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{{interwiki
{{Interwiki
| en = Porcupine
| de = Porcupine
| de = Porcupine
| en = Porcupine
| es =  
| es =  
| ja =  
| ja =  
}}
}}
[[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]]
{{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 equates a stack of three 6/5s to a stack of two 4/3s. 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. As a result, the [[generator]] of porcupine is a [[10/9|minor whole tone (10/9)]] which is tuned flat to around 160–165 [[cent]]s such that two of them stack to a [[6/5|classic minor third (6/5)]]. This is obviously in stark contrast to [[12edo]]. The "equal tetrachord" formed by dividing 4/3 into 3 equal parts is a characteristic feature of many of porcupine's scales.
| 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)
In just intonation, a stack of three 6/5s is a classical diminished seventh, flat of the classical minor seventh 9/5 by 25/24, and a stack of two 4/3s 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]]. This 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]] (i.e. meantone, schismic), by [[tempering out]] [[250/243]], the porcupine [[comma]].  Its [[pergen]] is (P8, P4/3).
| 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.]]


To extend porcupine to the 7-limit and 11-limit, you may notice that both primes are found naturally in simple positions along the porcupine generator chain:
'''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.


* 7 is found at +6 generators (tuned to about 960-990 cents), because the 16/9 has already been flattened to merge it with (6/5)<sup>3</sup>. This makes porcupine an [[archy]] temperament.  
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]]).  
* 11 is found at -4 generators (tuned to about 540-560 cents), corresponding with the acute fourth of 27/20; 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. This can also be understood as the generator 10/9 being flattened.  


Neither, one, or both of these additional mappings may be used (in 2.3.5.11, the temperament is sometimes called ''porkypine''). Porcupine is one of the most efficient temperaments in the 2.3.5.11 subgroup of reasonable 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.


In the 7-limit, porcupine can be seen as as 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 ==
{{Main|Porcupine intervals}}
{{Main| Porcupine intervals }}


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


{| class="wikitable center-all right-2 left-3 right-6 left-7"
{| 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
|-
|-
! #
! #
! Cents
! Cents*
! Ratios
! Ratios
!Porcupine notation
! Porcupine<br>notation
! Ups and downs<br>notation
! Ups and downs<br>notation
! #
! #
! Cents
! Cents*
! Ratios
! Ratios
!Porcupine notation
! Porcupine<br>notation
! Ups and downs<br>notation
! Ups and downs<br>notation
|-
|-
Line 45: Line 57:
| 0.0
| 0.0
| '''1/1'''
| '''1/1'''
|P1
| P1
| P1
| P1
| 0
| 0
| 1200.0
| 1200.0
| '''2/1'''
| '''2/1'''
|P8
| P8
| P8
| P8
|-
|-
Line 56: Line 68:
| 162.8
| 162.8
| 10/9, 11/10, 12/11
| 10/9, 11/10, 12/11
|P2
| P2
| vM2 = ^^m2
| vM2 = ^^m2
| -1
| −1
| 1037.2
| 1037.2
| 9/5, 11/6, 20/11
| 9/5, 11/6, 20/11
|P7
| P7
| ^m7 = vvM7
| ^m7 = vvM7
|-
|-
Line 67: Line 79:
| 325.6
| 325.6
| 6/5, 11/9
| 6/5, 11/9
|m3
| m3
| ^m3 = vvM3
| ^m3 = vvM3
| -2
| −2
| 874.4
| 874.4
| 5/3, 18/11
| 5/3, 18/11
|M6
| M6
| vM6 = ^^m6
| vM6 = ^^m6
|-
|-
Line 78: Line 90:
| 488.4
| 488.4
| 4/3
| 4/3
|m4
| m4
| P4
| P4
| -3
| −3
| 711.6
| 711.6
| '''3/2'''
| '''3/2'''
|M5
| M5
| P5
| P5
|-
|-
Line 89: Line 101:
| 651.3
| 651.3
| 16/11, 22/15
| 16/11, 22/15
|m5
| m5
| v5 = ^^d5
| v5 = ^^d5
| -4
| −4
| 548.7
| 548.7
| '''11/8''', 15/11
| '''11/8''', 15/11
|M4
| M4
| ^4 = vvA4
| ^4 = vvA4
|-
|-
Line 100: Line 112:
| 814.1
| 814.1
| 8/5
| 8/5
|m6
| m6
| ^m6 = vvM6
| ^m6 = vvM6
| -5
| −5
| 385.9
| 385.9
| '''5/4'''
| '''5/4'''
|M3
| M3
| vM3 = ^^m3
| vM3 = ^^m3
|-
|-
Line 111: Line 123:
| 976.9
| 976.9
| '''7/4''', 16/9
| '''7/4''', 16/9
|d7
| d7
| m7
| m7
| -6
| −6
| 223.1
| 223.1
| 8/7, '''9/8'''
| 8/7, '''9/8'''
|A2
| A2
| M2
| M2
|-
|-
| 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
|-
|-
Line 133: Line 145:
| 102.5
| 102.5
| 16/15, 21/20
| 16/15, 21/20
|d2
| d2
| ^m2 = vvM2
| ^m2 = vvM2
| -8
| −8
| 1097.5
| 1097.5
| 15/8, 40/21
| 15/8, 40/21
|A7
| A7
| vM7 = ^^m7
| vM7 = ^^m7
|-
|-
Line 144: Line 156:
| 265.3
| 265.3
| 7/6
| 7/6
|d3
| d3
| m3
| m3
| -9
| −9
| 934.7
| 934.7
| 12/7
| 12/7
|A6
| A6
| M6
| M6
|-
|-
Line 155: Line 167:
| 428.2
| 428.2
| 14/11
| 14/11
|d4
| d4
| v4 = ^^d4
| v4 = ^^d4
| -10
| −10
| 771.8
| 771.8
| 11/7
| 11/7
|A5
| A5
| ^5 = vvA5
| ^5 = vvA5
|-
|-
Line 166: Line 178:
| 591.0
| 591.0
| 7/5
| 7/5
|d5
| d5
| ^d5 = vv5
| ^d5 = vv5
| -11
| −11
| 609.0
| 609.0
| 10/7
| 10/7
|A4
| A4
| vA4 = ^^4
| vA4 = ^^4
|-
|-
Line 177: Line 189:
| 753.8
| 753.8
| 14/9
| 14/9
|d6
| d6
| m6
| m6
| -12
| −12
| 446.2
| 446.2
| 9/7
| 9/7
|A3
| A3
| M3
| M3
|}
|}
<nowiki/>* In 11-limit [[CWE tuning]], octave reduced


