User:Overthink/13-limit interval flavors: Difference between revisions

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{{Editable user page|Please complete interval tables for flavors of 7, 11, 13, 7/5, 11/5, 11/7, 13/5, 13/7, and 13/11, with all perfect, major, and minor intervals, as well as A4 and d5. Note that the flavors of 7/5 and 5/7 should be placed in the same table.}}
In this article, we will cover the various flavors of 13-limit intervals. We consider intervals that differ by a pythagorean interval to have the same flavor. The flavor of an interval depends on the primes higher than 3 in its prime factorization.
In this article, we will cover the various flavors of 13-limit intervals. We consider intervals that differ by a pythagorean interval to have the same flavor. The flavor of an interval depends on the primes higher than 3 in its prime factorization.
{| class="wikitable"
{| class="wikitable"
Line 46: Line 47:


== The flavors of intervals ==
== The flavors of intervals ==
We first look at the pythagorean intervals:
{| class="wikitable"
{| class="wikitable"
|+Pythagorean (wa)
|+Pythagorean (wa)
Line 123: Line 125:
|wa 8ve
|wa 8ve
|}
|}
We then look at intervals of 5:
{| class="wikitable"
|+Classical (yo, gu)
!Cents
!Ratio
!FJS Name
!Color name
|-
|21.506
|81/80
|P1<sub>5</sub>
|gu 1sn
|-
|111.731
|16/15
|m2<sub>5</sub>
|gu 2nd
|-
|182.404
|10/9
|M2<sup>5</sup>
|yo 2nd
|-
|315.641
|6/5
|m3<sub>5</sub>
|gu 3rd
|-
|386.314
|5/4
|M3<sup>5</sup>
|yo 3rd
|-
|519.551
|27/20
|P4<sub>5</sub>
|gu 4th
|-
|590.224
|45/32
|A4<sup>5</sup>
|yo 4th
|-
|609.776
|64/45
|d5<sub>5</sub>
|gu 5th
|-
|680.449
|40/27
|P5<sup>5</sup>
|yo 5th
|-
|813.686
|8/5
|m6<sub>5</sub>
|gu 6th
|-
|884.359
|5/3
|M6<sup>5</sup>
|yo 6th
|-
|1017.596
|9/5
|m7<sub>5</sub>
|gu 7th
|-
|1088.269
|15/8
|M7<sup>5</sup>
|yo 7th
|-
|1178.494
|160/81
|P8<sup>5</sup>
|yo 8ve
|}
And intervals of 7.
{| class="wikitable"
|+Septimal (zo, ru)
!Cents
!Ratio
!FJS Name
!Color name
|-
|27.264
|64/63
|P1<sub>7</sub>
|ru 1sn
|-
|62.961
|28/27
|m2<sup>7</sup>
|zo 2nd
|-
|231.174
|8/7
|M2<sub>7</sub>
|ru 2nd
|-
|266.871
|7/6
|m3<sup>7</sup>
|zo 3rd
|-
|435.084
|9/7
|M3<sub>7</sub>
|ru 3rd
|-
|470.781
|21/16
|P4<sup>7</sup>
|zo 4th
|-
|561.006
|112/81
|d5<sup>7</sup>
|zo 5th
|-
|638.994
|81/56
|A4<sub>7</sub>
|ru 4th
|-
|729.219
|32/21
|P5<sub>7</sub>
|yo 5th
|-
|764.916
|14/9
|m6<sup>7</sup>
|gu 6th
|-
|933.129
|12/7
|M6<sub>7</sub>
|yo 6th
|-
|968.826
|7/4
|m7<sup>7</sup>
|gu 7th
|-
|1137.039
|27/14
|M7<sub>7</sub>
|yo 7th
|-
|1172.736
|63/32
|P8<sup>7</sup>
|yo 8ve
|}
== Draft section ==
Content in this section includes ideas and constructions that are not yet complete. This section will remain until the page is completed.
'''Drafts below this line'''
From what I see here it seems like I should add A1 and possibly also A5 to the tables.
Flavor 11: 352/243, 88/81, 44/27, 11/9, 11/6, 11/8, 33/32, (99/64, 297/256, 891/512, 2673/2048)
Flavor /11: 243/176, 81/44, 27/22, 18/11, 12/11, 16/11, 64/33, (128/99, ...)
Flavor 13: (104/81,) 52/27*, 13/9, 13/12, 13/8, 39/32, 117/64, 351/256, 1053/1024, (3159/2048)
Flavor /13: (81/52,) 27/26*, 18/13, 24/13, 16/13, 64/39, 128/117, 512/351, 2048/1053, (4096/3159)
Flavor 7/5: (<- Marvelous 45/28,) 15/14*, 7/5, 21/20, 63/40, 189/160, 567/320, 1701/1280, 5103/2560, ...
Flavor 5/7: (<- Marvelous 56/45,) 28/15*, 10/7, 40/21, 80/63, 320/189, 640/567, 2560/1701, 5120/5103, ...
Flavor 11/5: (176/135,) 88/45, 22/15, 11/10, 33/20, 99/80, 297/160, 891/640, (2673/2560, ...)
Flavor 13/5: 208/135, 52/45, 26/15, 13/10, 39/20, 117/80
Flavor 11/7:

Latest revision as of 05:06, 25 September 2025

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Author comments: Please complete interval tables for flavors of 7, 11, 13, 7/5, 11/5, 11/7, 13/5, 13/7, and 13/11, with all perfect, major, and minor intervals, as well as A4 and d5. Note that the flavors of 7/5 and 5/7 should be placed in the same table.

