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{{interwiki
{{Interwiki
| en = Porcupine
| de = Porcupine
| de = Porcupine
| en = Porcupine
| es =  
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}}
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[[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.]]
{{Infobox regtemp
[[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.]]
| Title = Porcupine
'''Porcupine''' is a [[linear temperament]] that tempers out [[250/243]], the porcupine [[comma]], and whose generator is a [[10/9|minor whole tone (10/9)]] which is tuned flat to around 160–170 [[cent]]s such that two of them stack to a [[6/5|classic minor third (6/5)]]. It can be thought of as a [[5-limit]], [[7-limit]], or [[11-limit]] temperament, or a 2.3.5.11 [[subgroup temperament]]. It is one of the best temperaments in the 2.3.5.11 subgroup, with a unique combination of efficiency and accuracy.
| 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: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.  


Porcupine's basic 5-limit harmonic structure can be understood by noting that tempering out 250/243 also makes (4/3)<sup>2</sup> equivalent to (6/5)<sup>3</sup>; or, in other words, this measn that two "perfect fourths" equals three "minor thirds". 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. This is obviously in stark contrast to [[12edo]], in which neither 4/3 nor 6/5 can be divided into any number of equal parts. 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 ==
== Interval chain ==
{{Main|Porcupine intervals}}
{{Main| Porcupine intervals }}


{| class="wikitable center-all right-2 left-3 right-6 left-7"
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
! Cents*
! Ratios
! Ratios
! Ups and Downs <br> Notation
! Porcupine<br>notation
! Ups and downs<br>notation
! #
! #
! 2/1 inverse
! Cents*
! Ratios
! Ratios
! Ups and Downs <br> Notation
! Porcupine<br>notation
! Ups and downs<br>notation
|-
|-
| 0
| 0
| 0.00
| 0.0
| 1/1
| '''1/1'''
| P1
| P1
| P1
| 0
| 0
| 1200.00
| 1200.0
| 2/1
| '''2/1'''
| P8
| P8
| P8
|-
|-
| 1
| 1
| 162.75
| 162.8
| 12/11, 11/10, 10/9
| 10/9, 11/10, 12/11
| P2
| vM2 = ^^m2
| vM2 = ^^m2
| -1
| −1
| 1037.25
| 1037.2
| 9/5, 20/11, 11/6
| 9/5, 11/6, 20/11
| P7
| ^m7 = vvM7
| ^m7 = vvM7
|-
|-
| 2
| 2
| 325.50
| 325.6
| 6/5, 11/9
| 6/5, 11/9
| m3
| ^m3 = vvM3
| ^m3 = vvM3
| -2
| −2
| 874.50
| 874.4
| 18/11, 5/3
| 5/3, 18/11
| M6
| vM6 = ^^m6
| vM6 = ^^m6
|-
|-
| 3
| 3
| 488.25
| 488.4
| 4/3
| 4/3
| m4
| P4
| P4
| -3
| −3
| 711.75
| 711.6
| 3/2
| '''3/2'''
| M5
| P5
| P5
|-
|-
| 4
| 4
| 651.00
| 651.3
| 16/11, 22/15
| 16/11, 22/15
| m5
| v5 = ^^d5
| v5 = ^^d5
| -4
| −4
| 549.00
| 548.7
| 15/11, 11/8
| '''11/8''', 15/11
| M4
| ^4 = vvA4
| ^4 = vvA4
|-
|-
| 5
| 5
| 813.75
| 814.1
| 8/5
| 8/5
| m6
| ^m6 = vvM6
| ^m6 = vvM6
| -5
| −5
| 386.25
| 385.9
| 5/4
| '''5/4'''
| M3
| vM3 = ^^m3
| vM3 = ^^m3
|-
|-
| 6
| 6
| 976.50
| 976.9
| 7/4, 16/9
| '''7/4''', 16/9
| d7
| m7
| m7
| -6
| −6
| 223.50
| 223.1
| 9/8, 8/7
| 8/7, '''9/8'''
| A2
| M2
| M2
|-
|-
| 7
| 7
| 1139.25
| 1139.7
| 48/25, 160/81
| 35/18, 48/25, 64/33
| d8
| v8 = ^^d8
| v8 = ^^d8
| -7
| −7
| 60.75
| 60.3
| 81/80, 25/24
| 25/24, 33/32, 36/35
| A1
| ^1 = vvA1
| ^1 = vvA1
|-
|-
| 8
| 8
| 102.00
| 102.5
| 16/15, 21/20
| 16/15, 21/20
| d2
| ^m2 = vvM2
| ^m2 = vvM2
| -8
| −8
| 1098.00
| 1097.5
| 40/21, 15/8
| 15/8, 40/21
| A7
| vM7 = ^^m7
| vM7 = ^^m7
|-
|-
| 9
| 9
| 264.75
| 265.3
| 7/6
| 7/6
| d3
| m3
| m3
| -9
| −9
| 935.25
| 934.7
| 12/7
| 12/7
| A6
| M6
| M6
|-
|-
| 10
| 10
| 427.50
| 428.2
| 14/11
| 14/11
| d4
| v4 = ^^d4
| v4 = ^^d4
| -10
| −10
| 772.50
| 771.8
| 11/7
| 11/7
| A5
| ^5 = vvA5
| ^5 = vvA5
|-
|-
| 11
| 11
| 590.25
| 591.0
| 7/5
| 7/5
| d5
| ^d5 = vv5
| ^d5 = vv5
| -11
| −11
| 609.75
| 609.0
| 10/7
| 10/7
| A4
| vA4 = ^^4
| vA4 = ^^4
|-
|-
| 12
| 12
| 753.00
| 753.8
| 14/9
| 14/9
| d6
| m6
| m6
| -12
| −12
| 447.00
| 446.2
| 9/7
| 9/7
| A3
| M3
| M3
|}
|}
<nowiki/>* In 11-limit [[CWE tuning]], octave reduced