The specific tuning shown is the full 11-limit [[CWE tuning]], but of course 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 ==
{{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", due to the extreme flatness of 10/9. An interval in the neutral third range is not found for a long time (until 17 generators up), and as a result that interval varies drastically depending on the tuning. 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).
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 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.
== 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 213: 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 219: 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" |  
! colspan="2" | Euclidean
! colspan="3" | Euclidean
|-
|-
! Constrained !! Constrained & skewed
! Constrained !! Constrained & skewed !! Destretched
|-
|-
! Equilateral
! Equilateral
| CEE: ~10/9 = 163.6049{{c}}
| CEE: ~10/9 = 163.6049{{c}}
| CSEE: ~10/9 = 163.2835{{c}}
| CSEE: ~10/9 = 163.2835{{c}}
| POEE: ~10/9 = 163.9280{{c}}
|-
|-
! Tenney
! Tenney
| CTE: ~10/9 = 164.1659{{c}}
| CTE: ~10/9 = 164.1659{{c}}
| CWE: ~10/9 = 164.0621{{c}}
| CWE: ~10/9 = 164.0621{{c}}
| POTE: ~10/9 = 163.9504{{c}}
|-
|-
! Benedetti, <br>Wilson
! Benedetti, <br>Wilson
| CBE: ~10/9 = 164.3761{{c}}
| CBE: ~10/9 = 164.3761{{c}}
| CSBE: ~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="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" |  
! colspan="2" | Euclidean
! colspan="3" | Euclidean
|-
|-
! Constrained !! Constrained & skewed
! Constrained !! Constrained & skewed !! Destretched
|-
|-
! Equilateral
! Equilateral
| CEE: ~11/10 = 163.1459{{c}}
| CEE: ~11/10 = 163.1459{{c}}
| CSEE: ~11/10 = 162.8445{{c}}
| CSEE: ~11/10 = 162.8445{{c}}
| POEE: ~11/10 = 164.1867{{c}}
|-
|-
! Tenney
! Tenney
| CTE: ~11/10 = 163.8867{{c}}
| CTE: ~11/10 = 163.8867{{c}}
| CWE: ~11/10 = 163.9951{{c}}
| CWE: ~11/10 = 163.9951{{c}}
| POTE: ~11/10 = 164.0777{{c}}
|-
|-
! Benedetti, <br>Wilson
! Benedetti, <br>Wilson
| CBE: ~11/10 = 164.2393{{c}}
| CBE: ~11/10 = 164.2393{{c}}
| CSBE: ~11/10 = 164.4623{{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}}
|}
|}


Line 263: Line 311:
{| 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 309: 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]]


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=== 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 413: 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 452: 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 469: 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]]
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