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5-limit section; complete first table
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Since we are using a pentatonic system of notation, and [[5edo]] represents the [[2.3.7 subgroup]] very well, we will investigate ratios with factors of 7 before ratios with factors of 5. Just like in the FJS, we will be using [[64/63]] as our formal comma.
Since we are using a pentatonic system of notation, and [[5edo]] represents the [[2.3.7 subgroup]] very well, we will investigate ratios with factors of 7 before ratios with factors of 5. Just like in the FJS, we will be using [[64/63]] as our formal comma.
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|+ Ratios with a factor of 7
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Interval classification would be much simpler if the Pythagorean intervals were equated with their simpler septimal counterparts; this occurs in [[superpyth]] temperament, where 64/63 is [[tempering out|tempered out]].
Interval classification would be much simpler if the Pythagorean intervals were equated with their simpler septimal counterparts; this occurs in [[superpyth]] temperament, where 64/63 is [[tempering out|tempered out]].
With similar constructions, larger chords can be constructed, including [[28:36:42:49|a version of the dominant seventh chord]]; however, this is beyond the scope of this page.
== Ratios of 5 ==
Now, we will look at ratios of 5. The most salient fact is that 5/4 and 6/5 are no longer in the same interval category; 6/5 is a <sub>5</sub>second, while 5/4 is a <sub>5</sub>third.
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{| class="wikitable"
|+ Ratios with a factor of 5
|-
! Ratio !! Cents !! Interval name<br>(Pentatonic)
|-
| 81/80 || 21.5 || <sub>5</sub>P1<sub>5</sub>
|-
| 16/15 || 111.7 || <sub>5</sub>A1<sub>5</sub>
|-
| 135/128 || 92.2 || <sub>5</sub>d2<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>
|-
| 5/4 || 386.3 || <sub>5</sub>d3<sup>5</sup>
|-
| 27/20 || 519.6 || <sub>5</sub>P3<sub>5</sub>
|-
| 64/45 || 609.8 || <sub>5</sub>A3<sub>5</sub>
|-
| 45/32 || 590.2 || <sub>5</sub>d4<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>
|-
| 5/3 || 884.4 || <sub>5</sub>m5<sup>5</sup>
|-
| 9/5 || 1017.6 || <sub>5</sub>M5<sub>5</sub>
|-
| 256/135 || 1107.8 || <sub>5</sub>A5<sub>5</sub>
|-
| 15/8 || 1088.3 || <sub>5</sub>d6<sup>5</sup>
|-
| 160/81 || 1178.5 || <sub>5</sub>P6<sup>5</sup>
|}
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{| class="wikitable"
|+ Ratios with two factors of 5 (incomplete)
|-
! Ratio !! Cents !! Interval name<br>(Pentatonic)
|-
| 4096/3969 || 54.5 || <sub>5</sub>P1<sub>7,7</sub>
|-
| 49/48 || 35.7 || <sub>5</sub>A1<sup>7,7</sup>
|-
| 54/49 || 168.2 || <sub>5</sub>d2<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>
|-
| 64/49 || 462.3 || <sub>5</sub>d3<sub>7,7</sub>
|-
| 1323/1024 || 443.5 || <sub>5</sub>P3<sup>7,7</sup>
|-
| 49/36 || 533.7 || <sub>5</sub>A3<sup>7,7</sup>
|-
| 72/49 || 666.3 || <sub>5</sub>d4<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>
|-
| 256/147 || 960.4 || <sub>5</sub>m5<sub>7,7</sub>
|-
| 441/256 || 941.6 || <sub>5</sub>M5<sup>7,7</sup>
|-
| 49/27 || 1031.8 || <sub>5</sub>A5<sup>7,7</sup>
|-
| 96/49 || 1164.3 || <sub>5</sub>d6<sub>7,7</sub>
|-
| 3969/2048 || 1145.5 || <sub>5</sub>P6<sup>7,7</sup>
|}
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