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A '''projection pair''' is a pair of two rational intervals which can be employed by the [[Scala]] "project" command to reduce a JI scale to a scale in a [[JI subgroup]] of the group generated by the scale, in such a way that tempered versions of each are equivalent. This is particularly useful for analyzing [[planar temperament]]s, as the projection can then be viewed in lattice form by Scala's "lattice" or "lattice and player" command.  
A '''projection pair''' is a pair of two rational intervals which can be employed by the [[Scala]] "project" command to reduce a JI scale to a scale in a [[JI subgroup]] of the group generated by the scale, in such a way that tempered versions of each are equivalent. This is particularly useful for analyzing [[planar temperament]]s, as the projection can then be viewed in lattice form by Scala's "lattice" or "lattice and player" command.  


An example of a projection pair is "7 225/32", which when applied by Scala's "project" to a 7-limit scale produces a 5-limit scale, which when tempered by marvel (225/224) temperament gives exactly the same result as the original scale does when also tempered. More than one such pair may be required to reduce to the desired subgroup; for instance "7 225/32, 11 4096/375" reduces an 11-limit JI scale to a 5-limit JI scale equivalent under (undecimal) marvel. This can happen even when only one comma is involved (codimension one temperaments). For instance, to project a 7-limit scale in the hemimean (3136/3125) reduction to the 2.5.7 subgroup requires "5 3136/625, 7 68841472/9765625".
An example of a projection pair is <code>7 225/32</code>, which when applied by Scala's "project" to a 7-limit scale produces a 5-limit scale, which when tempered by marvel (225/224) temperament gives exactly the same result as the original scale does when also tempered. This can be thought of as marvel temperament replacing 7 by 225/32.
 
More than one such pair may be required to reduce to the desired subgroup; for instance <code>7 225/32, 11 4096/375</code> reduces an 11-limit JI scale to a 5-limit JI scale equivalent under (undecimal) marvel. This can happen even when only one comma is involved (codimension one temperaments). For instance, to project a 7-limit scale in the hemimean (3136/3125) reduction to the 2.5.7 subgroup requires <code>5 3136/625, 7 68841472/9765625</code>.


Many projection pairs are given on the pages for various planar temperaments. When no subgroup is indicated, the default 2.3.5 5-limit subgroup is presumed. These lists of pairs can be copied and pasted into Scala and applied to any suitable JI scale.
Many projection pairs are given on the pages for various planar temperaments. When no subgroup is indicated, the default 2.3.5 5-limit subgroup is presumed. These lists of pairs can be copied and pasted into Scala and applied to any suitable JI scale.
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== List of projection pairs ==
== List of projection pairs ==
=== 5-limit ===
=== 5-limit ===
* [[Negri comma|16875/16384]]: 3 50625/16384, 5 16384/3375 to 2.15
{| class="wikitable"
* [[250/243]]: 3 729/250, 5 59049/12500 to 2.9/5
|+ Projection pairs for 5-limit temperaments
* [[Magic comma|3125/3072]]: 3 3125/1024
|-
* [[Tetracot comma|20000/19683]]: 3 20000/6561, 5 2000000000/387420489 to 2.9/5
! Temperament !! Associated comma !! Projection pair !! Target subgroup
* [[81/80]]: 5 81/16
|-
* [[Würschmidt comma|393216/390625]]: 3 390625/131072
| [[Negri]] || [[16875/16384]] || <code>3 50625/16384, 5 16384/3375</code> || 2.15
* [[Semicomma|2109375/2097152]]: 3 13348388671875/4398046511104, 5 2097152/421875 to 2.75
|-
* [[15625/15552]]: 3 46656/15625, 5 15552/3125 to 2.5/3
| [[Porcupine]] || [[250/243]] || <code>3 729/250, 5 59049/12500</code> || 2.9/5
* [[Schisma|32805/32768]]: 5 32768/6561  
|-
| [[Magic]] || [[3125/3072]] || <code>3 3125/1024</code> || 2.5
|-
| [[Tetracot]] || [[20000/19683]] || <code>3 20000/6561, 5 2000000000/387420489</code> || 2.9/5
|-
| [[Meantone]] || [[81/80]] || <code>5 81/16</code> || 2.3
|-
| [[Würschmidt]] || [[393216/390625]] || <code>3 390625/131072</code> || 2.5
|-
| [[Orson]] || [[2109375/2097152]] || <code>3 13348388671875/4398046511104, 5 2097152/421875</code> || 2.75
|-
| [[Hanson]] || [[15625/15552]] || <code>3 46656/15625, 5 15552/3125</code> || 2.5/3
|-
| [[Helmholtz]] || [[32805/32768]] || <code>5 32768/6561</code> || 2.3
|}


