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{{interwiki
{{Interwiki
| en = Porcupine
| de = Porcupine
| de = Porcupine
| en = Porcupine
| es =  
| es =  
| ja =  
| ja =  
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| Mapping = 1; -3 -5 6 -4
| Mapping = 1; -3 -5 6 -4
| Edo join 1 = 15 | Edo join 2 = 22
| Edo join 1 = 15 | Edo join 2 = 22
| Generator = 11/10
| Generators = 10/9
| Generator tuning = 164
| Generators tuning = 163
| Optimization method = CWE
| Optimization method = CWE
| MOS scales = [[1L 6s]], [[7L 1s]], [[7L 8s]], [[15L 7s]]
| MOS scales = [[1L 6s]], [[7L 1s]], [[7L 8s]]
| Ploidacot = omega-tricot
| Pergen = (P8, P4/3)
| Pergen = (P8, P4/3)
| Color name = Triyoti
| Color name = Triyoti
| Odd limit 1 = 5 | Mistuning 1 = ? | Complexity 1 = 15
| Odd limit 1 = 5 | Mistuning 1 = 9.8 | Complexity 1 = 7
| Odd limit 2 = (11-limit) 15 | Mistuning 2 = ? | Complexity 2 = 37
| Odd limit 2 = 11-limit 15 | Mistuning 2 = 19.9 | Complexity 2 = 15
}}
}}
[[File:porcupine.png|thumb|Porcupine equates three minor thirds (6/5, in red) with two perfect fourths (4/3, in green). To do so, it tempers out 250/243, which implies a generator of a flat 10/9.|600x600px]]
[[File:porcupine.png|thumb|Porcupine equates three minor thirds (6/5, in red) with two perfect fourths (4/3, in green). To do so, it tempers out 250/243, which implies a generator of a flat 10/9.|600x600px]]
[[File:porcupinesymmetricminor22edo.mp3|thumb|Symmetric minor mode of the Porcupine[7] scale, containing two equal tetrachords with a major wholetone between them, in [[22edo]] tuning.]]
[[File:porcupinesymmetricminor22edo.mp3|thumb|Symmetric minor mode of the Porcupine[7] scale, containing two equal tetrachords with a major wholetone between them, in [[22edo]] tuning.]]


'''Porcupine''' is a [[regular temperament|temperament]] that is [[generator|generated]] by a minor whole tone which is tuned flat to around 160–165 [[cent]]s, and the porcupine [[comma]] ([[250/243]]) is [[tempering out|tempered out]]. Two generators (stacked) represent [[6/5]], and three represent [[4/3]]; from this, the generator itself represents a (severely flattened) [[10/9]]. This is obviously in stark contrast to [[meantone]] temperaments, including [[12edo]], where 10/9 is tuned sharp and equated with [[9/8]] so that two of them reach a ''major'' third of [[5/4]]. The "equal tetrachord" formed by dividing 4/3 into 3 equal parts is a characteristic feature of many of porcupine's scales.  
'''Porcupine''' is a [[regular temperament|temperament]] that is [[generator|generated]] by a [[10/9|minor whole tone]] which is tuned flat to around 160–165 [[cent]]s. Two generators (stacked) represent [[6/5]], and three represent [[4/3]], so that the [[250/243|porcupine comma (250/243)]] is [[tempering out|tempered out]]; from this, the generator itself represents a very flat 10/9. This is in stark contrast to [[meantone]] temperaments, including [[12edo]], where 10/9 is tuned sharp and equated with [[9/8]] so that two of them reach a ''major'' third of [[5/4]]. The "equal tetrachord" formed by dividing 4/3 into 3 equal parts is a characteristic feature of many of porcupine's scales.  


