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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. | |||
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. | |||
[[File:porcupinesymmetricminor22edo.mp3]] | |||
[[ | Porcupine symmetric minor scale, containing two equal tetrachords with a major wholetone between them. (Tuning in [[22edo]]) | ||
[[File:porcupine.png]] | |||
== Interval chain == | |||
{{Main|Porcupine intervals}} | |||
Main | |||
{| class="wikitable | {| class="wikitable center-all right-2 right-6" | ||
! Generators | ! Generators | ||
! Cents | ! Cents | ||
| Line 25: | Line 30: | ||
|- | |- | ||
| 0 | | 0 | ||
| 0.00 | |||
| 1/1 | | 1/1 | ||
| P1 | | P1 | ||
| 0 | | 0 | ||
| 1200.00 | |||
| 2/1 | | 2/1 | ||
| P8 | | P8 | ||
|- | |- | ||
| 1 | | 1 | ||
| 162.75 | |||
| 12/11~11/10~10/9 | | 12/11~11/10~10/9 | ||
| vM2 = ^^m2 | | vM2 = ^^m2 | ||
| -1 | | -1 | ||
| 1037.25 | |||
| 9/5~20/11~11/6 | | 9/5~20/11~11/6 | ||
| ^m7 = vvM7 | | ^m7 = vvM7 | ||
|- | |- | ||
| 2 | | 2 | ||
| 325.50 | |||
| 6/5~11/9 | | 6/5~11/9 | ||
| ^m3 = vvM3 | | ^m3 = vvM3 | ||
| -2 | | -2 | ||
| 874.50 | |||
| 18/11~5/3 | | 18/11~5/3 | ||
| vM6 = ^^m6 | | vM6 = ^^m6 | ||
|- | |- | ||
| 3 | | 3 | ||
| 488.25 | |||
| 4/3 | | 4/3 | ||
| P4 | | P4 | ||
| -3 | | -3 | ||
| 711.75 | |||
| 3/2 | | 3/2 | ||
| P5 | | P5 | ||
|- | |- | ||
| 4 | | 4 | ||
| 651.00 | |||
| 16/11~22/15 | | 16/11~22/15 | ||
| v5 = ^^d5 | | v5 = ^^d5 | ||
| -4 | | -4 | ||
| 549.00 | |||
| 15/11~11/8 | | 15/11~11/8 | ||
| ^4 = vvA4 | | ^4 = vvA4 | ||
|- | |- | ||
| 5 | | 5 | ||
| 813.75 | |||
| 8/5 | | 8/5 | ||
| ^m6 = vvM6 | | ^m6 = vvM6 | ||
| -5 | | -5 | ||
| 386.25 | |||
| 5/4 | | 5/4 | ||
| vM3 = ^^m3 | | vM3 = ^^m3 | ||
|- | |- | ||
| 6 | | 6 | ||
| 976.50 | |||
| 7/4~16/9 | | 7/4~16/9 | ||
| m7 | | m7 | ||
| -6 | | -6 | ||
| 223.50 | |||
| 9/8~8/7 | | 9/8~8/7 | ||
| M2 | | M2 | ||
|- | |- | ||
| 7 | | 7 | ||
| 1139.25 | |||
| 48/25~160/81 | | 48/25~160/81 | ||
| v8 = ^^d8 | | v8 = ^^d8 | ||
| -7 | | -7 | ||
| 60.75 | |||
| 81/80~25/24 | | 81/80~25/24 | ||
| ^1 = vvA1 | | ^1 = vvA1 | ||
|- | |- | ||
| 8 | | 8 | ||
| 102.00 | |||
| 16/15~21/20 | | 16/15~21/20 | ||
| ^m2 = vvM2 | | ^m2 = vvM2 | ||
| -8 | | -8 | ||
| 1098.00 | |||
| 40/21~15/8 | | 40/21~15/8 | ||
| vM7 = ^^m7 | | vM7 = ^^m7 | ||
|- | |- | ||
| 9 | | 9 | ||
| 264.75 | |||
| 7/6 | | 7/6 | ||
| m3 | | m3 | ||
| -9 | | -9 | ||
| 935.25 | |||
| 12/7 | | 12/7 | ||
| M6 | | M6 | ||
|- | |- | ||
| 10 | | 10 | ||
| 427.50 | |||
| 14/11 | | 14/11 | ||
| v4 = ^^d4 | | v4 = ^^d4 | ||
| -10 | | -10 | ||
| 772.50 | |||
| 11/7 | | 11/7 | ||
| ^5 = vvA5 | | ^5 = vvA5 | ||
|- | |- | ||
| 11 | | 11 | ||
| 590.25 | |||
| 7/5 | | 7/5 | ||
| ^d5 = vv5 | | ^d5 = vv5 | ||
| -11 | | -11 | ||
| 609.75 | |||
| 10/7 | | 10/7 | ||
| vA4 = ^^4 | | vA4 = ^^4 | ||
|- | |- | ||
| 12 | | 12 | ||
| 753.00 | |||
| 14/9 | | 14/9 | ||
