Equable heptatonic: Difference between revisions

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In the theory of [[Margo Schulter]], '''equable heptatonic''' is a category of intervals which occupy regions intermediate between [[11/10]] and [[10/9]], or [[9/5]] and [[20/11]]. There are two heartland regions given below, with approximate cents ranges from Schulter's article [http://www.bestii.com/%7Emschulter/IntervalSpectrumRegions.txt Regions of the Interval Spectrum]:
In the theory of [[Margo Schulter]], '''equable heptatonic''' is a category of intervals which occupy regions intermediate between [[11/10]] and [[10/9]], or [[9/5]] and [[20/11]]. There are two heartland regions given below, with approximate cents ranges from Schulter's article [http://www.bestii.com/%7Emschulter/IntervalSpectrumRegions.txt Regions of the Interval Spectrum]:


* Neut2–Maj2 – intermediate between [[11/10]] and [[10/9]] – 160¢–182¢
* Neut2–Maj2 – intermediate between [[11/10]] and [[10/9]] – 160–182{{c}} (~submajor second)
* min7–Neut7 – intermediate between [[9/5]] and [[20/11]] – 1018¢–1040¢
* min7–Neut7 – intermediate between [[9/5]] and [[20/11]] – 1018–1040{{c}} (~supraminor seventh)


Equable heptatonic intervals are well-represented in [[7edo]] at 171.429¢ (1＼7) and 1028.571¢ (6＼7). They also appear in [[27edo]], [[34edo]] and [[41edo]]. As they fall in ambiguous zones between simpler categories, they are inevitably xenharmonic.
Equable heptatonic intervals are well-represented in [[7edo]] at 171.429{{c}} (1\7) and 1028.571{{c}} (6\7). They also appear in [[27edo]], [[34edo]] and [[41edo]]. As they fall in ambiguous zones between simpler categories, they are inevitably xenharmonic.


== Examples ==
== Examples ==
Some equable heptatonic intervals in all two ranges, both just and tempered, are listed below.
Some equable heptatonic intervals in all two ranges, both just and tempered, are listed below.


