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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 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).
| 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's basic 5-limit harmonic structure can be understood by sharpening the porcupine generator step by a tempered chroma. This represents a) an augmented second of 125/108 (sharp by 25/24 from the generator) i.e. the octave-reduced form of (5/3)<sup>3</sup>, and b) a major second of 9/8 (sharp by 81/80 from the generator) i.e. the octave-reduced form of (3/2)<sup>2</sup>. (5/3)<sup>3</sup> is an octave above (3/2)<sup>2</sup>, so by taking the octave-complements of everything, we find that (6/5)<sup>3</sup> is equated to (4/3)<sup>2</sup>, without any octave-reduction necessary.
'''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.  


This means that two "perfect fourths" are equivalent to 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. 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.
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]]).  


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, with 7 found at +6 generators (tuned to about 960-990 cents), and 11 found at -4 generators (tuned to about 540-560 cents). 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 42: 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 53: 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 64: 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 75: 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 86: 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 97: 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 108: 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 130: 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 141: 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 152: 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 163: 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 174: 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
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]].


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]].
== 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 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" |  
! 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 260: 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 306: 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 397: 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 410: 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 449: 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 466: 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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