| m6 | | m6 | ||
| -12 | | -12 | ||
| 447.00 | |||
| 9/7 | | 9/7 | ||
| M3 | | M3 | ||
|} | |} | ||
The specific tuning shown is the full 11-limit [[ | 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, not annihilating [[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. | ||
{| class="wikitable" | {| class="wikitable" | ||
| Line 151: | Line 156: | ||
|- | |- | ||
| 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 [[ | | 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 [[ | | 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 [[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. | 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]]. | ||
==Spectrum of Porcupine Tunings by Eigenmonzos== | == Spectrum of Porcupine Tunings by Eigenmonzos == | ||
{| class="wikitable" | {| class="wikitable" | ||
| Line 255: | Line 260: | ||
[8/5 12/7] eigenmonzos: [[porcupinewoo15|porcupinewoo15]] [[porcupinewoo22|porcupinewoo22]] | [8/5 12/7] eigenmonzos: [[porcupinewoo15|porcupinewoo15]] [[porcupinewoo22|porcupinewoo22]] | ||
===Spectrum of Porcupinefish Tunings=== | === Spectrum of Porcupinefish Tunings === | ||
{| class="wikitable" | {| class="wikitable" | ||
| Line 346: | Line 351: | ||
|} | |} | ||
==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. | |||
== See also == | |||
==Musical examples== | * [[Chords of porcupine]] | ||
* [[Porcupine Notation]] | |||
* [[Porcupine modes]] | |||
* [[Porcupine Album Project]] | |||
== Musical examples == | |||
* ''[http://sites.google.com/site/teamouse/home#TOC-Mizarian-music Mizarian Porcupine Overture]'', [[Herman Miller]], 1999. (15edo, namesake of the temperament) | * ''[http://sites.google.com/site/teamouse/home#TOC-Mizarian-music Mizarian Porcupine Overture]'', [[Herman Miller]], 1999. (15edo, namesake of the temperament) | ||
* ''[http://www.myspace.com/paulerlich/music/songs/glassic-in-22-tone-equal-temperament-45202095 Glassic]'', [[Paul Erlich]], [[22edo]] (at least the beginning part is in porcupine). | * ''[http://www.myspace.com/paulerlich/music/songs/glassic-in-22-tone-equal-temperament-45202095 Glassic]'', [[Paul Erlich]], [[22edo]] (at least the beginning part is in porcupine). | ||
* ''[http://www.archive.org/download/NightOnPorcupineMountain/Genewardsmithmussorgsky-NightOnPorcupineMountain.mp3 Night on Porcupine Mountain]'', Gene Ward Smith and Modest Mussorgsky, [[22edo]]. | * ''[http://www.archive.org/download/NightOnPorcupineMountain/Genewardsmithmussorgsky-NightOnPorcupineMountain.mp3 Night on Porcupine Mountain]'', [[Gene Ward Smith]] and Modest Mussorgsky, [[22edo]]. | ||
* ''[http://soundclick.com/share.cfm?id=8839060 being a]'', Andrew Heathwaite, 2010, 22edo, mode 3 1 3 3 3 3 3 3 of Porcupine[8]. | * ''[http://soundclick.com/share.cfm?id=8839060 being a]'', [[Andrew Heathwaite]], 2010, 22edo, mode 3 1 3 3 3 3 3 3 of Porcupine[8]. | ||