The specific tuning shown is the full 11-limit [[POTE tuning]], but of course there is a range of acceptible 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 ==
{{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.
[[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"
{| 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 [[11/9]] interval, usually considered a "neutral third", is in porcupine identical to the [[6/5]] "minor third". This 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).
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.


The characteristic small interval of porcupine, which is 60.75 cents in this tuning but can range from &lt;50 to 80 cents in general, represents both [[25/24]] and [[81/80]].
== Scales ==
[[File:porcupine8.jpg|thumb|Porcupine[8]]]  


== Chords ==
{{Main| Porcupine scales }}
{{Main| Chords of porcupine }}


== Scales ==
; Mos scales, tuning optimized on the 2.3.5.11 subgroup
; Mos scales, tuning optimized on the 2.3.5.11 subgroup
* [[Porkypine7]]
* [[Porkypine7]]
Line 169: 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 175: Line 237:
== Tunings ==
== Tunings ==
{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | 5-limit Prime-Optimized Tunings
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings
|-
! Weight-skew\Order !! Euclidean
|-
| Tenney || CTE<br>~10/9 = 164.1659¢
|-
|-
| Tenney-Weil || CTWE<br>~10/9 = 164.0621¢
! rowspan="2" |  
! colspan="3" | Euclidean
|-
|-
| Equilateral || CEE<br>~10/9 = 163.6049¢<br>[[Eigenmonzo basis]] ([[unchanged-interval basis]]): 2.84375
! Constrained !! Constrained & skewed !! Destretched
|-
|-
| Equilateral-Weil || CEWE<br>~10/9 = 163.2835¢<br>Eigenmonzo basis (unchanged-interval basis): 2.375
! Equilateral
| CEE: ~10/9 = 163.6049{{c}}
| CSEE: ~10/9 = 163.2835{{c}}
| POEE: ~10/9 = 163.9280{{c}}
|-
|-
| Benedetti || CBE<br>~10/9 = 164.3761¢<br>Eigenmonzo basis (unchanged-interval basis): 2.30375
! Tenney
| CTE: ~10/9 = 164.1659{{c}}
| CWE: ~10/9 = 164.0621{{c}}
| POTE: ~10/9 = 163.9504{{c}}
|-
|-
| Benedetti-Weil || CBWE<br>~10/9 = 164.3761¢<br>Eigenmonzo basis (unchanged-interval basis): 2.30375
! 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"
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="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
|-
! Weight-skew\Order !! Euclidean
|-
|-
| Tenney || CTE<br>~11/10 = 163.8867¢
! rowspan="2" |  
! colspan="3" | Euclidean
|-
|-
| Tenney-Weil || CTWE<br>~11/10 = 163.9951¢
! Constrained !! Constrained & skewed !! Destretched
|-
|-
| Equilateral || CEE<br>~11/10 = 163.1459¢<br>Eigenmonzo basis (unchanged-interval basis): 2.{{monzo| 0 3 5 4 }}
! Equilateral
| CEE: ~11/10 = 163.1459{{c}}
| CSEE: ~11/10 = 162.8445{{c}}
| POEE: ~11/10 = 164.1867{{c}}
|-
|-
| Equilateral-Weil || CEWE<br>~11/10 = 162.8445¢<br>Eigenmonzo basis (unchanged-interval basis): 2.{{monzo| 0 3 13 8 }}
! Tenney
| CTE: ~11/10 = 163.8867{{c}}
| CWE: ~11/10 = 163.9951{{c}}
| POTE: ~11/10 = 164.0777{{c}}
|-
|-
| Benedetti || CBE<br>~11/10 = 164.2393¢<br>Eigenmonzo basis (unchanged-interval basis): 2.{{monzo| 0 605 363 60 }}
! Benedetti, <br>Wilson
|-
| CBE: ~11/10 = 164.2393{{c}}
| Benedetti-Weil || CBWE<br>~11/10 = 164.4623¢<br>Eigenmonzo basis (unchanged-interval basis): 2.{{monzo| 0 -1595 -957 90 }}
| CSBE: ~11/10 = 164.4623{{c}}
| POBE: ~11/10 = 164.2221{{c}}
|}
|}


