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→Ratios of 5: + explanation
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Adopt sub and super all around; expand 5-limit explanation
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Traditionally, we use a [[5L 2s|diatonic]] system of interval classification. This works well in the [[5-limit]] and in [[meantone]]. However, in other systems like [[superpyth]], a pentatonic system of classification based on the [[2L 3s]] [[MOS scale]] may be preferred. We will develop a pentatonic version of the [[FJS]], starting from the [[3-limit]] and using [[formal comma]]s to reach higher limits.
Traditionally, we use a [[5L 2s|diatonic]] system of interval classification. This works well in the [[5-limit]] and in [[meantone]]. However, in other systems like [[superpyth]], a pentatonic system of classification based on the [[2L 3s]] [[MOS scale]] may be preferred. In this page, we will develop a pentatonic version of the [[FJS]], starting from the [[3-limit]] and using [[formal comma]]s to reach higher limits.


== The 3-limit ==
== The 3-limit ==
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| 2/1 || 1200.0 || <sub>5</sub>P6
| 2/1 || 1200.0 || <sub>5</sub>P6
|}
|}
In contrast to diatonic, [[256/243]] is a chroma interval, separating major and minor intervals of the same category. Interestingly, only pentatonic seconds and fifths now have major/minor, and augmented and diminished intervals come way eariler.
In contrast to diatonic, [[256/243]] is a chroma interval, separating major and minor intervals of the same category. Interestingly, only pentatonic seconds and fifths now have major/minor, and augmented and diminished intervals show up way more often. From here on we will refer to augmented and diminished as "super" and "sub" (not to be confused with "supermajor" and "subminor"), with symbols "S" and "s" respectively.


== Ratios of 7 ==
== Ratios of 7 ==
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| 64/63 || 27.3 || <sub>5</sub>P1<sub>7</sub>
| 64/63 || 27.3 || <sub>5</sub>P1<sub>7</sub>
|-
|-
| 28/27 || 63.0 || <sub>5</sub>A1<sup>7</sup>
| 28/27 || 63.0 || <sub>5</sub>S1<sup>7</sup>
|-
|-
| 243/224 || 140.9 || <sub>5</sub>d2<sub>7</sub>
| 243/224 || 140.9 || <sub>5</sub>s2<sub>7</sub>
|-
|-
| 8/7 || 231.2 || <sub>5</sub>m2<sub>7</sub>
| 8/7 || 231.2 || <sub>5</sub>m2<sub>7</sub>
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| 7/6 || 266.9 || <sub>5</sub>M2<sup>7</sup>
| 7/6 || 266.9 || <sub>5</sub>M2<sup>7</sup>
|-
|-
| 896/729 || 357.1 || <sub>5</sub>A2<sup>7</sup>
| 896/729 || 357.1 || <sub>5</sub>S2<sup>7</sup>
|-
|-
| 9/7 || 435.1 || <sub>5</sub>d3<sub>7</sub>
| 9/7 || 435.1 || <sub>5</sub>s3<sub>7</sub>
|-
|-
| 21/16 || 470.8 || <sub>5</sub>P3<sup>7</sup>
| 21/16 || 470.8 || <sub>5</sub>P3<sup>7</sup>
|-
|-
| 112/81 || 561.0 || <sub>5</sub>A3<sup>7</sup>
| 112/81 || 561.0 || <sub>5</sub>S3<sup>7</sup>
|-
|-
| 81/56 || 639.0 || <sub>5</sub>d4<sub>7</sub>