=== 7-limit ===
=== 7-limit ===
* [[1029/1000]]: 3 1000/343 to 2.5.7
{| class="wikitable"
* [[36/35]]: 7 36/5  
|+ Projection pairs for 7-limit temperaments
* [[525/512]]: 7 512/75
|-
* [[49/48]]: 3 49/16 to 2.5.7
! Temperament !! Associated comma !! Projection pair !! Target subgroup
* [[686/675]]: 5 3375/686, 7 675/98 to 2.3.7/5
|-
* [[64/63]]: 7 64/9
| [[Keegic]] || [[1029/1000]] || <code>3 1000/343</code> || 2.5.7
* [[Blackjackisma|854296875/843308032]]: 5 843308032/170859375, 7 5903156224/854296875 to 2.3.7/5
|-
* [[Squalentine comma|64827/64000]]: 5 320000/64827, 7 64000/9261 to 2.3.7/5
| [[Mint]] || [[36/35]] || <code>7 36/5</code> || 2.3.5
* [[875/864]]: 7 864/125
|-
* [[3125/3087]]: 5 15625/3087, 7 9765625/1361367 to 2.3.25/7
| [[Avicennmic]] || [[525/512]] || <code>7 512/75</code> || 2.3.5
* [[2430/2401]]: 5 2401/486 to 2.3.7
|-
* [[Trimyna comma|50421/50000]]: 3 50000/16807 to 2.5.7
| [[Semaphoresmic]] || [[49/48]] || <code>3 49/16</code> || 2.5.7
* [[245/243]]: 5 243/49 to 2.3.7
|-
* [[126/125]]: 7 125/18
| [[Sengic]] || [[686/675]] || <code>5 3375/686, 7 675/98</code> || 2.3.7/5
* [[4000/3969]]: 5 3969/800, 7 27783/4000 to 2.3.7/5
|-
* [[1728/1715]]: 5 1728/343 to 2.3.7
| [[Archytas]] || [[64/63]] || <code>7 64/9</code> || 2.3.5
* [[1029/1024]]: 3 1024/343 to 2.5.7
|-
* [[225/224]]: 7 225/32
| [[Blackjackismic]] || [[Blackjackisma|854296875/843308032]] || <code>5 843308032/170859375, 7 5903156224/854296875</code> || 2.3.7/5
* [[Cataharry comma|19683/19600]]: 3 19600/6561, 7 1033052339200000000/150094635296999121 to 2.5.81/7
|-
* [[Mirkwai comma|16875/16807]]: 5 84375/16807, 7 16875/2401 to 2.3.7/5
| [[Squalentine]] || [[64827/64000]] || <code>5 320000/64827, 7 64000/9261</code> || 2.3.7/5
* [[Hemimage comma|10976/10935]]: 5 10976/2187 to 2.3.7
|-
* [[3136/3125]]: 5 3136/625, 7 68841472/9765625 to 2.3.25/7
| [[Keemic]] || [[875/864]] || <code>7 864/125</code> || 2.3.5
* [[5120/5103]]: 7 5120/729
|-
* [[6144/6125]]: 3 6125/2048 to 2.5.7
| [[Gariboh]] || [[3125/3087]] || <code>5 15625/3087, 7 9765625/1361367</code> || 2.3.25/7
* [[Garischisma|33554432/33480783]]: 7 33554432/4782969  
|-
* [[Wadisma|201768035/201326592]]: 5 201326592/40353607 to 2.3.7
| [[Nuwell]] || [[2430/2401]] || <code>5 2401/486</code> || 2.3.7
* [[Quasiorwellisma|29360128/29296875]]: 7 29296875/4194304
|-
* [[Horwell comma|65625/65536]]: 7 65536/9375
| [[Trimyna]] || [[50421/50000]] || <code>3 50000/16807</code> || 2.5.7
* [[Meter|703125/702464]]: 5 702464/140625, 7 3454189699072/494384765625 to 2.3.25/7
|-
* [[Wizma|420175/419904]]: 5 882735153125/176319369216, 7 419904/60025 to 2.3.245
| [[Sensamagic]] || [[245/243]] || <code>5 243/49</code> || 2.3.7
* [[2401/2400]]: 3 2401/800 to 2.5.7
|-
* [[4375/4374]]: 7 4374/625
| [[Starling]] || [[126/125]] || <code>7 125/18</code> || 2.3.5
|-
| [[Octagar]] || [[4000/3969]] || <code>5 3969/800, 7 27783/4000</code> || 2.3.7/5
|-
| [[Orwellismic]] || [[1728/1715]] || <code>5 1728/343</code> || 2.3.7
|-
| [[Gamelismic]] || [[1029/1024]] || <code>3 1024/343</code> || 2.5.7
|-
| [[Marvel]] || [[225/224]] || <code>7 225/32</code> || 2.3.5
|-
| [[Cataharry]] || [[19683/19600]] || <code>3 19600/6561, 7 1033052339200000000/150094635296999121</code> || 2.5.81/7
|-
| [[Canopic]] || [[16875/16807]] || <code>5 84375/16807, 7 16875/2401</code> || 2.3.7/5
|-
| [[Hemimage]] || [[10976/10935]] || <code>5 10976/2187</code> || 2.3.7
|-
| [[Hemimean]] || [[3136/3125]] || <code>5 3136/625, 7 68841472/9765625</code> || 2.3.25/7
|-
| [[Aberschismic]] || [[5120/5103]] || <code>7 5120/729</code> || 2.3.5
|-
| [[Porwell]] || [[6144/6125]] || <code>3 6125/2048</code> || 2.5.7
|-
| [[Garischismic]] || [[33554432/33480783]] || <code>7 33554432/4782969</code> || 2.3.5
|-
| [[Wadismic]] || [[201768035/201326592]] || <code>5 201326592/40353607</code> || 2.3.7
|-
| [[Quasiorwellismic]] || [[29360128/29296875]] || <code>7 29296875/4194304</code> || 2.3.5
|-
| [[Horwell]] || [[65625/65536]] || <code>7 65536/9375</code> || 2.3.5
|-
| [[Metric]] || [[703125/702464]] || <code>5 702464/140625, 7 3454189699072/494384765625</code> || 2.3.25/7
|-
| [[Wizmic]] || [[420175/419904]] || <code>5 882735153125/176319369216, 7 419904/60025</code> || 2.3.245
|-
| [[Breed]] || [[2401/2400]] || <code>3 2401/800</code> || 2.5.7
|-
| [[Ragismic]] || [[4375/4374]] || <code>7 4374/625</code> || 2.3.5
|}


