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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Interwiki
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| en = Porcupine
: This revision was by author [[User:guest|guest]] and made on <tt>2011-08-16 15:19:06 UTC</tt>.<br>
| de = Porcupine
: The original revision id was <tt>246298959</tt>.<br>
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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<h4>Original Wikitext content:</h4>
{{Infobox regtemp
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">I could write an article here about porcupine if I wanted to, even though I'm not an admin.</pre></div>
| Title = Porcupine
<h4>Original HTML content:</h4>
| Subgroups = 2.3.5, 2.3.5.11, 2.3.5.7.11
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Porcupine&lt;/title&gt;&lt;/head&gt;&lt;body&gt;I could write an article here about porcupine if I wanted to, even though I'm not an admin.&lt;/body&gt;&lt;/html&gt;</pre></div>
| 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: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 [[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]]).
 
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{{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 and alternative 7-limit extensions. See [[Porcupine extensions]] for a discussion on [[13-limit]] [[extension]]s.
 
== Interval chain ==
{{Main| Porcupine intervals }}
 
In the following table, odd harmonics 1–11 are in '''bold'''.
 
{| class="wikitable center-all right-2 left-3 right-7 left-8"
|-
! colspan="5" | Up from the tonic, and fourthward
! colspan="5" | Down from the octave, and fifthward
|-
! #
! Cents*
! Ratios
! Porcupine<br>notation
! Ups and downs<br>notation
! #
! Cents*
! Ratios
! Porcupine<br>notation
! Ups and downs<br>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
|}
<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.
 
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 ==
{{Main| 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.
 
{| class="wikitable"
|-
| [[File:OtonalPentad_JI.mp3]]
| [[File:OtonalPentad_22edo.mp3]]
| [[File:OtonalPentad_29edo.mp3]]
|-
| 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 [[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{{c}} depending on the tuning. This can be considered the "chroma" of porcupine temperament.
 
== Scales ==
[[File:porcupine8.jpg|thumb|Porcupine[8]]] 
 
{{Main| Porcupine scales }}
 
; Mos scales, tuning optimized on the 2.3.5.11 subgroup
* [[Porkypine7]]
* [[Porkypine8]]
* [[Porkypine15]]
 
; Mos scales, 8/5.12/7 [[Eigenmonzo|eigenmonzo (unchanged interval)]] tuning:
* [[Porcupinewoo15]]
* [[Porcupinewoo22]]
 
== Tunings ==
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained !! Constrained & skewed !! Destretched
|-
! Equilateral
| CEE: ~10/9 = 163.6049{{c}}
| CSEE: ~10/9 = 163.2835{{c}}
| POEE: ~10/9 = 163.9280{{c}}
|-
! Tenney
| CTE: ~10/9 = 164.1659{{c}}
| CWE: ~10/9 = 164.0621{{c}}
| POTE: ~10/9 = 163.9504{{c}}
|-
! Benedetti, <br>Wilson
| CBE: ~10/9 = 164.3761{{c}}
| CSBE: ~10/9 = 164.3761{{c}}
| POBE: ~10/9 = 164.1610{{c}}
|}
 
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 2.3.5.11-subgroup norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained !! Constrained & skewed !! Destretched
|-
! Equilateral
| CEE: ~11/10 = 163.1459{{c}}
| CSEE: ~11/10 = 162.8445{{c}}
| POEE: ~11/10 = 164.1867{{c}}
|-
! Tenney
| CTE: ~11/10 = 163.8867{{c}}
| CWE: ~11/10 = 163.9951{{c}}
| POTE: ~11/10 = 164.0777{{c}}
|-
! Benedetti, <br>Wilson
| CBE: ~11/10 = 164.2393{{c}}
| CSBE: ~11/10 = 164.4623{{c}}
| POBE: ~11/10 = 164.2221{{c}}
|}
 
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained !! Constrained & skewed !! Destretched
|-
! Equilateral
| CEE: ~11/10 = 162.4448{{c}}
| CSEE: ~11/10 = 162.2333{{c}}
| POEE: ~11/10 = 162.2522{{c}}
|-
! Tenney
| CTE: ~11/10 = 163.1055{{c}}
| CWE: ~11/10 = 162.8156{{c}}
| POTE: ~11/10 = 162.7474{{c}}
|-
! Benedetti, <br>Wilson
| CBE: ~11/10 = 163.5299{{c}}
| CSBE: ~11/10 = 163.2310{{c}}
| POBE: ~11/10 = 163.0304{{c}}
|}
 
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! EDO<br>generator
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Comments
|-
| '''[[8edo|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
|-
| '''[[15edo|2\15]]'''
|
| '''160.000'''
| '''Lower bound of 7-odd-limit to 11-limit 15-odd-limit diamond monotone'''
|-
|
| [[7/4]]
| 161.471
|
|-
| [[52edo|7\52]]
|
| 161.538
| 52b val
|-
|
| [[14/11]]
| 161.751
|
|-
|
| [[7/5]]
| 162.047
|
|-
| [[37edo|5\37]]
|
| 162.162
|
|-
|
| [[16/11]]
| 162.171
|
|-
| [[96edo|13\96]]
|
| 162.500
| 96b val
|-
| [[59edo|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
|
|-
| '''[[22edo|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
|-
| [[51edo|7\51]]
|
| 164.706
| 51d val
|-
|
| [[11/10]]
| 165.004
|
|-
| [[29edo|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
|-
| [[36edo|5\36]]
|
| 166.667
| 36cde val
|-
| '''[[7edo|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
|}
<nowiki/>* 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 ==
* [[Porcupine notation]]
* [[Porcupine modes]]
* [[Porcupine temperament modal harmony]]
* [[Porcupine Album Project]]
 
== Music ==
=== 20th century ===
; [[Herman Miller]]
* [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 ===
; [[Flora Canou]]
* [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]]
* [https://cellularautomaton.bandcamp.com/track/minnow ''Minnow''] (2024) – in [[29edo]] tuning
 
; [[Paul Erlich]]
* [https://web.archive.org/web/20070928093239/http://66.98.148.43/~xenharmo/mp3/erlich/glassic.mp3 ''Glassic''] – in [[22edo]] tuning (at least the beginning part is in porcupine.)
 