=== Neut2–Maj2 ===
=== Neut2–Maj2 (submajor second) ===
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|-
! Interval
! Interval
! Cents Value
! Size<br />(cents)
! Prime Limit (if applicable)
! Prime limit<br />(if applicable)
|-
|-
| [[34/31]]
| [[34/31]]
Line 19: Line 20:
| 31
| 31
|-
|-
| 2＼[[15edo|15]]
| 2\[[15edo|15]]
| 160.000
| 160.000
| -
| —
|-
|-
| [[79/72]]
| [[79/72]]
Line 31: Line 32:
| 41
| 41
|-
|-
| 7＼[[52edo|52]]
| 7\[[52edo|52]]
| 161.538
| 161.538
| -
| —
|-
|-
| [[101/92]]
| [[101/92]]
Line 43: Line 44:
| 17
| 17
|-
|-
| 5＼[[37edo|37]]
| 5\[[37edo|37]]
| 162.162
| 162.162
| -
| —
|-
|-
| [[67/61]]
| [[67/61]]
Line 63: Line 64:
| 13
| 13
|-
|-
| 3＼[[22edo|22]]
| 3\[[22edo|22]]
| 163.636
| 163.636
| -
| —
|-
|-
| 7＼[[51edo|51]]
| 7\[[51edo|51]]
| 164.706
| 164.706
| -
| —
|-
|-
| [[11/10]]
| [[11/10]]
Line 75: Line 76:
| 11
| 11
|-
|-
| 4＼[[29edo|29]]
| 4\[[29edo|29]]
| 165.517
| 165.517
| -
| —
|-
|-
| 5＼[[36edo|36]]
| 5\[[36edo|36]]
| 166.667
| 166.667
| -
| —
|-
|-
| [[98/89]]
| [[98/89]]
Line 95: Line 96:
| 23
| 23
|-
|-
| 6＼[[43edo|43]]
| 6\[[43edo|43]]
| 167.442
| 167.442
| -
| —
|-
|-
| [[65/59]]
| [[65/59]]
Line 103: Line 104:
| 59
| 59
|-
|-
| 7＼[[50edo|50]]
| 7\[[50edo|50]]
| 168.000
| 168.000
| -
| —
|-
|-
| [[54/49]]
| [[54/49]]
Line 131: Line 132:
| 17
| 17
|-
|-
| 1＼[[7edo|7]]
| 1\[[7edo|7]]
| 171.429
| 171.429
| -
| —
|-
|-
| [[53/48]]
| [[53/48]]
Line 159: Line 160:
| 77
| 77
|-
|-
| 7＼[[48edo|48]]
| 7\[[48edo|48]]
| 175.000
| 175.000
| -
| —
|-
|-
| [[52/47]]
| [[52/47]]
Line 171: Line 172:
| 83
| 83
|-
|-
| 6＼[[41edo|41]]
| 6\[[41edo|41]]
| 175.610
| 175.610
| -
| —
|-
|-
| [[31/28]]
| [[31/28]]
Line 179: Line 180:
| 31
| 31
|-
|-
| 5＼[[34edo|34]]
| 5\[[34edo|34]]
| 176.471
| 176.471
| -
| —
|-
|-
| [[72/65]]
| [[72/65]]
Line 191: Line 192:
| 41
| 41
|-
|-
| 4＼[[27edo|27]]
| 4\[[27edo|27]]
| 177.778
| 177.778
| -
| —
|-
|-
| [[92/83]]
| [[92/83]]
Line 203: Line 204:
| 23
| 23
|-
|-
| 7＼[[47edo|47]]
| 7\[[47edo|47]]
| 178.723
| 178.723
| -
| —
|-
|-
| [[61/55]]
| [[61/55]]
Line 215: Line 216:
| 71
| 71
|-
|-
| 3＼[[20edo|20]]
| 3\[[20edo|20]]
| 180.000
| 180.000
| -
| —
|-
|-
| [[81/73]]
| [[81/73]]
Line 227: Line 228:
| 41
| 41
|-
|-
| 5＼[[33edo|33]]
| 5\[[33edo|33]]
| 181.818
| 181.818
| -
| —
|-
|-
| [[10/9]]
| [[10/9]]
Line 235: Line 236:
| 5
| 5
|-
|-
| 7＼[[46edo|46]]
| 7\[[46edo|46]]
| 182.609
| 182.609
| -
| —
|-
|-
| 2＼[[13edo|13]]
| 2\[[13edo|13]]
| 184.615
| 184.615
| -
| —
|}
|}


