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{{interwiki
{{Interwiki
| en = Porcupine
| de = Porcupine
| de = Porcupine
| en = Porcupine
| es =  
| es =  
| ja =  
| ja =  
}}
{{Infobox regtemp
| Title = Porcupine
| Subgroups = 2.3.5, 2.3.5.11, 2.3.5.7.11
| Comma basis = [[250/243]] (2.3.5);<br>[[55/54]], [[100/99]] (2.3.5.11);<br>[[55/54]], [[64/63]], [[100/99]] (2.3.5.7.11)
| Mapping = 1; -3 -5 6 -4
| Edo join 1 = 15 | Edo join 2 = 22
| Generators = 10/9
| Generators tuning = 163
| Optimization method = CWE
| MOS scales = [[1L&nbsp;6s]], [[7L&nbsp;1s]], [[7L&nbsp;8s]]
| Pergen = (P8, P4/3)
| Color name = Triyoti
| Odd limit 1 = 5 | Mistuning 1 = 9.8 | Complexity 1 = 7
| 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, tuned flat to around 160–165 [[cent]]s, two of which represent [[6/5]] and three of which represent [[4/3]], so that the generator represents [[10/9]], the difference between the two, and [[250/243]], the porcupine [[comma]], is [[tempered out]]. As a consequence of this, 4/3 is divided into 3 equal parts, and 6/5 is divided into 2 of those same equal parts. Its [[pergen]] is (P8, P4/3). This is obviously in stark contrast to [[meantone]] temperaments, including [[12edo]], where the 10/9 interval is sharpened to merge with [[9/8]]. 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]]).  


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 the [[27/20]] acute fourth of the JI diatonic scale 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.  
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 ==
Line 24: Line 39:


{| class="wikitable center-all right-2 left-3 right-7 left-8"
{| class="wikitable center-all right-2 left-3 right-7 left-8"
! colspan="5" | Up from the tonic, aka fourthward
|-
! colspan="5" | Down from the octave, aka fifthward
! colspan="5" | Up from the tonic, and fourthward
! colspan="5" | Down from the octave, and fifthward
|-
|-
! #
! #
Line 54: Line 70:
| P2
| P2
| vM2 = ^^m2
| vM2 = ^^m2
| -1
| −1
| 1037.2
| 1037.2
| 9/5, 11/6, 20/11
| 9/5, 11/6, 20/11
Line 65: Line 81:
| m3
| m3
| ^m3 = vvM3
| ^m3 = vvM3
| -2
| −2
| 874.4
| 874.4
| 5/3, 18/11
| 5/3, 18/11
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| m4
| m4
| P4
| P4
| -3
| −3
| 711.6
| 711.6
| '''3/2'''
| '''3/2'''
Line 87: Line 103:
| m5
| m5
| v5 = ^^d5
| v5 = ^^d5
| -4
| −4
| 548.7
| 548.7
| '''11/8''', 15/11
| '''11/8''', 15/11
Line 98: Line 114:
| m6
| m6
| ^m6 = vvM6
| ^m6 = vvM6
| -5
| −5
| 385.9
| 385.9
| '''5/4'''
| '''5/4'''
Line 109: Line 125:
| d7
| d7
| m7
| m7
| -6
| −6
| 223.1
| 223.1
| 8/7, '''9/8'''
| 8/7, '''9/8'''
Line 117: Line 133:
| 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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| d2
| d2
| ^m2 = vvM2
| ^m2 = vvM2
| -8
| −8
| 1097.5
| 1097.5
| 15/8, 40/21
| 15/8, 40/21
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| d3
| d3
| m3
| m3
| -9
| −9
| 934.7
| 934.7
| 12/7
| 12/7
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| d4
| d4
| v4 = ^^d4
| v4 = ^^d4
| -10
| −10
| 771.8
| 771.8
| 11/7
| 11/7
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| d5
| d5
| ^d5 = vv5
| ^d5 = vv5
| -11
| −11
| 609.0
| 609.0
| 10/7
| 10/7
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| d6
| d6
| m6
| m6
| -12
| −12
| 446.2
| 446.2
| 9/7
| 9/7
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<nowiki/>* In 11-limit [[CWE tuning]], octave reduced
<nowiki/>* In 11-limit [[CWE tuning]], octave reduced


