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Porcupine family: Difference between revisions

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{{interwiki
| de = Porcupine
| en = Porcupine family
| es =
| ja =
}}
{{Technical data page}}
{{Technical data page}}
The '''porcupine family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[porcupine comma]], [[250/243]], also called the maximal diesis.  
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|legend=1| 1 2 3 | 0 -3 -5 }}
: mapping generators: ~2, ~10/9
: mapping generators: ~2, ~10/9


Line 46: Line 39:
Those all share the same generator with porcupine.  
Those all share the same generator with porcupine.  


[[#Nautilus|nautilus]] tempers out [[49/48]] and splits the generator in two. [[#Hedgehog|hedgehog]] tempers out [[50/49]] with a semi-octave period. Finally, [[#Ammonite|ammonite]] tempers out [[686/675]] and [[#Ceratitid|ceratitid]] tempers out [[1728/1715]]. Those split the generator in three.  
[[#Nautilus|Nautilus]] tempers out [[49/48]] and splits the generator in two. Hedgehog tempers out [[50/49]] with a semi-octave period. Finally, [[#Ammonite|ammonite]] tempers out [[686/675]] and [[#Ceratitid|ceratitid]] tempers out [[1728/1715]]. Those split the generator in three.  


Temperaments discussed elsewhere include:  
Temperaments discussed elsewhere include:  
* [[Oxygen]] → [[Very low accuracy temperaments #Oxygen|Very low accuracy temperaments]].
* ''[[Oxygen]]'' (+21/20) → [[Very low accuracy temperaments #Oxygen|Very low accuracy temperaments]]
* [[Jamesbond]] → [[7th-octave temperaments #Jamesbond|7th-octave temperaments]].
* ''[[Hedgehog]]'' (+50/49 or 245/243) → [[Stearnsmic clan #Hedgehog|Stearnsmic clan]]
* ''[[Jamesbond]]'' (+25/24) → [[Whitewood family #Jamesbond|Whitewood family]]
 
The rest are considered in this page.  


==== Subgroup extensions ====
==== Subgroup extensions ====
Line 60: Line 56:
Comma list: 55/54, 100/99
Comma list: 55/54, 100/99


Subgroup-val mapping: {{mapping| 1 2 3 4 | 0 -3 -5 -4 }}
{{Mapping|legend=2| 1 2 3 4 | 0 -3 -5 -4 }}


Gencom mapping: {{mapping| 1 2 3 0 4 | 0 -3 -5 0 -4 }}
{{Mapping|legend=4| 1 2 3 0 4 | 0 -3 -5 0 -4 }}


Optimal tunings:  
Optimal tunings:  
Line 71: Line 67:


Badness (Sintel): 0.303
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}}
{{Optimal ET sequence|legend=0| 7, 23bc, 30, 37, 44 }}
Badness (Sintel): 1.21


== Septimal porcupine ==
== Septimal porcupine ==
Line 126: Line 105:
Comma list: 55/54, 64/63, 100/99
Comma list: 55/54, 64/63, 100/99


Mapping: {{mapping| 1 2 3 2 4 | 0 -3 -5 6 -4 }}
{{Mapping|legend=0| 1 2 3 2 4 | 0 -3 -5 6 -4 }}


Optimal tunings:  
Optimal tunings:  
Line 151: Line 130:
Comma list: 40/39, 55/54, 64/63, 66/65
Comma list: 40/39, 55/54, 64/63, 66/65


Mapping: {{mapping| 1 2 3 2 4 4 | 0 -3 -5 6 -4 -2 }}
{{Mapping|legend=0| 1 2 3 2 4 4 | 0 -3 -5 6 -4 -2 }}


Optimal tunings:  
Optimal tunings:  
Line 177: Line 156:
Comma list: 55/54, 64/63, 91/90, 100/99
Comma list: 55/54, 64/63, 91/90, 100/99


Mapping: {{mapping| 1 2 3 2 4 6 | 0 -3 -5 6 -4 -17 }}
{{Mapping|legend=0| 1 2 3 2 4 6 | 0 -3 -5 6 -4 -17 }}


Optimal tunings:  
Optimal tunings:  
Line 201: Line 180:
Comma list: 55/54, 64/63, 100/99, 196/195
Comma list: 55/54, 64/63, 100/99, 196/195


Mapping: {{mapping| 1 2 3 2 4 1 | 0 -3 -5 6 -4 20 }}
{{Mapping|legend=0| 1 2 3 2 4 1 | 0 -3 -5 6 -4 20 }}


Optimal tunings:  
Optimal tunings:  
Line 220: Line 199:
Comma list: 55/54, 64/63, 65/63, 100/99
Comma list: 55/54, 64/63, 65/63, 100/99