=== min7–Neut7 ===
=== min7–Neut7 (supraminor seventh) ===
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|-
! Interval
! Interval
! Cents Value
! Size<br />(cents)
! Prime Limit (if applicable)
! Prime limit<br />(if applicable)
|-
|-
| 11＼[[13edo|13]]
| 11\[[13edo|13]]
| 1015.385
| 1015.385
| -
| —
|-
|-
| 39＼[[46edo|46]]
| 39\[[46edo|46]]
| 1017.391
| 1017.391
| -
| —
|-
|-
| [[9/5]]
| [[9/5]]
Line 262: Line 264:
| 5
| 5
|-
|-
| 28＼[[33edo|33]]
| 28\[[33edo|33]]
| 1018.182
| 1018.182
| -
| —
|-
|-
| 17＼[[20edo|20]]
| 17\[[20edo|20]]
| 1020.000
| 1020.000
| -
| —
|-
|-
| 40＼[[47edo|47]]
| 40\[[47edo|47]]
| 1021.277
| 1021.277
| -
| —
|-
|-
| [[92/51]]
| [[92/51]]
Line 282: Line 284:
| 83
| 83
|-
|-
| 23＼[[27edo|27]]
| 23\[[27edo|27]]
| 1022.222
| 1022.222
| -
| —
|-
|-
| [[74/41]]
| [[74/41]]
Line 294: Line 296:
| 13
| 13
|-
|-
| 29＼[[34edo|34]]
| 29\[[34edo|34]]
| 1023.529
| 1023.529
| -
| —
|-
|-
| [[56/31]]
| [[56/31]]
Line 302: Line 304:
| 31
| 31
|-
|-
| 35＼[[41edo|41]]
| 35\[[41edo|41]]
| 1024.390
| 1024.390
| -
| —
|-
|-
| [[47/26]]
| [[47/26]]
Line 310: Line 312:
| 47
| 47
|-
|-
| 41＼[[48edo|48]]
| 41\[[48edo|48]]
| 1025.000
| 1025.000
| -
| —
|-
|-
| [[85/47]]
| [[85/47]]
Line 320: Line 322:
| [[38/21]]
| [[38/21]]
| 1026.732
| 1026.732
|  
| 19
|-
|-
| [[67/37]]
| [[67/37]]
Line 326: Line 328:
| 67
| 67
|-
|-
| 6＼[[7edo|7]]
| 6\[[7edo|7]]
| 1028.571
| 1028.571
| -
| —
|-
|-
| [[29/16]]
| [[29/16]]
| 1029.577
| 1029.577
| 19
| 29
|-
|-
| [[78/43]]
| [[78/43]]
Line 342: Line 344:
| 7
| 7
|-
|-
| 43＼[[50edo|50]]
| 43\[[50edo|50]]
| 1032.000
| 1032.000
| -
| —
|-
|-
| [[69/38]]
| [[69/38]]
Line 350: Line 352:
| 23
| 23
|-
|-
| 37＼[[43edo|43]]
| 37\[[43edo|43]]
| 1032.558
| 1032.558
| -
| —
|-
|-
| [[89/49]]
| [[89/49]]
Line 358: Line 360:
| 89
| 89
|-
|-
| 31＼[[36edo|36]]
| 31\[[36edo|36]]
| 1033.333
| 1033.333
| -
| —
|-
|-
| 25＼[[29edo|29]]
| 25\[[29edo|29]]
| 1034.483
| 1034.483
| -
| —
|-
|-
| [[20/11]]
| [[20/11]]
Line 370: Line 372:
| 11
| 11
|-
|-
| 44＼[[51edo|51]]
| 44\[[51edo|51]]
| 1035.294
| 1035.294
| -
| —
|-
|-
| 19＼[[22edo|22]]
| 19\[[22edo|22]]
| 1036.364
| 1036.364
| -
| —
|-
|-
| [[91/50]]
| [[91/50]]
| 1036.726
| 1036.726
|  
| 13
|-
|-
| [[71/39]]
| [[71/39]]
Line 386: Line 388:
| 71
| 71
|-
|-
| 32＼[[37edo|37]]
| 32\[[37edo|37]]
| 1037.838
| 1037.838
| -
| —
|-
|-
| [[51/28]]
| [[51/28]]
| 1038.085
| 1038.085
|  
| 17
|-
|-
| 45＼[[52edo|52]]
| 45\[[52edo|52]]
| 1038.462
| 1038.462
| -
| —
|-
|-
| [[82/45]]
| [[82/45]]
Line 402: Line 404:
| 41
| 41
|-
|-
| 13＼[[15edo|15]]
| 13\[[15edo|15]]
| 1040.000
| 1040.000
| -
| —
|-
|-
| [[31/17]]
| [[31/17]]
Line 412: Line 414:


== See also ==
== See also ==
* [[Interseptimal interval|Interseptimal regions]]
* [[Interseptimal interval]]
* [[Gentle region]]
* [[Gentle region]]
* [[Gallery of just intervals]]
* [[Gallery of just intervals]]