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]].
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{{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 ==
Line 191: Line 209:


{| class="wikitable"
{| class="wikitable"
|-
| [[File:OtonalPentad_JI.mp3]]
| [[File:OtonalPentad_JI.mp3]]
| [[File:OtonalPentad_22edo.mp3]]
| [[File:OtonalPentad_22edo.mp3]]
| [[File:OtonalPentad_29edo.mp3]]
| [[File:OtonalPentad_29edo.mp3]]
|-
|-
| 8:9:10:11:12 chord, in just intonation. <br> All intervals are slightly different.
| 8:9:10:11:12 chord, in just intonation.<br>All intervals are slightly different.
| Porcupine-tempered 8:9:10:11:12 chord, in [[22edo]]. <br> Except the first, the intervals are the same.
| Porcupine-tempered 8:9:10:11:12 chord, in [[22edo]].<br>Except the first, the intervals are the same.
| Porcupine-tempered 8:9:10:11:12 chord, in [[29edo]]. <br> Except the first, the intervals are the same.
| Porcupine-tempered 8:9:10:11:12 chord, in [[29edo]].<br>Except the first, the intervals are the same.
|}
|}


The interval representing both [[25/24]] and [[81/80]] can be found in this interval chain at -7 steps, and ranges from about 45 to 80 cents depending on the tuning. This can be considered the "chroma" of porcupine temperament.
The interval representing both [[25/24]] and [[81/80]] can be found in this interval chain at −7 steps, and ranges from about 45 to 80{{c}} depending on the tuning. This can be considered the "chroma" of porcupine temperament.


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


Line 210: Line 231:
* [[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]]
Line 216: Line 237:
== 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" |  
Line 240: Line 261:


{| 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" |  
Line 264: Line 285:


{| 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
Line 336: Line 362:
|-
|-
|  
|  
| 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 ==
* [[Porcupine notation]]
* [[Porcupine notation]]
* [[Porcupine modes]]
* [[Porcupine modes]]
* [[Porcupine temperament modal harmony]]
* [[Porcupine Album Project]]
* [[Porcupine Album Project]]


Line 427: Line 465:
=== 20th century ===
=== 20th century ===
; [[Herman Miller]]
; [[Herman Miller]]
* [https://sites.google.com/site/teamouse/home#TOC-Mizarian-music ''Mizarian Porcupine Overture''] (1999) – in [[15edo]] tuning, namesake of the temperament
* [https://sites.google.com/site/teamouse/home#TOC-Mizarian-music ''Mizarian Porcupine Overture''] (1999) – [https://web.archive.org/web/20201127014859/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/MizarianPorcupineOverture.mp3 play] – in [[15edo]] tuning, namesake of the temperament


=== 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 440: 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]]
Line 479: Line 517:


; [[Chris Vaisvil]]
; [[Chris Vaisvil]]
* [https://web.archive.org/web/20231228102528/http://micro.soonlabel.com/15-ET/daily20110619_millers_porcupine_7a.mp3 ''Playing Gently with Miller's Porcupine'']
* ''Gently Playing With Miller's Porcupine'' (2011) – [https://www.chrisvaisvil.com/four-pieces-in-porcupine-temperament/ blog] | [https://web.archive.org/web/20231228102528/http://micro.soonlabel.com/15-ET/daily20110619_millers_porcupine_7a.mp3 play] – in Porcupine[7], mode 3|3, 15edo tuning
* [https://web.archive.org/web/20231121064756/http://micro.soonlabel.com/15-ET/daily20111231-porcupine15-indian.mp3 ''15 Porcupines in India''] – sarangi, tambura and sitar improvisation
* [https://web.archive.org/web/20231121064756/http://micro.soonlabel.com/15-ET/daily20111231-porcupine15-indian.mp3 ''15 Porcupines in India''] – sarangi, tambura and sitar improvisation
* [https://web.archive.org/web/20240118050711/http://micro.soonlabel.com/15-ET/daily20111231-porcupine15-piano.mp3 ''15 Quills''] – piano solo
* [https://web.archive.org/web/20240118050711/http://micro.soonlabel.com/15-ET/daily20111231-porcupine15-piano.mp3 ''15 Quills''] – piano solo
Line 496: Line 534:
* [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:Temperaments]]
[[Category:Porcupine| ]] <!-- Main article -->
[[Category:Porcupine| ]] <!-- Main article -->
[[Category:Rank-2 temperaments]]
[[Category:Rank-2 temperaments]]