One may also note that in [[just intonation]], a stack of three 6/5's is flat of the classical minor seventh [[9/5]] by [[25/24]], and a stack of two 4/3's is the Pythagorean minor seventh [[16/9]], which is flat of 9/5 by [[81/80]]. Thus, it can be determined that porcupine equates the syntonic comma 81/80 with the 5-limit chromatic semitone [[25/24]], which simplifies the 5-limit to a rank-2 structure in a simple way distinct from temperaments that reduce it to a strong extension of [[pythagorean]] (such as [[meantone]] and [[schismic]]).  
One may also note that in [[just intonation]], a stack of three 6/5's is flat of the classical minor seventh [[9/5]] by [[25/24]], and a stack of two 4/3's is the Pythagorean minor seventh [[16/9]], which is flat of 9/5 by [[81/80]]. Thus, it can be determined that porcupine equates the syntonic comma 81/80 with the 5-limit chromatic semitone [[25/24]], which simplifies the 5-limit to a rank-2 structure in a simple way distinct from temperaments that reduce it to a strong extension of [[pythagorean]] (such as [[meantone]] and [[schismic]]).  
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Porcupine can be thought of as a [[2.3.5.11 subgroup|2.3.5.11-subgroup]] temperament (sometimes called ''porkypine'') without much additional damage compared to the 5-limit; the generator here represents not only 10/9, but also [[11/10]] and [[12/11]] (equivalently, [[55/54]], [[100/99]], and [[121/120]] are tempered out), with the consequence that the [[11/9]] interval, usually considered a neutral third, is in porcupine identical to the 6/5 minor third, due to the extreme flatness of 10/9. This also means that [[27/20]], the 5-limit "acute fourth", is equivalent to [[11/8]] (rather than becoming 4/3 as in meantone), found at −4 generators (tuned to about 540–560 cents). This is because as the syntonic comma has been expanded, sharpening a fourth by a comma now leads to a significantly sharp interval close to the 11th harmonic. Porcupine is one of the most efficient temperaments in the 2.3.5.11 subgroup at a certain standard of accuracy.
Porcupine can be thought of as a [[2.3.5.11 subgroup|2.3.5.11-subgroup]] temperament (sometimes called ''porkypine'') without much additional damage compared to the 5-limit; the generator here represents not only 10/9, but also [[11/10]] and [[12/11]] (equivalently, [[55/54]], [[100/99]], and [[121/120]] are tempered out), with the consequence that the [[11/9]] interval, usually considered a neutral third, is in porcupine identical to the 6/5 minor third, due to the extreme flatness of 10/9. This also means that [[27/20]], the 5-limit "acute fourth", is equivalent to [[11/8]] (rather than becoming 4/3 as in meantone), found at −4 generators (tuned to about 540–560 cents). This is because as the syntonic comma has been expanded, sharpening a fourth by a comma now leads to a significantly sharp interval close to the 11th harmonic. Porcupine is one of the most efficient temperaments in the 2.3.5.11 subgroup at a certain standard of accuracy.


It is also very easy to extend porcupine to prime 7, because the 16/9, found at +6 generators (tuned to about 960–990 cents), has already been flattened to merge it with (6/5)<sup>3</sup>, and therefore can be equated to [[7/4]]. This makes porcupine a weak extension of [[archy]], splitting its generator into three parts; its Pythagorean major third is mapped to [[9/7]], and its fifth is tuned sharp, ranging from around 705–720 cents, with the best tunings around 711–712 cents, which roughly splits the damage on 7/4 and 9/7.
It is also very easy to extend porcupine to prime 7, because the 16/9, found at +6 generators (tuned to about 960–990{{c}}), has already been flattened to merge it with (6/5)<sup>3</sup>, and therefore can be equated to [[7/4]]. This makes porcupine a weak extension of [[archy]], splitting its generator into three parts; its Pythagorean major third is mapped to [[9/7]], and its fifth is tuned sharp, ranging from around 705–720{{c}}, with the best tunings around 711–712{{c}}, which roughly splits the damage on 7/4 and 9/7. This extension sets [[7/6]], 6/5, 5/4, and 9/7 equidistant, thus tempering out [[875/864]], making porcupine a [[keemic temperaments|keemic temperament]].


See [[Porcupine family #Porcupine]] for technical data. See [[Porcupine extensions]] for a discussion on [[13-limit]] [[extension]]s and alternative 7-limit extensions.
See [[Porcupine family #Porcupine]] for technical data and alternative 7-limit extensions. See [[Porcupine extensions]] for a discussion on [[13-limit]] [[extension]]s.


== Interval chain ==
== Interval chain ==
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| 7
| 7
| 1139.7
| 1139.7
| 48/25, 160/81
| 35/18, 48/25, 64/33
| d8
| d8
| v8 = ^^d8
| v8 = ^^d8
| −7
| −7
| 60.3
| 60.3
| 25/24, 81/80
| 25/24, 33/32, 36/35
| A1
| A1
| ^1 = vvA1
| ^1 = vvA1
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| M3
| M3
|}
|}
<nowiki>*</nowiki> In 11-limit [[CWE tuning]], octave reduced
<nowiki/>* In 11-limit [[CWE tuning]], octave reduced


In the ups and downs notation, the [[enharmonic unison]] is the trudsharp, the triple-down augmented unison. The porcupine notation does not have an enharmonic unison.
In the ups and downs notation, the [[enharmonic unison]] is the trudsharp, the triple-down augmented unison. The porcupine notation does not have an enharmonic unison.