; [[Jake Freivald]]
* ''[https://soundcloud.com/jdfreivald/porcupine-comma-pump Porcupine Comma Pump]''
 
; [[Cody Hallenbeck]]
* ''Porcupine Walk'' (2019)
** [https://soundcloud.com/cody-hallenbeck/porcupine-walk 15edo version] · [https://soundcloud.com/cody-hallenbeck/porcupine-walk-22edo 22edo version]
 
; [[Lillian Hearne]]
* [https://soundcloud.com/lillianhearne/mass-in-22edo-sanctus ''Sanctus''] (2015)
 
; [[Andrew Heathwaite]]
* [https://soundclick.com/share.cfm?id=8839060 ''being a''] (2010) – in Porcupine[8], mode 1|6, 22edo tuning
 
; [[Jollybard]]
* [https://soundcloud.com/jollybard/porcupeen ''Porcupeen''] (2017)
* [https://jollybard.bandcamp.com/track/porcupine "Porcupine"], from ''pato, with friends'' (2019)
 
; [[Igliashon Jones]]
* [https://cityoftheasleep.bandcamp.com/track/second-breakfast-15edo ''Second Breakfast (15edo)''] (2018){{dead link}}
 
; [[Löis Lancaster]]
* [https://soundcloud.com/lois-lancaster/porcupine-experience ''Porcupine Experience''] (2012) – in 22edo tuning
 
; [[John Moriarty]]
* [https://www.youtube.com/watch?v=se79rdp705Y ''Flying Straight Down''] (2020) – in 22edo tuning
 
; [[Omega9]]
* [https://www.youtube.com/watch?v=DSao0Yg3Tck ''Life on Mars''] (2014)
 
; [[Petr Pařízek]]
* [[:File:AmongOtherThings2.mp3|''Among Other Things 2'']]
 
; [[Ray Perlner]]
* [https://www.youtube.com/watch?v=8reCr2nDGbw ''Porcupine Lullaby''] (2020) – in 37edo tuning
* [https://www.youtube.com/playlist?list=PLkW9S8bpltfw464vJg3CAJJbV4IR6ggPd ''Porcupine{{lbrack}}7{{rbrack}} Modal Fugues''] – 7-piece playlist
 
; [[Gene Ward Smith]] and {{w|Modest Mussorgsky}}
* [https://www.archive.org/download/NightOnPorcupineMountain/Genewardsmithmussorgsky-NightOnPorcupineMountain.mp3 ''Night on Porcupine Mountain''] (archived 2010) – in 22edo tuning
 
; [[Chris Vaisvil]]
* ''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/20240118050711/http://micro.soonlabel.com/15-ET/daily20111231-porcupine15-piano.mp3 ''15 Quills''] – piano solo
* [https://web.archive.org/web/20231121043724/http://micro.soonlabel.com/15-ET/daily20111231-porcupine15-prickly-side-of-love.mp3 ''Prickly Side of Love''] – rock band with vocals
* [https://web.archive.org/web/20221221154102/http://micro.soonlabel.com/15-ET/daily20120102-porcupine-organ.mp3 ''Porcupine Organ Composition'']
 
; [[Nick Vuci]]
* [https://en.xen.wiki/images/0/0b/NickVuci-20230426-22edo-PorcupinePrelude1.mp3 ''Porcupine Prelude 1''] – in 22edo tuning
* [https://en.xen.wiki/images/3/39/NickVuci-20230518-22edo-PorcupinePrelude2.mp3 ''Porcupine Prelude 2''] – in 22edo tuning
* [https://en.xen.wiki/images/b/bd/NickVuci-20230521-22edo-PorcupinePrelude3.mp3 ''Porcupine Prelude 3''] – in 22edo tuning
* [https://en.xen.wiki/images/0/0b/NickVuci-20230523-22edo-Praeambulum.mp3 ''Porcupine Praeambulum''] – in 22edo tuning
* [https://en.xen.wiki/images/2/26/NickVuci-20230531-22edo-PorcupineChoraleWithPrelude.mp3 ''Porcupine Chorale with Prelude "Nature's Lament"''] – in 22edo tuning
 
; [[Well-Tempered Fox]]
* [https://www.youtube.com/watch?v=INM6J9pS_xE ''Porcupine Major Overture''] (2015) – in 22edo tuning
* [https://soundcloud.com/pianodog/waltzing-in-candyland-15-edo ''Waltzing in Candyland''] (2015) – in Porcupine[8], 15edo tuning
 
; [[Juhani Nuorvala]]
* [https://www.youtube.com/watch?v=aAHkjOvplVg ''Kellot (Bells)''] (2025) – in 96edo tuning
 
[[Category:Porcupine| ]] <!-- Main article -->
[[Category:Rank-2 temperaments]]
[[Category:Porcupine family]]
[[Category:Archytas clan]]
[[Category:Keemic temperaments]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/Porcupine"