Latest revision as of 00:22, 30 April 2026

Porcupine
Subgroups 2.3.5, 2.3.5.11, 2.3.5.7.11
Comma basis 250/243 (2.3.5);
55/54, 100/99 (2.3.5.11);
55/54, 64/63, 100/99 (2.3.5.7.11)
Reduced mapping ⟨1; -3 -5 6 -4]
ET join 15 & 22
Generators (CWE) ~10/9 = 163 ¢
MOS scales 1L 6s, 7L 1s, 7L 8s
Ploidacot omega-tricot
Pergen (P8, P4/3)
Color name Triyoti
Minimax error 5-odd-limit: 9.8 ¢;
11-limit 15-odd-limit: 19.9 ¢
Target scale size 5-odd-limit: 7 notes;
11-limit 15-odd-limit: 15 notes
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.
Symmetric minor mode of the Porcupine[7] scale, containing two equal tetrachords with a major wholetone between them, in 22edo tuning.

Porcupine is a temperament that is generated by a minor whole tone which is tuned flat to around 160–165 cents. Two generators (stacked) represent 6/5, and three represent 4/3, so that the porcupine comma (250/243) is 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).

Porcupine can be thought of as a 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 ¢), has already been flattened to merge it with (6/5)3, 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 ¢, with the best tunings around 711–712 ¢, 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 temperament.

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

Interval chain

Main article: Porcupine intervals

In the following table, odd harmonics 1–11 are in bold.

Up from the tonic, and fourthward Down from the octave, and fifthward
# Cents* Ratios Porcupine
notation 		Ups and downs
notation 		# Cents* Ratios Porcupine
notation 		Ups and downs
notation
0 0.0 1/1 P1 P1 0 1200.0 2/1 P8 P8
1 162.8 10/9, 11/10, 12/11 P2 vM2 = ^^m2 −1 1037.2 9/5, 11/6, 20/11 P7 ^m7 = vvM7
2 325.6 6/5, 11/9 m3 ^m3 = vvM3 −2 874.4 5/3, 18/11 M6 vM6 = ^^m6
3 488.4 4/3 m4 P4 −3 711.6 3/2 M5 P5
4 651.3 16/11, 22/15 m5 v5 = ^^d5 −4 548.7 11/8, 15/11 M4 ^4 = vvA4
5 814.1 8/5 m6 ^m6 = vvM6 −5 385.9 5/4 M3 vM3 = ^^m3
6 976.9 7/4, 16/9 d7 m7 −6 223.1 8/7, 9/8 A2 M2
7 1139.7 35/18, 48/25, 64/33 d8 v8 = ^^d8 −7 60.3 25/24, 33/32, 36/35 A1 ^1 = vvA1
8 102.5 16/15, 21/20 d2 ^m2 = vvM2 −8 1097.5 15/8, 40/21 A7 vM7 = ^^m7
9 265.3 7/6 d3 m3 −9 934.7 12/7 A6 M6
10 428.2 14/11 d4 v4 = ^^d4 −10 771.8 11/7 A5 ^5 = vvA5
11 591.0 7/5 d5 ^d5 = vv5 −11 609.0 10/7 A4 vA4 = ^^4
12 753.8 14/9 d6 m6 −12 446.2 9/7 A3 M3

* 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.

Besides the specific tuning shown here, there is a range of acceptable porcupine tunings that includes generators as small as 160 ¢ (15edo) and as large as 165.5 ¢ (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

Main article: Chords of porcupine

12/11, 11/10, and 10/9 are all represented by the same interval, the generator. This makes chords such as 8:9:10:11:12 exceptionally common and easy to find.

8:9:10:11:12 chord, in just intonation.
All intervals are slightly different. 		Porcupine-tempered 8:9:10:11:12 chord, in 22edo.
Except the first, the intervals are the same. 		Porcupine-tempered 8:9:10:11:12 chord, in 29edo.
Except the first, the intervals are the same.

