/7 = 435.2737{{c}}
 * CWE: ~7/5 = 600.0000{{c}}, ~9/7 = 435.4047{{c}}
-<!-- * CTE: ~7/5 = 600.000{{c}}, ~9/7 = 435.329{{c}}
-* POTE: ~7/5 = 600.000{{c}}, ~9/7 = 435.425{{c}} -->
 {{Optimal ET sequence|legend=0| 22 }}
@@ Line 621: / Line 551: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 82.2758{{c}}
 : error map: {{val| 0.000 +4.390 -9.072 -15.653 }}
-<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~21/20 = 81.914{{c}}
-: [[error map]]: {{val| 0.000 +6.559 -5.457 -14.569 }}
-* [[POTE]]: ~2 = 1200.000{{c}}, ~21/20 = 82.505{{c}}
-: error map: {{val| 0.000 +3.012 -11.368 -16.342 }} -->
 {{Optimal ET sequence|legend=1| 14c, 15, 29 }}
@@ Line 640: / Line 566: @@
 * WE: ~2 = 1202.3781{{c}}, ~21/20 = 82.6673{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~21/20 = 82.2434{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~21/20 = 81.802{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~21/20 = 82.504{{c}} -->
 {{Optimal ET sequence|legend=0| 14c, 15, 29 }}
@@ Line 657: / Line 581: @@
 * WE: ~2 = 1202.4145{{c}}, ~21/20 = 82.6963{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~21/20 = 82.3130{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~21/20 = 81.912{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~21/20 = 82.530{{c}} -->
 {{Optimal ET sequence|legend=0| 14cf, 15, 29 }}
@@ Line 674: / Line 596: @@
 * WE: ~2 = 1199.0072{{c}}, ~21/20 = 81.6911{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~21/20 = 81.8576{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~21/20 = 82.034{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~21/20 = 81.759{{c}} -->
 {{Optimal ET sequence|legend=0| 14c, 15 }}
@@ Line 685: / Line 605: @@
 == Ammonite ==
+{{See also|Subgroup temperaments #Ammon}}
 Ammonite adds 686/675 to the comma list and may be described as the {{nowrap| 8d & 29 }} temperament. Its ploidacot is epsilon-enneacot. [[37edo]] provides an obvious tuning.
@@ Line 700: / Line 621: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 745.4994{{c}}
 : error map: {{val| 0.000 +7.540 -3.823 -4.337 }}
-<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~14/9 = 745.450{{c}}
-: [[error map]]: {{val| 0.000 +7.095 -4.564 -5.276 }}
-* [[POTE]]: ~2 = 1200.000{{c}}, ~14/9 = 745.552{{c}}
-: error map: {{val| 0.000 +8.009 -3.040 -3.346 }} -->
 {{Optimal ET sequence|legend=1| 8d, 21cd, 29, 37, 66 }}
@@ Line 719: / Line 636: @@
 * WE: ~2 = 1200.0141{{c}}, ~14/9 = 745.4971{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~14/9 = 745.4894{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~14/9 = 745.495{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~14/9 = 745.488{{c}} -->
 {{Optimal ET sequence|legend=0| 8d, 21cde, 29, 37, 66 }}
@@ Line 736: / Line 651: @@
 * WE: ~2 = 1200.2478{{c}}, ~14/9 = 745.6252{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~14/9 = 745.4904{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~14/9 = 745.520{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~14/9 = 745.471{{c}} -->
 {{Optimal ET sequence|legend=0| 8d, 21cdef, 29, 37, 66 }}
@@ Line 759: / Line 672: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 54.5489{{c}}
 : error map: {{val| 0.000 +7.105 -4.548 +12.978 }}
-<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~36/35 = 54.804{{c}}
-: [[error map]]: {{val| 0.000 +4.809 -8.374 +11.958 }}
-* [[POTE]]: ~2 = 1200.000{{c}}, ~36/35 = 54.384{{c}}
-: error map: {{val| 0.000 +8.585 -2.081 +13.636 }} -->
 {{Optimal ET sequence|legend=1| 1c, 21c, 22 }}
@@ Line 778: / Line 687: @@
 * WE: ~2 = 1198.2851{{c}}, ~36/35 = 54.2986{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~36/35 = 54.4992{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~36/35 = 54.702{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~36/35 = 54.376{{c}} -->
 {{Optimal ET sequence|legend=0| 1ce, 21ce, 22 }}
@@ Line 795: / Line 702: @@
 * WE: ~2 = 1200.3864{{c}}, ~36/35 = 54.6830{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~36/35 = 54.6396{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~36/35 = 54.575{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~36/35 = 54.665{{c}} -->
 {{Optimal ET sequence|legend=0| 1ce, 21cef, 22 }}
@@ Line 803: / Line 708: @@
 [[Category:Temperament families]]
-[[Category:Pages with mostly numerical content]]
 [[Category:Porcupine family| ]] <!-- main article -->
-[[Category:Porcupine| ]] <!-- key article -->
 [[Category:Rank 2]]