=== Tuning spectra ===
{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable center-all left-4"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
|+ Tuning spectrum of 13-limit porcupine
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-Interval)]]
! Generator (¢)
! Comments
|-
|-
|  
! rowspan="2" |  
| 13/12
! colspan="3" | Euclidean
| 138.573
|
|-
|-
|
! Constrained !! Constrained & skewed !! Destretched
| 13/11
| 144.605
|
|-
|-
| 1\8
! Equilateral
|  
| CEE: ~11/10 = 162.4448{{c}}
| 150.000
| CSEE: ~11/10 = 162.2333{{c}}
| Lower bound of 5-odd-limit diamond monotone
| POEE: ~11/10 = 162.2522{{c}}
|-
|
| 12/11
| 150.637
| Lower bound of 11-odd-limit diamond tradeoff
|-
|-
|
! Tenney
| 13/10
| CTE: ~11/10 = 163.1055{{c}}
| 151.405
| CWE: ~11/10 = 162.8156{{c}}
|  
| POTE: ~11/10 = 162.7474{{c}}
|-
|-
|
! Benedetti, <br>Wilson
| 6/5
| CBE: ~11/10 = 163.5299{{c}}
| 157.821
| CSBE: ~11/10 = 163.2310{{c}}
| Lower bound of 5-, 7-, and 9-odd-limit diamond tradeoff
| POBE: ~11/10 = 163.0304{{c}}
|-
|
| 15/13
| 158.710
|
|-
|
| 18/13
| 159.154
|
|-
| 2\15
|
| 160.000
| Lower bound of 7-, 9-, and 11-odd-limit diamond monotone
|-
|
| 8/7
| 161.471
|
|-
|
| 14/11
| 161.751
|
|-
|
| 7/5
| 162.047
|
|-
| 5\37
|
| 162.162
|
|-
|
| 11/8
| 162.171
| 13- and 15-odd-limit minimax
|-
| 8\59
|
| 162.712
|
|-
|
| 5/4
| 162.737
| 5- and 7-odd-limit minimax
|-
|
| 15/14
| 162.897
|
|-
|
| 7/6
| 162.986
|
|-
| 3\22
|
| 163.636
| Upper bound of 7-, 9-, and 11-odd-limit diamond monotone
|-
|
| 9/7
| 163.743
| 9- and 11-odd-limit minimax
|-
|
| 16/15
| 163.966
|
|-
| 7\51
|
| 164.706
|
|-
|
| 11/10
| 165.004
|
|-
| 4\29
|
| 165.517
|
|-
|
| 15/11
| 165.762
|
|-
|
| 4/3
| 166.015
| Upper bound of 5- and 7-odd-limit diamond tradeoff
|-
|
| 14/13
| 166.037
|
|-
| 1\7
| 11/9
| 171.429
| Upper bound of 5-odd-limit diamond monotone
|-
|
| 11/9
| 173.704
|
|-
|
| 16/13
| 179.736
|
|-
|
| 10/9
| 182.404
| Upper bound of 9- and 11-odd-limit diamond tradeoff
|}
|}