| 81/56 || 639.0 || <sub>5</sub>s4<sub>7</sub>
|-
|-
| 32/21 || 729.2 || <sub>5</sub>P4<sub>7</sub>
| 32/21 || 729.2 || <sub>5</sub>P4<sub>7</sub>
|-
|-
| 14/9 || 764.9 || <sub>5</sub>A4<sup>7</sup>
| 14/9 || 764.9 || <sub>5</sub>S4<sup>7</sup>
|-
|-
| 729/448 || 842.9 || <sub>5</sub>A2<sub>7</sub>
| 729/448 || 842.9 || <sub>5</sub>s5<sub>7</sub>
|-
|-
| 12/7 || 933.1 || <sub>5</sub>m5<sub>7</sub>
| 12/7 || 933.1 || <sub>5</sub>m5<sub>7</sub>
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| 7/4 || 968.8 || <sub>5</sub>M5<sup>7</sup>
| 7/4 || 968.8 || <sub>5</sub>M5<sup>7</sup>
|-
|-
| 448/243 || 1059.1 || <sub>5</sub>A5<sup>7</sup>
| 448/243 || 1059.1 || <sub>5</sub>S5<sup>7</sup>
|-
|-
| 27/14 || 1137.0 || <sub>5</sub>d6<sub>7</sub>
| 27/14 || 1137.0 || <sub>5</sub>s6<sub>7</sub>
|-
|-
| 63/32 || 1200.0 || <sub>5</sub>P6<sup>7</sup>
| 63/32 || 1200.0 || <sub>5</sub>P6<sup>7</sup>
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| 49/48 || 35.7 || <sub>5</sub>A1<sup>7,7</sup>
| 49/48 || 35.7 || <sub>5</sub>A1<sup>7,7</sup>
|-
|-
| 54/49 || 168.2 || <sub>5</sub>d2<sub>7,7</sub>
| 54/49 || 168.2 || <sub>5</sub>s2<sub>7,7</sub>
|-
|-
| 512/441 || 258.4 || <sub>5</sub>m2<sub>7,7</sub>
| 512/441 || 258.4 || <sub>5</sub>m2<sub>7,7</sub>
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| 147/128 || 239.6 || <sub>5</sub>M2<sup>7,7</sup>
| 147/128 || 239.6 || <sub>5</sub>M2<sup>7,7</sup>
|-
|-
| 98/81 || 329.8 || <sub>5</sub>A2<sup>7,7</sup>
| 98/81 || 329.8 || <sub>5</sub>S2<sup>7,7</sup>
|-
|-
| 64/49 || 462.3 || <sub>5</sub>d3<sub>7,7</sub>
| 64/49 || 462.3 || <sub>5</sub>s3<sub>7,7</sub>
|-
|-
| 1323/1024 || 443.5 || <sub>5</sub>P3<sup>7,7</sup>
| 1323/1024 || 443.5 || <sub>5</sub>P3<sup>7,7</sup>
|-
|-
| 49/36 || 533.7 || <sub>5</sub>A3<sup>7,7</sup>
| 49/36 || 533.7 || <sub>5</sub>S3<sup>7,7</sup>
|-
|-
| 72/49 || 666.3 || <sub>5</sub>d4<sub>7,7</sub>
| 72/49 || 666.3 || <sub>5</sub>s4<sub>7,7</sub>
|-
|-
| 2048/1323 || 756.5 || <sub>5</sub>P4<sub>7,7</sub>
| 2048/1323 || 756.5 || <sub>5</sub>P4<sub>7,7</sub>
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| 49/32 || 737.7 || <sub>5</sub>A4<sup>7,7</sup>
| 49/32 || 737.7 || <sub>5</sub>A4<sup>7,7</sup>
|-
|-
| 81/49 || 870.2 || <sub>5</sub>d5<sub>7,7</sub>
| 81/49 || 870.2 || <sub>5</sub>s5<sub>7,7</sub>
|-
|-
| 256/147 || 960.4 || <sub>5</sub>m5<sub>7,7</sub>
| 256/147 || 960.4 || <sub>5</sub>m5<sub>7,7</sub>
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| 441/256 || 941.6 || <sub>5</sub>M5<sup>7,7</sup>
| 441/256 || 941.6 || <sub>5</sub>M5<sup>7,7</sup>
|-
|-
| 49/27 || 1031.8 || <sub>5</sub>A5<sup>7,7</sup>
| 49/27 || 1031.8 || <sub>5</sub>S5<sup>7,7</sup>
|-
|-
| 96/49 || 1164.3 || <sub>5</sub>d6<sub>7,7</sub>
| 96/49 || 1164.3 || <sub>5</sub>s6<sub>7,7</sub>
|-
|-
| 3969/2048 || 1145.5 || <sub>5</sub>P6<sup>7,7</sup>
| 3969/2048 || 1145.5 || <sub>5</sub>P6<sup>7,7</sup>
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| 81/80 || 21.5 || <sub>5</sub>P1<sub>5</sub>