Besides the specific tuning shown here, there is a range of acceptable porcupine tunings that includes generators as small as 160 cents ([[15edo]]) and as large as 165.5 cents ([[29edo]]). However, the 29edo patent val does not support 11-limit porcupine proper, since it does not temper out [[64/63]].
Besides the specific tuning shown here, there is a range of acceptable porcupine tunings that includes generators as small as 160{{c}} ([[15edo]]) and as large as 165.5{{c}} ([[29edo]]). However, the 29edo patent val does not support full 11-limit porcupine proper, since it does not temper out [[64/63]].


== Chords and harmony ==
== Chords and harmony ==
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== Scales ==
== Scales ==
[[File:porcupine8.jpg|thumb|Porcupine[8]]] 
{{Main| Porcupine scales }}
{{Main| Porcupine scales }}


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* [[Porkypine15]]
* [[Porkypine15]]


; Mos scales, 8/5.12/7 [[Eigenmonzo|eigenmonzo (unchanged-interval)]] tuning:  
; Mos scales, 8/5.12/7 [[Eigenmonzo|eigenmonzo (unchanged interval)]] tuning:  
* [[Porcupinewoo15]]
* [[Porcupinewoo15]]
* [[Porcupinewoo22]]
* [[Porcupinewoo22]]
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== Tunings ==
== Tunings ==
{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit prime-optimized tunings
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings
|-
|-
! rowspan="2" |  
! rowspan="2" |  
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{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 2.3.5.11-subgroup prime-optimized tunings
|+ style="font-size: 105%; white-space: nowrap;" | 2.3.5.11-subgroup norm-based tunings
|-
|-
! rowspan="2" |  
! rowspan="2" |  
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{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit prime-optimized tunings
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
|-
|-
! rowspan="2" |  
! rowspan="2" |  
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{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
|-
! Edo<br>generator
! EDO<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|-
| 1\8
| '''[[8edo|1\8]]'''
|  
|  
| 150.000
| '''150.000'''
| Lower bound of 5-odd-limit diamond monotone
| '''Lower bound of 5-odd-limit diamond monotone'''
|-
|-
|  
|  
| 12/11
| [[12/11]]
| 150.637
| 150.637
| Lower bound of 11-odd-limit diamond tradeoff
| Lower bound of 11-odd-limit and 11-limit 15-odd-limit diamond tradeoff
|-
|-
|  
|  
| 6/5
| [[6/5]]
| 157.821
| 157.821
| Lower bound of 5-, 7-, and 9-odd-limit diamond tradeoff
| 1/2-comma; lower bound of 5-, 7-, and 9-odd-limit diamond tradeoff
|-
|-
| 2\15
| '''[[15edo|2\15]]'''
|  
|  
| 160.000
| '''160.000'''
| Lower bound of 7-, 9-, and 11-odd-limit diamond monotone
| '''Lower bound of 7-odd-limit to 11-limit 15-odd-limit diamond monotone'''
|-
|-
|  
|  
| 8/7
| [[7/4]]
| 161.471
| 161.471
|  
|  
|-
|-
| [[52edo|7\52]]
|  
|  
| 14/11
| 161.538
| 52b val
|-
|
| [[14/11]]
| 161.751
| 161.751
|  
|  
|-
|-
|  
|  
| 7/5
| [[7/5]]
| 162.047
| 162.047
|  
|  
|-
|-
| 5\37
| [[37edo|5\37]]
|  
|  
| 162.162
| 162.162
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|-
|-
|  
|  
| 11/8
| [[16/11]]
| 162.171
| 162.171
|  
|  
|-
|-
| 8\59
| [[96edo|13\96]]
|
| 162.500
| 96b val
|-
| [[59edo|8\59]]
|  
|  
| 162.712
| 162.712
|
|  
|-
|-
|  
|  
| 5/4
| [[8/5]]
| 162.737
| 162.737
| 5- and 7-odd-limit minimax
| 2/5-comma, 5- and 7-odd-limit minimax
|-
|-
|  
|  
| 15/14
| [[28/15]]
| 162.897
| 162.897
|  
|  
|-
|-
|  
|  
| 7/6
| [[7/6]]
| 162.986
| 162.986
|  
|  
|-
|-
| 3\22
| '''[[22edo|3\22]]'''
|  
|  
| 163.636
| '''163.636'''
| Upper bound of 7-, 9-, and 11-odd-limit diamond monotone
| '''Upper bound of 7-odd-limit to 11-limit 15-odd-limit diamond monotone'''
|-
|-
|  
|  
| 9/7
| [[14/9]]
| 163.743
| 163.743
| 9- and 11-odd-limit minimax
| 9-, 11-, and 11-limit 15-odd-limit minimax
|-
|-
|  
|  
| 16/15
| [[16/15]]
| 163.966
| 163.966
|  
| 3/8-comma
|-
|-
| 7\51
| [[51edo|7\51]]
|  
|  
| 164.706
| 164.706
|  
| 51d val
|-
|-
|  
|  
| 11/10
| [[11/10]]
| 165.004
| 165.004
|  
|  
|-
|-
| 4\29
| [[29edo|4\29]]
|  
|  
| 165.517
| 165.517
|  
| 29d val
|-
|-
|  
|  
| 15/11
| [[22/15]]
| 165.762
| 165.762
|  
|  
|-
|-
|  
|  
| 4/3
| [[4/3]]
| 166.015
| 166.015
| Upper bound of 5- and 7-odd-limit diamond tradeoff
| 1/3-comma; upper bound of 5- and 7-odd-limit diamond tradeoff
|-
| [[36edo|5\36]]
|
| 166.667
| 36cde val
|-
|-
| 1\7
| '''[[7edo|1\7]]'''
|  
|  
| 171.429
| '''171.429'''
| Upper bound of 5-odd-limit diamond monotone
| '''Upper bound of 5-odd-limit diamond monotone'''
|-
|-
|  
|  
| 11/9
| [[11/9]]
| 173.704
| 173.704
|  
|  
|-
|-
|  
|  
| 10/9
| [[10/9]]
| 182.404
| 182.404
| Upper bound of 9- and 11-odd-limit diamond tradeoff
| Untempered generator; upper bound of 9- to 15-odd-limit diamond tradeoff
|}
|}
<nowiki/>* Besides the octave