=== 11-limit ===
=== 11-limit ===
* [[33/32]]: 11 32/3
{| class="wikitable"
* [[45/44]]: 11 45/4
|+ Projection pairs for 11-limit temperaments
* [[55/54]]: 11 54/5
|-
* [[56/55]]: 11 56/5
! Temperament !! Associated comma !! Projection pair !! Target subgroup
* [[245/242]]: 5 242/49 to 2.3.7.11
|-
* [[99/98]]: 11 98/9
|  || [[33/32]] || <code>11 32/3</code> || 2.3.5.7
* [[100/99]]: 11 100/9
|-
* [[121/120]]: 5 121/24 to 2.3.7.11
| [[Cake]] || [[45/44]] || <code>11 45/4</code> || 2.3.5.7
* [[1331/1323]]: 7 9261/1331, 11 1323/121 to 2.3.5.11/7
|-
* [[176/175]]: 11 175/16
|  || [[55/54]] || <code>11 54/5</code> || 2.3.5.7
* [[896/891]]: 11 896/81
|-
* [[4375/4356]]: 7 4356/625 to 2.3.5.11
|  || [[56/55]] || <code>11 56/5</code> || 2.3.5.7
* [[Semicanousma|14641/14580]]: 5 14641/2916 to 2.3.7.11
|-
* [[243/242]]: 3 242/81, 11 644204/59049 to 2.5.7.11/9
| [[Frostmic]] || [[245/242]] || <code>5 242/49</code> || 2.3.7.11
* [[3388/3375]]: 7 3375/484 to 2.3.5.11
|-
* [[385/384]]: 11 384/35
| [[Mothwellsmic]] || [[99/98]] || <code>11 98/9</code> || 2.3.5.7
* [[8019/8000]]: 11 8000/729
|-
* [[441/440]]: 11 441/40
| [[Ptolemismic]] || [[100/99]] || <code>11 100/9</code> || 2.3.5.7
* [[1375/1372]]: 11 1372/125
|-
* [[6250/6237]]: 11 6250/567
| [[Biyatismic]] || [[121/120]] || <code>5 121/24</code> || 2.3.7.11
* [[540/539]]: 11 540/49
|-
* [[4000/3993]]: 3 4000/1331 to 2.5.7.11
|  || [[1331/1323]] || <code>7 9261/1331, 11 1323/121</code> || 2.3.5.11/7
* [[Symbiotic comma|19712/19683]]: 11 19683/1792
|-
* [[5632/5625]]: 11 5625/512
| [[Valinorsmic]] || [[176/175]] || <code>11 175/16</code> || 2.3.5.7
* [[Argyria|41503/41472]]: 7 41472/5929, 11 456533/41472 to 2.3.5.77
|-
* [[3025/3024]]: 7 3025/432 to 2.3.5.11
| [[Pentacircle]] || [[896/891]] || <code>11 896/81</code> || 2.3.5.7
|-
|  || [[4375/4356]] || <code>7 4356/625</code> || 2.3.5.11
|-
| [[Semicanousmic]] || [[14641/14580]] || <code>5 14641/2916</code> || 2.3.7.11
|-
| [[Rastmic]] || [[243/242]] || <code>3 242/81, 11 644204/59049</code> || 2.5.7.11/9
|-
|  || [[3388/3375]] || <code>7 3375/484</code> || 2.3.5.11
|-
| [[Keenanismic]] || [[385/384]] || <code>11 384/35</code> || 2.3.5.7
|-
| [[Trimitone]] || [[8019/8000]] || <code>11 8000/729</code> || 2.3.5.7
|-
| [[Werckismic]] || [[441/440]] || <code>11 441/40</code> || 2.3.5.7
|-
|  || [[1375/1372]] || <code>11 1372/125</code> || 2.3.5.7
|-
|  || [[6250/6237]] || <code>11 6250/567</code> || 2.3.5.7
|-
| [[Swetismic]] || [[540/539]] || <code>11 540/49</code> || 2.3.5.7
|-
|  || [[4000/3993]] || <code>3 4000/1331</code> || 2.5.7.11
|-
| [[Symbiotic]] || [[19712/19683]] || <code>11 19683/1792</code> || 2.3.5.7
|-
|  || [[5632/5625]] || <code>11 5625/512</code> || 2.3.5.7
|-
| [[Argyric]] || [[41503/41472]] || <code>7 41472/5929, 11 456533/41472</code> || 2.3.5.77
|-
| [[Lehmerismic]] || [[3025/3024]] || <code>7 3025/432</code> || 2.3.5.11
|}