| 81/80 || 21.5 || <sub>5</sub>P1<sub>5</sub>
|-
|-
| 16/15 || 111.7 || <sub>5</sub>A1<sub>5</sub>
| 16/15 || 111.7 || <sub>5</sub>S1<sub>5</sub>
|-
|-
| 135/128 || 92.2 || <sub>5</sub>d2<sup>5</sup>
| 135/128 || 92.2 || <sub>5</sub>s2<sup>5</sup>
|-
|-
| 10/9 || 182.4 || <sub>5</sub>m2<sup>5</sup>
| 10/9 || 182.4 || <sub>5</sub>m2<sup>5</sup>
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| 6/5 || 315.6 || <sub>5</sub>M2<sub>5</sub>
| 6/5 || 315.6 || <sub>5</sub>M2<sub>5</sub>
|-
|-
| 512/405 || 405.9 || <sub>5</sub>A2<sub>5</sub>
| 512/405 || 405.9 || <sub>5</sub>S2<sub>5</sub>
|-
|-
| 5/4 || 386.3 || <sub>5</sub>d3<sup>5</sup>
| 5/4 || 386.3 || <sub>5</sub>s3<sup>5</sup>
|-
|-
| 27/20 || 519.6 || <sub>5</sub>P3<sub>5</sub>
| 27/20 || 519.6 || <sub>5</sub>P3<sub>5</sub>
|-
|-
| 64/45 || 609.8 || <sub>5</sub>A3<sub>5</sub>
| 64/45 || 609.8 || <sub>5</sub>S3<sub>5</sub>
|-
|-
| 45/32 || 590.2 || <sub>5</sub>d4<sup>5</sup>
| 45/32 || 590.2 || <sub>5</sub>s4<sup>5</sup>
|-
|-
| 40/27 || 680.4 || <sub>5</sub>P4<sup>5</sup>
| 40/27 || 680.4 || <sub>5</sub>P4<sup>5</sup>
|-
|-
| 8/5 || 813.7 || <sub>5</sub>A4<sub>5</sub>
| 8/5 || 813.7 || <sub>5</sub>S4<sub>5</sub>
|-
|-
| 405/256 || 794.1 || <sub>5</sub>d5<sup>5</sup>
| 405/256 || 794.1 || <sub>5</sub>s5<sup>5</sup>
|-
|-
| 5/3 || 884.4 || <sub>5</sub>m5<sup>5</sup>
| 5/3 || 884.4 || <sub>5</sub>m5<sup>5</sup>
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| 9/5 || 1017.6 || <sub>5</sub>M5<sub>5</sub>
| 9/5 || 1017.6 || <sub>5</sub>M5<sub>5</sub>
|-
|-
| 256/135 || 1107.8 || <sub>5</sub>A5<sub>5</sub>
| 256/135 || 1107.8 || <sub>5</sub>S5<sub>5</sub>
|-
|-
| 15/8 || 1088.3 || <sub>5</sub>d6<sup>5</sup>
| 15/8 || 1088.3 || <sub>5</sub>s6<sup>5</sup>
|-
|-
| 160/81 || 1178.5 || <sub>5</sub>P6<sup>5</sup>
| 160/81 || 1178.5 || <sub>5</sub>P6<sup>5</sup>
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| 6561/6400 || 43.0 || <sub>5</sub>P1<sub>5,5</sub>
| 6561/6400 || 43.0 || <sub>5</sub>P1<sub>5,5</sub>
|-
|-
| 27/25 || 133.2 || <sub>5</sub>A1<sub>5,5</sub>
| 27/25 || 133.2 || <sub>5</sub>S1<sub>5,5</sub>
|-
|-
| 25/24 || 70.7 || <sub>5</sub>d2<sup>5,5</sup>
| 25/24 || 70.7 || <sub>5</sub>s2<sup>5,5</sup>
|-
|-
| 800/729 || 160.9 || <sub>5</sub>m2<sup>5,5</sup>
| 800/729 || 160.9 || <sub>5</sub>m2<sup>5,5</sup>
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| 243/200 || 337.1 || <sub>5</sub>M2<sub>5,5</sub>
| 243/200 || 337.1 || <sub>5</sub>M2<sub>5,5</sub>
|-
|-
| 32/25 || 427.4 || <sub>5</sub>A2<sub>5,5</sub>
| 32/25 || 427.4 || <sub>5</sub>S2<sub>5,5</sub>
|-
|-
| 100/81 || 364.8 || <sub>5</sub>d3<sup>5,5</sup>
| 100/81 || 364.8 || <sub>5</sub>s3<sup>5,5</sup>
|-
|-
| 2187/1600 || 541.1 || <sub>5</sub>P3<sub>5,5</sub>
| 2187/1600 || 541.1 || <sub>5</sub>P3<sub>5,5</sub>
|-
|-
| 36/25 || 631.3 || <sub>5</sub>A3<sub>5,5</sub>
| 36/25 || 631.3 || <sub>5</sub>S3<sub>5,5</sub>
|-
|-
| 25/18 || 568.7 || <sub>5</sub>d4<sup>5,5</sup>
| 25/18 || 568.7 || <sub>5</sub>s4<sup>5,5</sup>
