Interseptimal interval: Difference between revisions

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m →Maj2-min3 - 240-260¢: links to interval pages
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→Examples: links to (potential) intervals pages (up to 31-limit) added
Line 24: Line 24:
=== Maj2-min3 - 240-260¢ ===
=== Maj2-min3 - 240-260¢ ===


{| class="wikitable"
{| class="wikitable center-1 right-2"
|-
! Interval
! Interval
! Cents Value
! Cents Value
Line 34: Line 33:
| 7
| 7
|-
|-
| 1\[[5edo]]
| 1\[[5edo|5]]
| 240.000
| 240.000
| -
| -
|-
|-
| [[54/47]]
| 54/47
| 240.358
| 240.358
| 47
| 47
Line 62: Line 61:
| 13
| 13
|-
|-
| 6\[[29edo]]
| 6\[[29edo|29]]
| 248.276
| 248.276
| -
| -
|-
|-
| 5\[[24edo]]
| 5\[[24edo|24]]
| 250.000
| 250.000
| -
| -
Line 74: Line 73:
| 13
| 13
|-
|-
| [[37/32]]
| 37/32
| 251.344
| 251.344
| 37
| 37
Line 82: Line 81:
| 7
| 7
|-
|-
| 4\[[19edo|19edo]]
| 4\[[19edo|19]]
| 252.632
| 252.632
| -
| -
Line 94: Line 93:
| 29
| 29
|-
|-
| 3\[[14edo]]
| 3\[[14edo|14]]
| 257.143
| 257.143
| -
| -
Line 106: Line 105:
| 31
| 31
|-
|-
| 5\[[23edo]]
| 5\[[23edo|23]]
| 260.870
| 260.870
| -
| -
Line 113: Line 112:
=== Maj3-4 - 440-468¢ ===
=== Maj3-4 - 440-468¢ ===


{| class="wikitable"
{| class="wikitable center-1 right-2"
|-
! Interval
! Interval
! Cents Value
! Cents Value
! Prime Limit (if applicable)
! Prime Limit (if applicable)
|-
|-
| 5\[[88cET]] or 11\[[30edo]]
| 5\[[88cET]] or 11\[[30edo|30]]
| 440.000
| 440.000
| -
| -
|-
|-
| 40/31
| [[40/31]]
| 441.278
| 441.278
| 31
| 31
|-
|-
| 7\[[19edo]]
| 7\[[19edo|19]]
| 442.015
| 442.015
| -
| -
|-
|-
| 31/24
| [[31/24]]
| 443.081
| 443.081
| 31
| 31
|-
|-
| 10\[[27edo]]
| 10\[[27edo|27]]
| 444.444
| 444.444
| -
| -
Line 147: Line 145:
| 7
| 7
|-
|-
| 3\[[8edo]]
| 3\[[8edo|8]]
| 450.000
| 450.000
| -
| -
Line 159: Line 157:
| 13
| 13
|-
|-
| 11\[[29edo]]
| 11\[[29edo|29]]
| 455.172
| 455.172
| -
| -
|-
|-
| 125/96
| [[125/96]]
| 456.986
| 456.986
| 5
| 5
|-
|-
| 8\[[21edo]]
| 8\[[21edo|21]]
| 457.143
| 457.143
| -
| -
Line 179: Line 177:
| 43
| 43
|-
|-
| 30/23
| [[30/23]]
| 459.994
| 459.994
| 23
| 23
|-
|-
| 5\[[13edo]]
| 5\[[13edo|13]]
| 461.538
| 461.538
| -
| -
Line 195: Line 193:
| 7
| 7
|-
|-
| 98/75
| [[98/75]]
| 463.069
| 463.069
| 7
| 7
Line 203: Line 201:
| 17
| 17
|-
|-
| 12\[[31edo]]
| 12\[[31edo|31]]
| 464.516
| 464.516
| -
| -
|-
|-
| 7\[[18edo]]
| 7\[[18edo|18]]
| 466.667
| 466.667
| -
| -
|-
|-
| 38/29
| [[38/29]]
| 467.936
| 467.936
| 29
| 29
Line 218: Line 216:
=== 5-min6 - 732-760¢ ===
=== 5-min6 - 732-760¢ ===