[[Category:Lists of intervals]]
[[Category:Lists of intervals]]
[[Category:Just intonation]]
[[Category:Just intonation]]
[[Category:Method]]
[[Category:Method]]

Latest revision as of 06:52, 7 June 2026

A projection pair is a pair of two rational intervals which can be employed by the Scala "project" command to reduce a JI scale to a scale in a JI subgroup of the group generated by the scale, in such a way that tempered versions of each are equivalent. This is particularly useful for analyzing planar temperaments, as the projection can then be viewed in lattice form by Scala's "lattice" or "lattice and player" command.

An example of a projection pair is 7 225/32, which when applied by Scala's "project" to a 7-limit scale produces a 5-limit scale, which when tempered by marvel (225/224) temperament gives exactly the same result as the original scale does when also tempered. This can be thought of as marvel temperament replacing 7 by 225/32.

More than one such pair may be required to reduce to the desired subgroup; for instance 7 225/32, 11 4096/375 reduces an 11-limit JI scale to a 5-limit JI scale equivalent under (undecimal) marvel. This can happen even when only one comma is involved (codimension one temperaments). For instance, to project a 7-limit scale in the hemimean (3136/3125) reduction to the 2.5.7 subgroup requires 5 3136/625, 7 68841472/9765625.

Many projection pairs are given on the pages for various planar temperaments. When no subgroup is indicated, the default 2.3.5 5-limit subgroup is presumed. These lists of pairs can be copied and pasted into Scala and applied to any suitable JI scale.