[[Category:Equable heptatonic]]
[[Category:Equable heptatonic| ]]
[[Category:Intervals]]
[[Category:Intervals]]
[[Category:Interval naming]]
[[Category:Interval naming]]

Latest revision as of 13:47, 5 March 2025

In the theory of Margo Schulter, equable heptatonic is a category of intervals which occupy regions intermediate between 11/10 and 10/9, or 9/5 and 20/11. There are two heartland regions given below, with approximate cents ranges from Schulter's article Regions of the Interval Spectrum:

  • Neut2–Maj2 – intermediate between 11/10 and 10/9 – 160–182 ¢ (~submajor second)
  • min7–Neut7 – intermediate between 9/5 and 20/11 – 1018–1040 ¢ (~supraminor seventh)

Equable heptatonic intervals are well-represented in 7edo at 171.429 ¢ (1\7) and 1028.571 ¢ (6\7). They also appear in 27edo, 34edo and 41edo. As they fall in ambiguous zones between simpler categories, they are inevitably xenharmonic.

Examples

Some equable heptatonic intervals in all two ranges, both just and tempered, are listed below.

Neut2–Maj2 (submajor second)

Interval Size
(cents) 		Prime limit
(if applicable)
34/31 159.920 31
2\15 160.000 —
79/72 160.627 79
45/41 161.161 41
7\52 161.538 —
101/92 161.579 101
56/51 161.915 17
5\37 162.162 —
67/61 162.422 67
78/71 162.786 71
89/81 163.060 89
100/91 163.274 13
3\22 163.636 —
7\51 164.706 —
11/10 165.004 11
4\29 165.517 —
5\36 166.667 —
98/89 166.772 89
87/79 166.995 79
76/69 167.284 23
6\43 167.442 —
65/59 167.670 59
7\50 168.000 —
54/49 168.213 7
97/88 168.577 97
43/39 169.035 43
75/68 169.627 17
32/29 170.423 29
85/77 171.125 17
1\7 171.429 —
53/48 171.550 53
74/67 172.037 67
95/86 172.309 43
21/19 173.268 19
94/85 174.237 47
73/66 174.517 77
7\48 175.000 —
52/47 175.021 47
83/75 175.465 83
6\41 175.610 —
31/28 176.210 31
5\34 176.471 —
72/65 177.069 13
41/37 177.718 41
4\27 177.778 —
92/83 178.227 83
51/46 178.636 23
7\47 178.723 —
61/55 179.253 61
71/64 179.697 71
3\20 180.000 —
81/73 180.031 73
91/82 180.291 41
5\33 181.818 —
10/9 182.404 5
7\46 182.609 —
2\13 184.615 —

min7–Neut7 (supraminor seventh)

Interval Size
(cents) 		Prime limit
(if applicable)
11\13 1015.385 —
39\46 1017.391 —
9/5 1017.596 5
28\33 1018.182 —
17\20 1020.000 —
40\47 1021.277 —
92/51 1021.364 23
83/46 1021.773 83
23\27 1022.222 —
74/41 1022.282 41
65/36 1022.931 13
29\34 1023.529 —
56/31 1023.790 31
35\41 1024.390 —
47/26 1024.979 47
41\48 1025.000 —
85/47 1025.763 47
38/21 1026.732 19
67/37 1027.963 67
6\7 1028.571 —
29/16 1029.577 29
78/43 1030.965 43
49/27 1031.787 7
43\50 1032.000 —
69/38 1032.716 23
37\43 1032.558 —
89/49 1033.228 89
31\36 1033.333 —
25\29 1034.483 —
20/11 1034.996 11
44\51 1035.294 —
19\22 1036.364 —
91/50 1036.726 13
71/39 1037.214 71
32\37 1037.838 —
51/28 1038.085 17
45\52 1038.462 —
82/45 1038.839 41
13\15 1040.000 —
31/17 1040.080 31

See also

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