== History ==
== History ==
Porcupine temperament/scales were discovered by [[Dave Keenan]], but did not have a name until [[Herman Miller]] mentioned that his ''Mizarian Porcupine Overture'' in 15et had a section that pumps the 250/243 comma. Although this music did not use a porcupine mos or [[modmos]] (which would have 7 or 8 notes), the name was adopted for such scales as well, once the essentially one-to-one relationship between vanishing commas and sequences of [[DE]] scales was fully evident. It was clear that even though Herman's piece was in 15edo, 22edo was a porcupine tuning par excellence, and that was an interesting development in itself.
Porcupine temperament/scales were discovered by [[Dave Keenan]], but did not have a name until [[Herman Miller]] mentioned that his ''Mizarian Porcupine Overture'' in 15et had a section that pumps the 250/243 comma. Although this music did not use a porcupine mos or [[modmos]] (which would have 7 or 8 notes), the name was adopted for such scales as well, once the essentially one-to-one relationship between vanishing commas and sequences of [[MOS]] scales was fully evident. It was clear that even though Herman's piece was in 15edo, 22edo was a porcupine tuning par excellence, and that was an interesting development in itself.


== See also ==
== See also ==
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=== 21st century ===
=== 21st century ===
; [[Flora Canou]]
; [[Flora Canou]]
* [https://soundcloud.com/floracanou/april-porkfest?in=floracanou/sets/totmc-suite-vol-1 "April Porkfest"] from [https://soundcloud.com/floracanou/sets/totmc-suite-vol-1 ''TOTMC Suite Vol. 1''] (2023) – in 11-limit CTE tuning
* [https://soundcloud.com/floracanou/april-porkfest?in=floracanou/sets/totmc-suite "April Porkfest"] from [https://soundcloud.com/floracanou/sets/totmc-suite ''TOTMC Suite''] (2023–2025) – in 11-limit CTE tuning


; [[User:CellularAutomaton|CellularAutomaton]]
; [[User:CellularAutomaton|CellularAutomaton]]
Line 461: Line 478:


; [[Jake Freivald]]
; [[Jake Freivald]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/porcupine-comma-pump.mp3 ''Porcupine Comma Pump'']{{dead link}}
* ''[https://soundcloud.com/jdfreivald/porcupine-comma-pump Porcupine Comma Pump]''


; [[Cody Hallenbeck]]
; [[Cody Hallenbeck]]
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* [https://soundcloud.com/pianodog/waltzing-in-candyland-15-edo ''Waltzing in Candyland''] (2015) – in Porcupine[8], 15edo tuning
* [https://soundcloud.com/pianodog/waltzing-in-candyland-15-edo ''Waltzing in Candyland''] (2015) – in Porcupine[8], 15edo tuning


== Diagrams ==
; [[Juhani Nuorvala]]
[[File:porcupine8.jpg]]    
* [https://www.youtube.com/watch?v=aAHkjOvplVg ''Kellot (Bells)''] (2025) – in 96edo tuning


[[Category:Porcupine| ]] <!-- Main article -->
[[Category:Porcupine| ]] <!-- Main article -->
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