The interval representing both 25/24 and 81/80 can be found in this interval chain at −7 steps, and ranges from about 45 to 80 ¢ depending on the tuning. This can be considered the "chroma" of porcupine temperament.

Scales

Porcupine[8]
Main article: Porcupine scales
Mos scales, tuning optimized on the 2.3.5.11 subgroup
Mos scales, 8/5.12/7 eigenmonzo (unchanged interval) tuning

Tunings

5-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Equilateral CEE: ~10/9 = 163.6049 ¢ CSEE: ~10/9 = 163.2835 ¢ POEE: ~10/9 = 163.9280 ¢
Tenney CTE: ~10/9 = 164.1659 ¢ CWE: ~10/9 = 164.0621 ¢ POTE: ~10/9 = 163.9504 ¢
Benedetti,
Wilson 		CBE: ~10/9 = 164.3761 ¢ CSBE: ~10/9 = 164.3761 ¢ POBE: ~10/9 = 164.1610 ¢
2.3.5.11-subgroup norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Equilateral CEE: ~11/10 = 163.1459 ¢ CSEE: ~11/10 = 162.8445 ¢ POEE: ~11/10 = 164.1867 ¢
Tenney CTE: ~11/10 = 163.8867 ¢ CWE: ~11/10 = 163.9951 ¢ POTE: ~11/10 = 164.0777 ¢
Benedetti,
Wilson 		CBE: ~11/10 = 164.2393 ¢ CSBE: ~11/10 = 164.4623 ¢ POBE: ~11/10 = 164.2221 ¢
11-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Equilateral CEE: ~11/10 = 162.4448 ¢ CSEE: ~11/10 = 162.2333 ¢ POEE: ~11/10 = 162.2522 ¢
Tenney CTE: ~11/10 = 163.1055 ¢ CWE: ~11/10 = 162.8156 ¢ POTE: ~11/10 = 162.7474 ¢
Benedetti,
Wilson 		CBE: ~11/10 = 163.5299 ¢ CSBE: ~11/10 = 163.2310 ¢ POBE: ~11/10 = 163.0304 ¢

Tuning spectrum

EDO
generator 		Unchanged interval
(eigenmonzo)* 		Generator (¢) Comments
1\8 150.000 Lower bound of 5-odd-limit diamond monotone
12/11 150.637 Lower bound of 11-odd-limit and 11-limit 15-odd-limit diamond tradeoff
6/5 157.821 1/2-comma; lower bound of 5-, 7-, and 9-odd-limit diamond tradeoff
2\15 160.000 Lower bound of 7-odd-limit to 11-limit 15-odd-limit diamond monotone
7/4 161.471
7\52 161.538 52b val
14/11 161.751
7/5 162.047
5\37 162.162
16/11 162.171
13\96 162.500 96b val
8\59 162.712
8/5 162.737 2/5-comma, 5- and 7-odd-limit minimax
28/15 162.897
7/6 162.986
3\22 163.636 Upper bound of 7-odd-limit to 11-limit 15-odd-limit diamond monotone
14/9 163.743 9-, 11-, and 11-limit 15-odd-limit minimax
16/15 163.966 3/8-comma
7\51 164.706 51d val
11/10 165.004
4\29 165.517 29d val
22/15 165.762
4/3 166.015 1/3-comma; upper bound of 5- and 7-odd-limit diamond tradeoff
5\36 166.667 36cde val
1\7 171.429 Upper bound of 5-odd-limit diamond monotone
11/9 173.704
10/9 182.404 Untempered generator; upper bound of 9- to 15-odd-limit diamond tradeoff

* Besides the octave

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 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

Music

20th century

Herman Miller

21st century

Flora Canou
CellularAutomaton
Paul Erlich
  • Glassic – in 22edo tuning (at least the beginning part is in porcupine.)
Jake Freivald
Cody Hallenbeck
Lillian Hearne
Andrew Heathwaite
  • being a (2010) – in Porcupine[8], mode 1|6, 22edo tuning
Jollybard
Igliashon Jones
Löis Lancaster
John Moriarty
Omega9
Petr Pařízek
Ray Perlner
Gene Ward Smith and Modest Mussorgsky
Chris Vaisvil
Nick Vuci
Well-Tempered Fox
Juhani Nuorvala
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