In this article, we will cover the various flavors of 13-limit intervals. We consider intervals that differ by a pythagorean interval to have the same flavor. The flavor of an interval depends on the primes higher than 3 in its prime factorization.

13-limit flavors
D\N 1/3/9 5 7 11 13
1/3/9 1/1 (wa) 5/4 (yo) 7/4 (zo) 11/8 (ilo) 13/8 (tho)
5 8/5 (gu) 1/1 7/5 (zogu) 11/10 (logu) 13/10 (thogu)
7 8/7 (ru) 10/7 (yoru) 1/1 11/7 (loru) 13/7 (thoru)
11 16/11 (lu) 20/11 (yolu) 14/11 (zolu) 1/1 13/11 (tholu)
13 16/13 (thu) 20/13 (yothu) 14/13 (zothu) 22/13 (lothu) 1/1

The flavors of intervals

We first look at the pythagorean intervals:

Pythagorean (wa)
Cents Ratio FJS Name Color name
0.000 1/1 P1 wa 1sn
90.225 256/243 m2 sawa 2nd
203.910 9/8 M2 wa 2nd
294.135 32/27 m3 wa 3rd
407.820 81/64 M3 lawa 3rd
498.045 4/3 P4 wa 4th
588.270 1024/729 d5 sawa 5th
611.730 729/512 A4 lawa 4th
701.955 3/2 P5 wa 5th
792.180 128/81 m6 sawa 6th
905.865 27/16 M6 wa 6th
996.090 16/9 m7 wa 7th
1109.775 243/128 M7 lawa 7th
1200.000 2/1 P8 wa 8ve

We then look at intervals of 5:

Classical (yo, gu)
Cents Ratio FJS Name Color name
21.506 81/80 P15 gu 1sn
111.731 16/15 m25 gu 2nd
182.404 10/9 M25 yo 2nd
315.641 6/5 m35 gu 3rd
386.314 5/4 M35 yo 3rd
519.551 27/20 P45 gu 4th
590.224 45/32 A45 yo 4th
609.776 64/45 d55 gu 5th
680.449 40/27 P55 yo 5th
813.686 8/5 m65 gu 6th
884.359 5/3 M65 yo 6th
1017.596 9/5 m75 gu 7th
1088.269 15/8 M75 yo 7th
1178.494 160/81 P85 yo 8ve

And intervals of 7.

Septimal (zo, ru)
Cents Ratio FJS Name Color name
27.264 64/63 P17 ru 1sn
62.961 28/27 m27 zo 2nd
231.174 8/7 M27 ru 2nd
266.871 7/6 m37 zo 3rd
435.084 9/7 M37 ru 3rd
470.781 21/16 P47 zo 4th
561.006 112/81 d57 zo 5th
638.994 81/56 A47 ru 4th
729.219 32/21 P57 yo 5th
764.916 14/9 m67 gu 6th
933.129 12/7 M67 yo 6th
968.826 7/4 m77 gu 7th
1137.039 27/14 M77 yo 7th
1172.736 63/32 P87 yo 8ve

Draft section

Content in this section includes ideas and constructions that are not yet complete. This section will remain until the page is completed.

Drafts below this line

From what I see here it seems like I should add A1 and possibly also A5 to the tables.

Flavor 11: 352/243, 88/81, 44/27, 11/9, 11/6, 11/8, 33/32, (99/64, 297/256, 891/512, 2673/2048)

Flavor /11: 243/176, 81/44, 27/22, 18/11, 12/11, 16/11, 64/33, (128/99, ...)

Flavor 13: (104/81,) 52/27*, 13/9, 13/12, 13/8, 39/32, 117/64, 351/256, 1053/1024, (3159/2048)

Flavor /13: (81/52,) 27/26*, 18/13, 24/13, 16/13, 64/39, 128/117, 512/351, 2048/1053, (4096/3159)

Flavor 7/5: (<- Marvelous 45/28,) 15/14*, 7/5, 21/20, 63/40, 189/160, 567/320, 1701/1280, 5103/2560, ...

Flavor 5/7: (<- Marvelous 56/45,) 28/15*, 10/7, 40/21, 80/63, 320/189, 640/567, 2560/1701, 5120/5103, ...

Flavor 11/5: (176/135,) 88/45, 22/15, 11/10, 33/20, 99/80, 297/160, 891/640, (2673/2560, ...)

Flavor 13/5: 208/135, 52/45, 26/15, 13/10, 39/20, 117/80

Flavor 11/7:

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