{| class="wikitable"
{| class="wikitable center-1 right-2"
|-
! Interval
! Interval
! Cents Value
! Cents Value
Line 232: Line 229:
| 29
| 29
|-
|-
| 11\[[18edo]]
| 11\[[18edo|18]]
| 733.333
| 733.333
| -
| -
|-
|-
| 19\[[31edo]]
| 19\[[31edo|31]]
| 735.484
| 735.484
| -
| -
Line 244: Line 241:
| 17
| 17
|-
|-
| 49/75
| [[49/75]]
| 736.931
| 736.931
| 7
| 7
Line 256: Line 253:
| 47
| 47
|-
|-
| 23/15
| [[23/15]]
| 740.006
| 740.006
| 23
| 23
Line 268: Line 265:
| 43
| 43
|-
|-
| 13\[[21edo]]
| 13\[[21edo|21]]
| 742.857
| 742.857
| -
| -
|-
|-
| 182/125
| [[182/125]]
| 743.014
| 743.014
| 5
| 5
|-
|-
| 18\[[29edo]]
| 18\[[29edo|29]]
| 744.828
| 744.828
| -
| -
Line 288: Line 285:
| 37
| 37
|-
|-
| 5\[[8edo]]
| 5\[[8edo|8]]
| 750.000
| 750.000
| -
| -
|-
|-
| 54/35
| [[54/35]]
| 750.725
| 750.725
| 7
| 7
Line 300: Line 297:
| 17
| 17
|-
|-
| 17\[[27edo]]
| 17\[[27edo|27]]
| 755.556
| 755.556
| -
| -
|-
|-
| 48/31
| [[48/31]]
| 756.919
| 756.919
| 31
| 31
|-
|-
| 12\[[19edo]]
| 12\[[19edo|19]]
| 757.895
| 757.895
| -
| -
|-
|-
| 31/20
| [[31/20]]
| 758.722
| 758.722
| 31
| 31
|-
|-
| 19\[[30edo]]
| 19\[[30edo|30]]
| 760.000
| 760.000
| -
| -
Line 323: Line 320:
=== Maj6-min7 - 940-960¢ ===
=== Maj6-min7 - 940-960¢ ===


{| class="wikitable"
{| class="wikitable center-1 right-2"
|-
! Interval
! Interval
! Cents Value
! Cents Value
! Prime Limit (if applicable)
! Prime Limit (if applicable)
|-
|-
| 18\[[23edo]]
| 18\[[23edo|23]]
| 939.130
| 939.130
| -
| -
|-
|-
| 31/18
| [[31/18]]
| 941.126
| 941.126
| 31
| 31
Line 341: Line 337:
| 11
| 11
|-
|-
| 11\[[14edo]]
| 11\[[14edo|14]]
| 942.857
| 942.857
| -
| -
|-
|-
| 50/29
| [[50/29]]
| 943.050
| 943.050
| 29
| 29
Line 353: Line 349:
| 19
| 19
|-
|-
| 140/81
| [[140/81]]
| 947.320
| 947.320
| 7
| 7
|-
|-
| 15\[[19edo]]
| 15\[[19edo|19]]
| 947.368
| 947.368
| -
| -
Line 365: Line 361:
| 37
| 37
|-
|-
| 45/26
| [[45/26]]
| 949.696
| 949.696
| 13
| 13
|-
|-
| 19\[[24edo]]
| 19\[[24edo|24]]
| 950.000
| 950.000
| -
| -
|-
|-
| 23\[[29edo]]
| 23\[[29edo|29]]
| 951.724
| 951.724
| -
| -
Line 381: Line 377:
| 13
| 13
|-
|-
| 125/72
| [[125/72]]
| 955.031
| 955.031
| 5
| 5
|-
|-
| 33/19
| [[33/19]]
| 955.760
| 955.760
| 19
| 19
Line 393: Line 389:
| 13
| 13
|-
|-
| 40/23
| [[40/23]]
| 958.039
| 958.039
| 23
| 23
Line 401: Line 397:
| 47
| 47
|-
|-
| 4\[[5edo]]
| 4\[[5edo|5]]
| 960.000
| 960.000
| -
| -

