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Pentatonic Functional Just System: Difference between revisions

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{{Editable user page|Please add the 11-limit tables if no one has done so already. Also feel free to add more content, but please do not delete existing content without my permission.}}
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.
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.


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! Ratio !! Cents !! Interval name<br>(Pentatonic)
! Ratio !! Cents !! Interval name<br>(Pentatonic)
|-
|-
| 1/1 || 0.0 || <sub>5</sub>P1
| [[1/1]] || 0.0 || <sub>5</sub>P1
|-
|-
| 256/243 || 90.2 || <sub>5</sub>A1
| [[256/243]] || 90.2 || <sub>5</sub>A1
|-
|-
| 2187/2048 || 113.7 || <sub>5</sub>d2
| [[2187/2048]] || 113.7 || <sub>5</sub>d2
|-
|-
| 9/8 || 203.9 || <sub>5</sub>m2
| [[9/8]] || 203.9 || <sub>5</sub>m2
|-
|-
| 32/27 || 294.1 || <sub>5</sub>M2
| [[32/27]] || 294.1 || <sub>5</sub>M2
|-
|-
| 8192/6561 || 384.4 || <sub>5</sub>A2
| [[8192/6561]] || 384.4 || <sub>5</sub>A2
|-
|-
| 81/64 || 407.8 || <sub>5</sub>d3
| [[81/64]] || 407.8 || <sub>5</sub>d3
|-
|-
| 4/3 || 498.0 || <sub>5</sub>P3
| [[4/3]] || 498.0 || <sub>5</sub>P3
|-
|-
| 1024/729 || 588.3 || <sub>5</sub>A3
| [[1024/729]] || 588.3 || <sub>5</sub>A3
|-
|-
| 729/512 || 611.7 || <sub>5</sub>d4
| [[729/512]] || 611.7 || <sub>5</sub>d4
|-
|-
| 3/2 || 702.0 || <sub>5</sub>P4
| [[3/2]] || 702.0 || <sub>5</sub>P4
|-
|-
| 128/81 || 792.2 || <sub>5</sub>A4
| [[128/81]] || 792.2 || <sub>5</sub>A4
|-
|-
| 6561/4096 || 815.6 || <sub>5</sub>d5
| [[6561/4096]] || 815.6 || <sub>5</sub>d5
|-
|-
| 27/16 || 905.9 || <sub>5</sub>m5
| [[27/16]] || 905.9 || <sub>5</sub>m5
|-
|-
| 16/9 || 996.1 || <sub>5</sub>M5
| [[16/9]] || 996.1 || <sub>5</sub>M5
|-
|-
| 4096/2187 || 1086.3 || <sub>5</sub>A5
| [[4096/2187]] || 1086.3 || <sub>5</sub>A5
|-
|-
| 243/128 || 1109.8 || <sub>5</sub>d6
| [[243/128]] || 1109.8 || <sub>5</sub>d6
|-
|-
| 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 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.
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.
