Pentatonic Functional Just System: Difference between revisions
→The 11- and 13-limits: new formal commas |
fix formal commas |
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== Ratios of 7 == | == Ratios of 7 == | ||
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|63/64]] as our formal comma. | ||
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== Ratios of 5 == | == 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. | Now, we will look at ratios of 5. Just like in the FJS, our formal comma is [[81/80|80/81]]. 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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== The 11- and 13-limits == | == The 11- and 13-limits == | ||
We now look at the entire [[15-odd-limit]] [[tonality diamond]]. Here, we will use more complex formal commas: The formal comma for 11 is [[8192/8019|8019/8192]] (11/8 is <sub>5</sub>S3<sup>11</sup>), and the formal comma for 13 is [[6656/6561]] (13/8 is <sub>5</sub>s5<sup>13</sup>). | We now look at the entire [[15-odd-limit]] [[tonality diamond]]. Here, we will use more complex formal commas than in the FJS: The formal comma for 11 is [[8192/8019|8019/8192]] (11/8 is <sub>5</sub>S3<sup>11</sup>), and the formal comma for 13 is [[6656/6561]] (13/8 is <sub>5</sub>s5<sup>13</sup>). | ||
{| class="wikitable right-all mw-collapsible" | {| class="wikitable right-all mw-collapsible" | ||
|+ {{nowrap|15-odd-limit by PFJS (last column is outdated)}} | |+ {{nowrap|15-odd-limit by PFJS (last column is outdated)}} | ||
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| 14/11 || 417.5 || <sub>5</sub>P3<sup>7</sup><sub>11</sub> | | 14/11 || 417.5 || <sub>5</sub>P3<sup>7</sup><sub>11</sub> | ||
|- | |- | ||
| 9/7 || 435.1|| | | 9/7 || 435.1 || | ||
|- | |- | ||
| 13/10 || 454.2|| | | 13/10 || 454.2 || | ||
|- | |- | ||
| 4/3 || 498.0|| | | 4/3 || 498.0 || | ||
|- | |- | ||
| 15/11 || 537.0|| | | 15/11 || 537.0 || | ||
|- | |- | ||
| 11/8 || 551.3|| | | 11/8 || 551.3 || | ||
|- | |- | ||
| 18/13 || 563.4|| | | 18/13 || 563.4 || | ||
|- | |- | ||
| 7/5 || 582.5|| | | 7/5 || 582.5 || | ||
|- | |- | ||
| 10/7 || 617.5|| | | 10/7 || 617.5 || | ||
|- | |- | ||
| 13/9 || 636.6|| | | 13/9 || 636.6 || | ||
|- | |- | ||
| 16/11 || 648.7|| | | 16/11 || 648.7 || | ||
|- | |- | ||
| 22/15 || 663.0|| | | 22/15 || 663.0 || | ||
|- | |- | ||
| 3/2 || 702.0|| | | 3/2 || 702.0 || | ||
|- | |- | ||
| 20/13 || 745.8|| | | 20/13 || 745.8 || | ||
|- | |- | ||
| 14/9 || 764.9|| | | 14/9 || 764.9 || | ||
|- | |- | ||
| 11/7 || 782.5|| | | 11/7 || 782.5 || | ||
|- | |- | ||
| 8/5 || 813.7|| | | 8/5 || 813.7 || | ||
|- | |- | ||
| 13/8 || 840.5|| | | 13/8 || 840.5 || | ||
|- | |- | ||
| 18/11 || 852.6|| | | 18/11 || 852.6 || | ||
|- | |- | ||
| 5/3 || 884.4|| | | 5/3 || 884.4 || | ||
|- | |- | ||
| 22/13 || 910.8|| | | 22/13 || 910.8 || | ||
|- | |- | ||
| 12/7 || 933.1|| | | 12/7 || 933.1 || | ||
|- | |- | ||
| 26/15 || 952.3|| | | 26/15 || 952.3 || | ||
|- | |- | ||
| 7/4 || 968.8|| | | 7/4 || 968.8 || | ||
|- | |- | ||
| 16/9 || 996.1|| | | 16/9 || 996.1 || | ||
|- | |- | ||
| 9/5 || 1017.6|| | | 9/5 || 1017.6 || | ||
|- | |- | ||
| 20/11 || 1035.0|| | | 20/11 || 1035.0 || | ||
|- | |- | ||
| 11/6 || 1049.4|| | | 11/6 || 1049.4 || | ||
|- | |- | ||
| 24/13 || 1061.4|| | | 24/13 || 1061.4 || | ||
|- | |- | ||
| 13/7 || 1071.7|| | | 13/7 || 1071.7 || | ||
|- | |- | ||
| 28/15 || 1080.6|| | | 28/15 || 1080.6 || | ||
|- | |- | ||
| 15/8 || 1088.3|| | | 15/8 || 1088.3 || | ||
|- | |- | ||
| 2/1 || 1200.0|| | | 2/1 || 1200.0 || | ||
|} | |} | ||