{| class="wikitable center-all"
=== Tuning spectrum ===
|+ Tuning spectrum of porcupinefish
{| class="wikitable center-all left-4"
! Edo<br>generator
|-
! Eigenmonzo<br>(Unchanged-Interval)
! EDO<br>generator
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|-
| '''[[8edo|1\8]]'''
|  
|  
| 12/11
| '''150.000'''
| '''Lower bound of 5-odd-limit diamond monotone'''
|-
|
| [[12/11]]
| 150.637
| 150.637
|  
| Lower bound of 11-odd-limit and 11-limit 15-odd-limit diamond tradeoff
|-
|-
|  
|  
| 6/5
| [[6/5]]
| 157.821
| 157.821
|  
| 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-odd-limit to 11-limit 15-odd-limit diamond monotone'''
|-
|-
|  
|  
| 18/13
| [[7/4]]
| 160.307
|
|-
|
| 15/13
| 160.860
|
|-
|
| 8/7
| 161.471
| 161.471
|  
|  
|-
|-
| [[52edo|7\52]]
|  
|  
| 13/12
| 161.538
| 161.531
| 52b val
|  
|-
|-
|  
|  
| 14/11
| [[14/11]]
| 161.751
| 161.751
|  
|  
|-
|-
|  
|  
| 7/5
| [[7/5]]
| 162.047
| 162.047
|  
|  
|-
|-
|  
| [[37edo|5\37]]
| 14/13
| 162.100
|
|-
|
| 13/10
| 162.149
|
|-
| 5\37
|  
|  
| 162.162
| 162.162
Line 442: Line 362:
|-
|-
|  
|  
| 11/8
| [[16/11]]
| 162.171
| 162.171
|  
|  
|-
|-
| [[96edo|13\96]]
|  
|  
| 16/13
| 162.500
| 162.322
| 96b val
|  
|-
|-
|  
| [[59edo|8\59]]
| 13/11
| 162.368
| 13- and 15-odd-limit minimax
|-
| 8\59
|  
|  
| 162.712
| 162.712
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|-
|-
|  
|  
| 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-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
| 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
| [[11/9]]
| 173.704
| 173.704
|  
|  
|-
|-
|  
|  
| 10/9
| [[10/9]]
| 182.404
| 182.404
|  
| 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 [[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.
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.


== See also ==
== See also ==
* [[Porcupine Notation]]
* [[Porcupine notation]]
* [[Porcupine family]]
* [[Porcupine modes]]
* [[Porcupine modes]]
* [[Porcupine temperament modal harmony]]
* [[Porcupine Album Project]]
* [[Porcupine Album Project]]


Line 540: 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]], 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) – 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]]
* [https://cellularautomaton.bandcamp.com/track/minnow ''Minnow''] (2024) – in [[29edo]] tuning


; [[Paul Erlich]]
; [[Paul Erlich]]
* [https://web.archive.org/web/20070928093239/http://66.98.148.43/~xenharmo/mp3/erlich/glassic.mp3 ''Glassic''] – in [[22edo]] (at least the beginning part is in porcupine.)
* [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]]
; [[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 557: Line 485:


; [[Lillian Hearne]]
; [[Lillian Hearne]]
* ''[https://soundcloud.com/lillianhearne/mass-in-22edo-sanctus Sanctus] (2015)''
* [https://soundcloud.com/lillianhearne/mass-in-22edo-sanctus ''Sanctus''] (2015)


; [[Andrew Heathwaite]]
; [[Andrew Heathwaite]]
* [https://soundclick.com/share.cfm?id=8839060 ''being a''] (2010) – in 22edo, mode 3 1 3 3 3 3 3 3 of porcupine[8]
* [https://soundclick.com/share.cfm?id=8839060 ''being a''] (2010) – in Porcupine[8], mode 1|6, 22edo tuning