List of projection pairs

5-limit

Projection pairs for 5-limit temperaments
Temperament Associated comma Projection pair Target subgroup
Negri 16875/16384 3 50625/16384, 5 16384/3375 2.15
Porcupine 250/243 3 729/250, 5 59049/12500 2.9/5
Magic 3125/3072 3 3125/1024 2.5
Tetracot 20000/19683 3 20000/6561, 5 2000000000/387420489 2.9/5
Meantone 81/80 5 81/16 2.3
Würschmidt 393216/390625 3 390625/131072 2.5
Orson 2109375/2097152 3 13348388671875/4398046511104, 5 2097152/421875 2.75
Hanson 15625/15552 3 46656/15625, 5 15552/3125 2.5/3
Helmholtz 32805/32768 5 32768/6561 2.3

7-limit

Projection pairs for 7-limit temperaments
Temperament Associated comma Projection pair Target subgroup
Keegic 1029/1000 3 1000/343 2.5.7
Mint 36/35 7 36/5 2.3.5
Avicennmic 525/512 7 512/75 2.3.5
Semaphoresmic 49/48 3 49/16 2.5.7
Sengic 686/675 5 3375/686, 7 675/98 2.3.7/5
Archytas 64/63 7 64/9 2.3.5
Blackjackismic 854296875/843308032 5 843308032/170859375, 7 5903156224/854296875 2.3.7/5
Squalentine 64827/64000 5 320000/64827, 7 64000/9261 2.3.7/5
Keemic 875/864 7 864/125 2.3.5
Gariboh 3125/3087 5 15625/3087, 7 9765625/1361367 2.3.25/7
Nuwell 2430/2401 5 2401/486 2.3.7
Trimyna 50421/50000 3 50000/16807 2.5.7
Sensamagic 245/243 5 243/49 2.3.7
Starling 126/125 7 125/18 2.3.5
Octagar 4000/3969 5 3969/800, 7 27783/4000 2.3.7/5
Orwellismic 1728/1715 5 1728/343 2.3.7
Gamelismic 1029/1024 3 1024/343 2.5.7
Marvel 225/224 7 225/32 2.3.5
Cataharry 19683/19600 3 19600/6561, 7 1033052339200000000/150094635296999121 2.5.81/7
Canopic 16875/16807 5 84375/16807, 7 16875/2401 2.3.7/5
Hemimage 10976/10935 5 10976/2187 2.3.7
Hemimean 3136/3125 5 3136/625, 7 68841472/9765625 2.3.25/7
Aberschismic 5120/5103 7 5120/729 2.3.5
Porwell 6144/6125 3 6125/2048 2.5.7
Garischismic 33554432/33480783 7 33554432/4782969 2.3.5
Wadismic 201768035/201326592 5 201326592/40353607 2.3.7
Quasiorwellismic 29360128/29296875 7 29296875/4194304 2.3.5
Horwell 65625/65536 7 65536/9375 2.3.5
Metric 703125/702464 5 702464/140625, 7 3454189699072/494384765625 2.3.25/7
Wizmic 420175/419904 5 882735153125/176319369216, 7 419904/60025 2.3.245
Breed 2401/2400 3 2401/800 2.5.7
Ragismic 4375/4374 7 4374/625 2.3.5

11-limit

Projection pairs for 11-limit temperaments
Temperament Associated comma Projection pair Target subgroup
33/32 11 32/3 2.3.5.7
Cake 45/44 11 45/4 2.3.5.7
55/54 11 54/5 2.3.5.7
56/55 11 56/5 2.3.5.7
Frostmic 245/242 5 242/49 2.3.7.11
Mothwellsmic 99/98 11 98/9 2.3.5.7
Ptolemismic 100/99 11 100/9 2.3.5.7
Biyatismic 121/120 5 121/24 2.3.7.11
1331/1323 7 9261/1331, 11 1323/121 2.3.5.11/7
Valinorsmic 176/175 11 175/16 2.3.5.7
Pentacircle 896/891 11 896/81 2.3.5.7
4375/4356 7 4356/625 2.3.5.11
Semicanousmic 14641/14580 5 14641/2916 2.3.7.11
Rastmic 243/242 3 242/81, 11 644204/59049 2.5.7.11/9
3388/3375 7 3375/484 2.3.5.11
Keenanismic 385/384 11 384/35 2.3.5.7
Trimitone 8019/8000 11 8000/729 2.3.5.7
Werckismic 441/440 11 441/40 2.3.5.7
1375/1372 11 1372/125 2.3.5.7
6250/6237 11 6250/567 2.3.5.7
Swetismic 540/539 11 540/49 2.3.5.7
4000/3993 3 4000/1331 2.5.7.11
Symbiotic 19712/19683 11 19683/1792 2.3.5.7
5632/5625 11 5625/512 2.3.5.7
Argyric 41503/41472 7 41472/5929, 11 456533/41472 2.3.5.77
Lehmerismic 3025/3024 7 3025/432 2.3.5.11
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