* ''[http://micro.soonlabel.com/15-ET/daily20110619_millers_porcupine_7a.mp3 Playing Gently with Miller's Porcupine]'' by [[Chris Vaisvil]] | * ''[http://micro.soonlabel.com/15-ET/daily20110619_millers_porcupine_7a.mp3 Playing Gently with Miller's Porcupine]'' by [[Chris Vaisvil]] | ||
* ''[http://micro.soonlabel.com/15-ET/daily20111231-porcupine15-indian.mp3 15 Porcupines in India]'', Sarangi, Tambura and Sitar improvisation by [[Chris Vaisvil]] | * ''[http://micro.soonlabel.com/15-ET/daily20111231-porcupine15-indian.mp3 15 Porcupines in India]'', Sarangi, Tambura and Sitar improvisation by [[Chris Vaisvil]] | ||
| Line 372: | Line 378: | ||
* [https://www.youtube.com/watch?v=se79rdp705Y Flying Straight Down] by [[John Moriarty]] | * [https://www.youtube.com/watch?v=se79rdp705Y Flying Straight Down] by [[John Moriarty]] | ||
==Images== | == Images == | ||
[[File: | |||
[[Category: | [[File:porcupine8.jpg]] | ||
[[Category:Porcupine]] | |||
[[Category:Temperament]] | |||
Revision as of 10:46, 25 October 2020
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. (Tuning in 22edo)
Interval chain
| Generators | Cents | Ratios | Ups and Downs notation |
Generators | 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, not annihilating 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.
Spectrum of Porcupine Tunings by Eigenmonzos
| Eigenmonzo | Neutral Second |
|---|---|
| 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-limit minimax |
| 8\59 | 162.712 |
| 5/4 | 162.737 5-limit minimax |
| 15/14 | 162.897 |
| 7/6 | 162.986 |
| 3\22 | 163.636 |
| 9/7 | 163.743 7- 9- and 11-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 |
[8/5 12/7] eigenmonzos: porcupinewoo15 porcupinewoo22
Spectrum of Porcupinefish Tunings
| 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-limit minimax |
| 8\59 | 162.712 |
| 5/4 | 162.737 |
| 15/14 | 162.897 |
| 7/6 | 162.986 |
| 3\22 | 163.636 |
| 9/7 | 163.743 |
| 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 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 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.
See also
Musical examples
- Mizarian Porcupine Overture, Herman Miller, 1999. (15edo, namesake of the temperament)
- Glassic, Paul Erlich, 22edo (at least the beginning part is in porcupine).
- Night on Porcupine Mountain, Gene Ward Smith and Modest Mussorgsky, 22edo.
- being a, Andrew Heathwaite, 2010, 22edo, mode 3 1 3 3 3 3 3 3 of Porcupine[8].
- Playing Gently with Miller's Porcupine by Chris Vaisvil
- 15 Porcupines in India, Sarangi, Tambura and Sitar improvisation by Chris Vaisvil
- 15 Quills piano solo by Chris Vaisvil
- Prickly Side of Love - rock band in Porcupine Temperament with vocals by Chris Vaisvil
- Porcupine Organ Composition by Chris Vaisvil
- Among Other Things 2 by Petr Pařízek
- Porcupine Comma Pump, by Jake Freivald
- Life on Mars by Omega9
- Flying Straight Down by John Moriarty
Images