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! Ratio !! Cents !! Interval name<br>(Pentatonic)
! Ratio !! Cents !! Interval name<br>(Pentatonic)
|-
|-
| 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>S1<sup>7</sup>
| [[28/27]] || 63.0 || <sub>5</sub>S1<sup>7</sup>
|-
|-
| 243/224 || 140.9 || <sub>5</sub>s2<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>
|-
|-
| 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>S2<sup>7</sup>
| [[896/729]] || 357.1 || <sub>5</sub>S2<sup>7</sup>
|-
|-
| 9/7 || 435.1 || <sub>5</sub>s3<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>S3<sup>7</sup>
| [[112/81]] || 561.0 || <sub>5</sub>S3<sup>7</sup>
|-
|-
| 81/56 || 639.0 || <sub>5</sub>s4<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>S4<sup>7</sup>
| [[14/9]] || 764.9 || <sub>5</sub>S4<sup>7</sup>
|-
|-
| 729/448 || 842.9 || <sub>5</sub>s5<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>
|-
|-
| 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>S5<sup>7</sup>
| [[448/243]] || 1059.1 || <sub>5</sub>S5<sup>7</sup>
|-
|-
| 27/14 || 1137.0 || <sub>5</sub>s6<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>
|}
|}
</div><div style="display: inline-grid; margin-right: 25px;">
</div><div style="display: inline-grid; margin-right: 25px;">
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! Ratio !! Cents !! Interval name<br>(Pentatonic)
! Ratio !! Cents !! Interval name<br>(Pentatonic)
|-
|-
| 4096/3969 || 54.5 || <sub>5</sub>P1<sub>7,7</sub>
| [[4096/3969]] || 54.5 || <sub>5</sub>P1<sub>7,7</sub>
|-
|-
| 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>s2<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>
|-
|-
| 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>S2<sup>7,7</sup>
| [[98/81]] || 329.8 || <sub>5</sub>S2<sup>7,7</sup>
|-
|-
| 64/49 || 462.3 || <sub>5</sub>s3<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>S3<sup>7,7</sup>
| [[49/36]] || 533.7 || <sub>5</sub>S3<sup>7,7</sup>
|-
|-
| 72/49 || 666.3 || <sub>5</sub>s4<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>
|-
|-
| 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>s5<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>
|-
|-
| 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>S5<sup>7,7</sup>
| [[49/27]] || 1031.8 || <sub>5</sub>S5<sup>7,7</sup>
|-
|-
| 96/49 || 1164.3 || <sub>5</sub>s6<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>
|}
|}
</div></div>
</div></div>
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! Ratio !! Cents !! Interval name<br>(Pentatonic)
! Ratio !! Cents !! Interval name<br>(Pentatonic)
|-
|-
| 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>S1<sub>5</sub>
| [[16/15]] || 111.7 || <sub>5</sub>S1<sub>5</sub>
|-
|-
| 135/128 || 92.2 || <sub>5</sub>s2<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>
|-
|-
| 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>S2<sub>5</sub>
| [[512/405]] || 405.9 || <sub>5</sub>S2<sub>5</sub>
|-
|-
| 5/4 || 386.3 || <sub>5</sub>s3<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>S3<sub>5</sub>
| [[64/45]] || 609.8 || <sub>5</sub>S3<sub>5</sub>
|-
|-
| 45/32 || 590.2 || <sub>5</sub>s4<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>S4<sub>5</sub>
| [[8/5]] || 813.7 || <sub>5</sub>S4<sub>5</sub>
|-
|-
| 405/256 || 794.1 || <sub>5</sub>s5<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>
|-
|-
| 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>S5<sub>5</sub>
| [[256/135]] || 1107.8 || <sub>5</sub>S5<sub>5</sub>
|-
|-
| 15/8 || 1088.3 || <sub>5</sub>s6<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>
|}
|}
</div><div style="display: inline-grid; margin-right: 25px;">
</div><div style="display: inline-grid; margin-right: 25px;">
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! Ratio !! Cents !! Interval name<br>(Pentatonic)
! Ratio !! Cents !! Interval name<br>(Pentatonic)
|-
|-
| 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>S1<sub>5,5</sub>
| [[27/25]] || 133.2 || <sub>5</sub>S1<sub>5,5</sub>
|-
|-
| 25/24 || 70.7 || <sub>5</sub>s2<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>
|-
|-
| 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>S2<sub>5,5</sub>
| [[32/25]] || 427.4 || <sub>5</sub>S2<sub>5,5</sub>
|-
|-
| 100/81 || 364.8 || <sub>5</sub>s3<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>S3<sub>5,5</sub>
| [[36/25]] || 631.3 || <sub>5</sub>S3<sub>5,5</sub>
|-
|-
| 25/18 || 568.7 || <sub>5</sub>s4<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>S4<sub>5,5</sub>
| [[81/50]] || 835.2 || <sub>5</sub>S4<sub>5,5</sub>
|-
|-
| 25/16 || 772.6 || <sub>5</sub>s5<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>
|-
|-
| 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>S5<sub>5,5</sub>
| [[48/25]] || 1129.3 || <sub>5</sub>S5<sub>5,5</sub>
|-
|-
| 50/27 || 1066.8 || <sub>5</sub>s6<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>
|}
|}
</div></div>
</div></div>
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