Mapping: {{mapping| 1 2 3 2 4 3 | 0 -3 -5 6 -4 5 }}
{{Mapping|legend=0| 1 2 3 2 4 3 | 0 -3 -5 6 -4 5 }}


Optimal tunings:  
Optimal tunings:  
Line 233: Line 212:


Badness (Sintel): 1.08
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|legend=0| 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 ==
== Opossum ==
Line 263: Line 260:
Comma list: 28/27, 55/54, 77/75
Comma list: 28/27, 55/54, 77/75


Mapping: {{mapping| 1 2 3 4 4 | 0 -3 -5 -9 -4 }}
{{Mapping|legend=0| 1 2 3 4 4 | 0 -3 -5 -9 -4 }}


Optimal tunings:  
Optimal tunings:  
Line 281: Line 278:
Comma list: 28/27, 40/39, 55/54, 66/65
Comma list: 28/27, 40/39, 55/54, 66/65


Mapping: {{mapping| 1 2 3 4 4 4 | 0 -3 -5 -9 -4 -2 }}
{{Mapping|legend=0| 1 2 3 4 4 4 | 0 -3 -5 -9 -4 -2 }}


Optimal tunings:  
Optimal tunings:  
Line 322: Line 319:
Comma list: 55/54, 100/99, 225/224
Comma list: 55/54, 100/99, 225/224


Mapping: {{mapping| 1 2 3 5 4 | 0 -3 -5 -16 -4 }}
{{Mapping|legend=0| 1 2 3 5 4 | 0 -3 -5 -16 -4 }}


Optimal tunings:  
Optimal tunings:  
Line 341: Line 338:
Comma list: 55/54, 65/64, 91/90, 100/99
Comma list: 55/54, 65/64, 91/90, 100/99


Mapping: {{mapping| 1 2 3 5 4 3 | 0 -3 -5 -16 -4 5 }}
{{Mapping|legend=0| 1 2 3 5 4 3 | 0 -3 -5 -16 -4 5 }}


Optimal tunings:  
Optimal tunings:  
Line 382: Line 379:
Comma list: 55/54, 100/99, 525/512
Comma list: 55/54, 100/99, 525/512


Mapping: {{mapping| 1 2 3 1 4 | 0 -3 -5 13 -4 }}
{{Mapping|legend=0| 1 2 3 1 4 | 0 -3 -5 13 -4 }}


Optimal tunings:  
Optimal tunings:  
Line 401: Line 398:
Comma list: 55/54, 65/64, 100/99, 105/104
Comma list: 55/54, 65/64, 100/99, 105/104


Mapping: {{mapping| 1 2 3 1 4 3 | 0 -3 -5 13 -4 5 }}
{{Mapping|legend=0| 1 2 3 1 4 3 | 0 -3 -5 13 -4 5 }}


Optimal tunings:  
Optimal tunings:  
Line 443: Line 440:
Comma list: 22/21, 36/35, 80/77
Comma list: 22/21, 36/35, 80/77


Mapping: {{mapping| 1 2 3 3 4 | 0 -3 -5 -1 -4 }}
{{Mapping|legend=0| 1 2 3 3 4 | 0 -3 -5 -1 -4 }}


Optimal tunings:  
Optimal tunings:  
Line 452: Line 449:


Badness (Sintel): 0.886
Badness (Sintel): 0.886
== Hedgehog ==
{{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 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.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 50/49, 245/243
{{Mapping|legend=1| 2 1 1 2 | 0 3 5 5 }}
: mapping generators: ~7/5, ~9/7
[[Optimal tuning]]s:
* [[WE]]: ~7/5 = 599.6061{{c}}, ~9/7 = 435.3620{{c}}
: [[error map]]: {{val| -0.788 +3.737 -9.897 +7.197 }}
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~9/7 = 435.4483{{c}}
: error map: {{val| 0.000 +4.390 -9.072 +8.416 }}
{{Optimal ET sequence|legend=1| 8d, 14c, 22 }}
[[Badness]] (Sintel): 1.11
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 50/49, 55/54, 99/98
Mapping: {{mapping| 2 1 1 2 4 | 0 3 5 5 4 }}
Optimal tunings:
* WE: ~7/5 = 600.1133{{c}}, ~9/7 = 435.4680{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~9/7 = 435.4431{{c}}
{{Optimal ET sequence|legend=0| 8d, 14c, 22, 58ce }}
Badness (Sintel): 0.764
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 50/49, 55/54, 65/63, 99/98
Mapping: {{mapping| 2 1 1 2 4 3 | 0 3 5 5 4 6 }}
Optimal tunings:
* WE: ~7/5 = 600.3651{{c}}, ~9/7 = 436.1258{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~9/7 = 436.0483{{c}}
{{Optimal ET sequence|legend=0| 8d, 14cf, 22 }}
Badness (Sintel): 0.889
==== Urchin ====
Subgroup: 2.3.5.7.11.13
Comma list: 40/39, 50/49, 55/54, 66/65
Mapping: {{mapping| 2 1 1 2 4 6 | 0 3 5 5 4 2 }}
Optimal tunings:
* WE: ~7/5 = 598.3303{{c}}, ~9/7 = 435.8617{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~9/7 = 436.3485{{c}}
{{Optimal ET sequence|legend=0| 14c, 22f }}
Badness (Sintel): 1.04
=== Hedgepig ===
Subgroup: 2.3.5.7.11
Comma list: 50/49, 245/243, 385/384
Mapping: {{mapping| 2 1 1 2 12 | 0 3 5 5 -7 }}
Optimal tunings:
* WE: ~7/5 = 599.7917{{c}}, ~9/7 = 435.2737{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~9/7 = 435.4047{{c}}
{{Optimal ET sequence|legend=0| 22 }}
Badness (Sintel): 2.26
; Music
* [https://web.archive.org/web/20240624173512/http://micro.soonlabel.com/22-ET/20120207-phobos-light-hedgehog14.mp3 ''Phobos Light''] by [[Chris Vaisvil]] – in [[hedgehog14|Hedgehog[14]]], 22edo tuning.


== Nautilus ==
== Nautilus ==
Line 567: Line 476:
Comma list: 49/48, 55/54, 245/242
Comma list: 49/48, 55/54, 245/242


Mapping: {{mapping| 1 2 3 3 4 | 0 -6 -10 -3 -8 }}
{{Mapping|legend=0| 1 2 3 3 4 | 0 -6 -10 -3 -8 }}


Optimal tunings:  
Optimal tunings:  
Line 582: Line 491:
Comma list: 49/48, 55/54, 91/90, 100/99
Comma list: 49/48, 55/54, 91/90, 100/99


Mapping: {{mapping| 1 2 3 3 4 5 | 0 -6 -10 -3 -8 -19 }}
{{Mapping|legend=0| 1 2 3 3 4 5 | 0 -6 -10 -3 -8 -19 }}


Optimal tunings:  
Optimal tunings:  
Line 597: Line 506:
Comma list: 40/39, 49/48, 55/54, 66/65
Comma list: 40/39, 49/48, 55/54, 66/65


Mapping: {{mapping| 1 2 3 3 4 4 | 0 -6 -10 -3 -8 -4 }}
{{Mapping|legend=0| 1 2 3 3 4 4 | 0 -6 -10 -3 -8 -4 }}


Optimal tunings:  
Optimal tunings:  
Line 637: Line 546:
Comma list: 55/54, 100/99, 686/675
Comma list: 55/54, 100/99, 686/675


Mapping: {{mapping| 1 -4 -7 -9 -4 | 0 9 15 19 12 }}
{{Mapping|legend=0| 1 -4 -7 -9 -4 | 0 9 15 19 12 }}


Optimal tunings:  
Optimal tunings:  
Line 652: Line 561:
Comma list: 55/54, 91/90, 100/99, 169/168
Comma list: 55/54, 91/90, 100/99, 169/168


Mapping: {{mapping| 1 -4 -7 -9 -4 -5 | 0 9 15 19 12 14 }}
{{Mapping|legend=0| 1 -4 -7 -9 -4 -5 | 0 9 15 19 12 14 }}


Optimal tunings:  
Optimal tunings:  
Line 688: Line 597:
Comma list: 55/54, 100/99, 352/343
Comma list: 55/54, 100/99, 352/343


Mapping: {{mapping| 1 2 3 3 4 | 0 -9 -15 -4 -12 }}
{{Mapping|legend=0| 1 2 3 3 4 | 0 -9 -15 -4 -12 }}


Optimal tunings:  
Optimal tunings:  
Line 703: Line 612:
Comma list: 55/54, 65/63, 100/99, 352/343
Comma list: 55/54, 65/63, 100/99, 352/343


Mapping: {{mapping| 1 2 3 3 4 4 | 0 -9 -15 -4 -12 -7 }}
{{Mapping|legend=0| 1 2 3 3 4 4 | 0 -9 -15 -4 -12 -7 }}


Optimal tunings:  
Optimal tunings:  
Line 713: Line 622:
Badness (Sintel): 1.85
Badness (Sintel): 1.85


[[Category:Porcupine family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Porcupine family| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Porcupine| ]] <!-- key article -->
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Porcupine_family"