|-
|-
| 3200/2187 || 658.9 || <sub>5</sub>P4<sup>5,5</sup>
| 3200/2187 || 658.9 || <sub>5</sub>P4<sup>5,5</sup>
|-
|-
| 81/50 || 835.2 || <sub>5</sub>A4<sub>5,5</sub>
| 81/50 || 835.2 || <sub>5</sub>S4<sub>5,5</sub>
|-
|-
| 25/16 || 772.6 || <sub>5</sub>d5<sup>5,5</sup>
| 25/16 || 772.6 || <sub>5</sub>s5<sup>5,5</sup>
|-
|-
| 400/243 || 862.9 || <sub>5</sub>m5<sup>5,5</sup>
| 400/243 || 862.9 || <sub>5</sub>m5<sup>5,5</sup>
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| 729/400 || 1039.1 || <sub>5</sub>M5<sub>5,5</sub>
| 729/400 || 1039.1 || <sub>5</sub>M5<sub>5,5</sub>
|-
|-
| 48/25 || 1129.3 || <sub>5</sub>A5<sub>5,5</sub>
| 48/25 || 1129.3 || <sub>5</sub>S5<sub>5,5</sub>
|-
|-
| 50/27 || 1066.8 || <sub>5</sub>d6<sup>5,5</sup>
| 50/27 || 1066.8 || <sub>5</sub>s6<sup>5,5</sup>
|-
|-
| 12800/6561 || 1157.0 || <sub>5</sub>P6<sup>5,5</sup>
| 12800/6561 || 1157.0 || <sub>5</sub>P6<sup>5,5</sup>
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One can see that the ratios of 5 are further from 5edo intervals than ratios of 7. Thus, the 5-limit intervals can now be considered "subminor" and "supermajor", compared to the intervals of 7 in diatonic. Using the <sub>5</sub>fifth construction, we get the [[3:5:9]] subminor and [[5:9:15|1/(9:5:3) = 5:9:15]] supermajor chords, the compact voicings of which are [[9:10:12]] and [[15:18:20]] respectively.
One can see that the ratios of 5 are further from 5edo intervals than ratios of 7. Thus, the 5-limit intervals can now be considered "subminor" and "supermajor", compared to the intervals of 7 in diatonic. Using the <sub>5</sub>fifth construction, we get the [[3:5:9]] subminor and [[5:9:15|1/(9:5:3) = 5:9:15]] supermajor chords, the compact voicings of which are [[9:10:12]] and [[15:18:20]] respectively.


If we try to construct 5-limit triads the normal way, the [[4:5:6]] major triad becomes <sub>5</sub>P1–<sub>5</sub>d3<sup>5</sup>–<sub>5</sub>P4, and the [[10:12:15]] minor triad becomes <sub>5</sub>P1–<sub>5</sub>M2<sub>5</sub>–<sub>5</sub>P4. Augmented and diminished intervals occur way more frequently; thus they are not the best names. From here on we will refer to augmented and diminished as "super" and "sub" (not to be confused with "supermajor" and "subminor"), with symbols "S" and "s" respectively.
If we try to construct 5-limit triads the normal way, the [[4:5:6]] major triad becomes <sub>5</sub>P1–<sub>5</sub>s3<sup>5</sup>–<sub>5</sub>P4, and the [[10:12:15]] minor triad becomes <sub>5</sub>P1–<sub>5</sub>M2<sub>5</sub>–<sub>5</sub>P4. No wonder it was a good idea refer to augmented and diminished as "super" and "sub"; otherwise 5/4 would be a diminished <sub>5</sub>third. However, now the [[4:5:6]] and [[10:12:15]] triads aren't classified by the same interval categories, while they are in diatonic.
 
In full 7-limit superpyth, 10/9 is a subsecond, 6/5 is a supersecond, and 5/4 is a subsubthird (9/7 is a subthird).
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