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'''Porcupine''' is a [[linear temperament]] in the [[porcupine family]] that tempers out [[250/243]], the porcupine [[comma]], and whose generator is somewhere around 160–165 cents. It can be thought of as a 5-[[Harmonic Limit|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.
'''Porcupine''' is a [[linear temperament]] in the [[porcupine family]] that tempers out [[250/243]], the porcupine [[comma]], and whose generator is somewhere around 160–165 cents. It can be thought of as a 5-[[Harmonic Limit|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.


The basic 5-limit harmonic structure of porcupine can be understood simply by noting that tempering out 250/243 makes (4/3)<sup>2</sup> equivalent to (6/5)<sup>3</sup>. In perhaps more familiar musical terms, this means 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|12EDO]], and to meantone, 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 porcupine scales.
The basic 5-limit harmonic structure of porcupine can be understood simply by noting that tempering out 250/243 makes (4/3)<sup>2</sup> equivalent to (6/5)<sup>3</sup>. In perhaps more familiar musical terms, this means 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]], and to meantone, 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 porcupine scales.


[[File:porcupinesymmetricminor22edo.mp3]]
[[File:porcupinesymmetricminor22edo.mp3]]


Porcupine symmetric minor scale, containing two equal tetrachords with a major wholetone between them. (Tuning in [[22edo|22EDO]])
Porcupine symmetric minor scale, containing two equal tetrachords with a major wholetone between them ([[22edo]] tuning).


[[File:porcupine.png]]
[[File:porcupine.png]]


== Interval chain ==
== Interval chain ==
 
{{Main| Porcupine intervals }}
{{Main|Porcupine intervals}}


{| class="wikitable center-all right-2 left-3 right-6 left-7"
{| class="wikitable center-all right-2 left-3 right-6 left-7"
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| M3
| M3
|}
|}
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|15EDO]]) and as large as 165.5 cents ([[29edo|29EDO]]). (However, the 29EDO patent val does not support 11-limit porcupine proper, since it doesn't temper out [[64/63]].)
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]].)


[[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.
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|-
|-
| 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|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|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.
|}
|}


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== Chords ==
== Chords ==
{{main| Chords of porcupine }}
{{Main| Chords of porcupine }}


== Scales ==
== Scales ==
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{| class="wikitable center-all"
{| class="wikitable center-all"
! ET<br>generator
! Edo<br>Generator
! [[eigenmonzo|eigenmonzo<br>(unchanged interval]])
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged Interval]])
! neutral<br>second
! Generator (¢)
! comments
! Comments
|-
|-
|  
|  
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{| class="wikitable center-all"
{| class="wikitable center-all"
! ET generator
! Edo Generator
! eigenmonzo<br>(unchanged interval)
! Eigenmonzo<br>(Unchanged Interval)
! neutral<br>second
! Generator (¢)
! comments
! Comments
|-
|-
|  
|  
Line 484: Line 483:
== History ==
== History ==


Porcupine temperament/scales were discovered by [[Dave Keenan]], but didn't have a name until [[Herman Miller]] mentioned that his Mizarian Porcupine Overture in 15-tET had a section that pumps the [[250/243|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 15, 22 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 ==
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* [[Porcupine Album Project]]
* [[Porcupine Album Project]]


== Musical examples ==
== Music ==
* ''[https://sites.google.com/site/teamouse/home#TOC-Mizarian-music Mizarian Porcupine Overture]'', [[Herman Miller]], 1999. (15EDO, namesake of the temperament)
* ''[https://sites.google.com/site/teamouse/home#TOC-Mizarian-music Mizarian Porcupine Overture]'', [[Herman Miller]], 1999. (15EDO, namesake of the temperament)
* ''[https://web.archive.org/web/20070928093239/http://66.98.148.43/~xenharmo/mp3/erlich/glassic.mp3 Glassic]'', [[Paul Erlich]], [[22edo|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]'', [[Paul Erlich]], [[22edo|22EDO]] (at least the beginning part is in porcupine).
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* [https://cityoftheasleep.bandcamp.com/track/second-breakfast-15edo Second Breakfast (15EDO)] by [[Igliashon Jones]]
* [https://cityoftheasleep.bandcamp.com/track/second-breakfast-15edo Second Breakfast (15EDO)] by [[Igliashon Jones]]


== Images ==
== Diagrams ==


[[File:porcupine8.jpg]]       
[[File:porcupine8.jpg]]       

Revision as of 18:26, 7 February 2023

Porcupine is a linear temperament in the porcupine family that tempers out 250/243, the porcupine comma, and whose generator is somewhere around 160–165 cents. 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.

The basic 5-limit harmonic structure of porcupine can be understood simply by noting that tempering out 250/243 makes (4/3)2 equivalent to (6/5)3. In perhaps more familiar musical terms, this means 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, and to meantone, 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 porcupine scales.

Porcupine symmetric minor scale, containing two equal tetrachords with a major wholetone between them (22edo tuning).

Interval chain

Main article: Porcupine intervals
# Cents Ratios Ups and Downs
notation 		# 2/1 inverse Ratios Ups and Downs
notation
0 0.00 1/1 P1 0 1200.00 2/1 P8
1 162.75 12/11, 11/10, 10/9 vM2 = ^^m2 -1 1037.25 9/5, 20/11, 11/6 ^m7 = vvM7
2 325.50 6/5, 11/9 ^m3 = vvM3 -2 874.50 18/11, 5/3 vM6 = ^^m6
3 488.25 4/3 P4 -3 711.75 3/2 P5
4 651.00 16/11, 22/15 v5 = ^^d5 -4 549.00 15/11, 11/8 ^4 = vvA4
5 813.75 8/5 ^m6 = vvM6 -5 386.25 5/4 vM3 = ^^m3
6 976.50 7/4, 16/9 m7 -6 223.50 9/8, 8/7 M2
7 1139.25 48/25, 160/81 v8 = ^^d8 -7 60.75 81/80, 25/24 ^1 = vvA1
8 102.00 16/15, 21/20 ^m2 = vvM2 -8 1098.00 40/21, 15/8 vM7 = ^^m7
9 264.75 7/6 m3 -9 935.25 12/7 M6
10 427.50 14/11 v4 = ^^d4 -10 772.50 11/7 ^5 = vvA5
11 590.25 7/5 ^d5 = vv5 -11 609.75 10/7 vA4 = ^^4
12 753.00 14/9 m6 -12 447.00 9/7 M3

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.)

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 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 characteristic small interval of porcupine, which is 60.75 cents in this tuning but can range from <50 to 80 cents in general, represents both 25/24 and 81/80.

Chords

Main article: Chords of porcupine

Scales

[8/5 12/7] eigenmonzos

Tuning spectra

Porcupine

Edo
Generator 		Eigenmonzo
(Unchanged Interval) 		Generator (¢) Comments
13/12 138.573
13/11 144.605
12/11 150.637
13/10 151.405
6/5 157.821
15/13 158.710
18/13 159.154
2\15 160.000
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
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
14/13 166.037
11/9 173.704
16/13 179.736
10/9 182.404

Porcupinefish

Edo Generator Eigenmonzo
(Unchanged Interval) 		Generator (¢) Comments
12/11 150.637
6/5 157.821
2\15 160.000
18/13 160.307
15/13 160.860
8/7 161.471
13/12 161.531
14/11 161.751
7/5 162.047
14/13 162.100
13/10 162.149
5\37 162.162
11/8 162.171
16/13 162.322
13/11 162.368 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
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
11/9 173.704
10/9 182.404

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.

See also

Music

Diagrams

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