; [[Jollybard]]
; [[Jollybard]]
* [https://soundcloud.com/jollybard/porcupeen ''Porcupeen''] (2017)
* [https://soundcloud.com/jollybard/porcupeen ''Porcupeen''] (2017)
* [https://jollybard.bandcamp.com/track/porcupine ''Porcupine''] from ''pato, with friends'' (2019)
* [https://jollybard.bandcamp.com/track/porcupine "Porcupine"], from ''pato, with friends'' (2019)


; [[Igliashon Jones]]
; [[Igliashon Jones]]
Line 570: Line 498:


; [[Löis Lancaster]]
; [[Löis Lancaster]]
* [https://soundcloud.com/lois-lancaster/porcupine-experience ''Porcupine Experience''] (2012) – in 22edo
* [https://soundcloud.com/lois-lancaster/porcupine-experience ''Porcupine Experience''] (2012) – in 22edo tuning


; [[John Moriarty]]
; [[John Moriarty]]
* [https://www.youtube.com/watch?v=se79rdp705Y ''Flying Straight Down''] (2020) – in 22edo
* [https://www.youtube.com/watch?v=se79rdp705Y ''Flying Straight Down''] (2020) – in 22edo tuning


; [[Omega9]]
; [[Omega9]]
Line 582: Line 510:


; [[Ray Perlner]]
; [[Ray Perlner]]
* [https://www.youtube.com/watch?v=8reCr2nDGbw ''Porcupine Lullaby''] – in 37edo
* [https://www.youtube.com/watch?v=8reCr2nDGbw ''Porcupine Lullaby''] (2020) – in 37edo tuning
* [https://www.youtube.com/playlist?list=PLkW9S8bpltfw464vJg3CAJJbV4IR6ggPd ''Porcupine(7) Modal Fugues''] – 7-piece playlist
* [https://www.youtube.com/playlist?list=PLkW9S8bpltfw464vJg3CAJJbV4IR6ggPd ''Porcupine{{lbrack}}7{{rbrack}} Modal Fugues''] – 7-piece playlist


; [[Gene Ward Smith]] and {{w|Modest Mussorgsky}}
; [[Gene Ward Smith]] and {{w|Modest Mussorgsky}}
* [https://www.archive.org/download/NightOnPorcupineMountain/Genewardsmithmussorgsky-NightOnPorcupineMountain.mp3 ''Night on Porcupine Mountain''] (archived 2010) – in 22edo
* [https://www.archive.org/download/NightOnPorcupineMountain/Genewardsmithmussorgsky-NightOnPorcupineMountain.mp3 ''Night on Porcupine Mountain''] (archived 2010) – in 22edo tuning


; [[Chris Vaisvil]]
; [[Chris Vaisvil]]
* [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
* [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
* [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
* [http://micro.soonlabel.com/15-ET/daily20111231-porcupine15-prickly-side-of-love.mp3 ''Prickly Side of Love''] - rock band in Porcupine Temperament with vocals
* [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
* [http://micro.soonlabel.com/15-ET/daily20120102-porcupine-organ.mp3 ''Porcupine Organ Composition'']
* [https://web.archive.org/web/20221221154102/http://micro.soonlabel.com/15-ET/daily20120102-porcupine-organ.mp3 ''Porcupine Organ Composition'']


; [[Nick Vuci]]
; [[Nick Vuci]]
* [https://en.xen.wiki/images/0/0b/NickVuci-20230426-22edo-PorcupinePrelude1.mp3 ''Porcupine Prelude 1''] – in 22edo
* [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
* [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
* [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
* [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
* [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]]
; [[Well-Tempered Fox]]
* [https://www.youtube.com/watch?v=INM6J9pS_xE ''Porcupine Major Overture''] (2015) – in 22edo
* [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 15edo porcupine[8]
* [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:Porcupine family]]
[[Category:Porcupine family]]
[[Category:Archytas clan]]
[[Category:Archytas clan]]
[[Category:Keemic temperaments]]
[[Category:Keemic temperaments]]
[[Category:Listen]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/Porcupine"