Revision as of 21:24, 10 June 2020

In the theory of Margo Schulter, interseptimal is a category of intervals which occupy regions intermediate between two septimal ratios such as 8/7 and 7/6, or 12/7 and 7/4. There are four interseptimal regions given below, with approximate cents ranges from Schulter's article Regions of the Interval Spectrum:

  • Maj2-min3 -- intermediate between 8/7 and 7/6 -- 240¢-260¢
  • Maj3-4 -- intermediate between 9/7 and 21/16 -- 440¢-468¢
  • 5-min6 -- intermediate between 32/21 and 14/9 -- 732¢-760¢
  • Maj6-min7 -- intermediate between 12/7 and 7/4 -- 940¢-960¢

Interseptimal intervals are well-represented in 24edo at 250¢, 450¢, 750¢ and 950¢. They also appear in 19edo and 29edo.

As they fall in ambiguous zones between simpler categories, they are inevitably xenharmonic. This also makes them difficult to name: do we classify a 250-cent interval as a second, a third, both, or neither? One option is to give each region a distinct name (analogous to using the word tritone rather than diminished fifth or augmented fourth). Possible names that could be used are:

  • 240¢-260¢ -- semifourth -- an interval of this size is around half the size of a perfect fourth.
  • 440¢-468¢ -- semisixth -- an interval of this size is around half the size of a major sixth.
  • 732¢-760¢ -- semitenth -- an interval of this size is around half the size of a major tenth (i. e., compound major third). Another possible name is sesquifourth (since this is also about one and a half times the size of a perfect fourth).
  • 940¢-960¢ -- semitwelfth -- an interval of this size is around half the size of a perfect twelfth (i e., a compound perfect fifth, or tritave). All even edts have a semitwelfth of approximately 951 cents, analogous to the 600 cent tritone shared by all even edos.

This makes notating these intervals very easy as long as we have an agreed-upon symbol for "semi".

By analogy the tritone could also be called a semioctave, although the term tritone is so well-established that seems is little reason to change it now. A key difference is that the tritone is intermediate between two septimal ratios separated by a jubilisma (50/49), whereas the other interseptimal ranges listed above are between two septimal ratios separated by a slendro diesis (49/48).

Examples

Some interseptimal intervals in all four ranges, both just and tempered, are listed below.

Maj2-min3 - 240-260¢

Interval Cents Value Prime Limit (if applicable)
147/128 239.607 7
1\5 240.000 -
54/47 240.358 47
23/20 241.961 23
1152/1001 243.238 13
38/33 244.240 19
144/125 244.969 5
15/13 247.741 13
6\29 248.276 -
5\24 250.000 -
52/45 250.304 13
37/32 251.344 37
81/70 252.680 7
4\19 252.632 -
22/19 253.805 19
29/25 256.950 29
3\14 257.143 -
297/256 257.183 11
36/31 258.874 31
5\23 260.870 -

Maj3-4 - 440-468¢

Interval Cents Value Prime Limit (if applicable)
5\88cET or 11\30 440.000 -
40/31 441.278 31
7\19 442.015 -
31/24 443.081 31
10\27 444.444 -
22/17 446.363 17
35/27 449.275 7
3\8 450.000 -
48/37 450.611 37
13/10 454.214 13
11\29 455.172 -
125/96 456.986 5
8\21 457.143 -
56/43 457.308 43
43/33 458.245 43
30/23 459.994 23
5\13 461.538 -
47/36 461.597 47
64/49 462.348 7
98/75 463.069 7
17/13 464.428 17
12\31 464.516 -
7\18 466.667 -
38/29 467.936 29

5-min6 - 732-760¢

Interval Cents Value Prime Limit (if applicable)
5\Bohlen-Pierce 731.521 -
29/19 732.064 29
11\18 733.333 -
19\31 735.484 -
26/17 735.572 17
49/75 736.931 7
49/32 737.652 7
72/47 738.403 47
23/15 740.006 23
66/43 741.755 43
43/28 742.692 43
13\21 742.857 -
182/125 743.014 5
18\29 744.828 -
20/13 745.786 13
37/24 749.389 37
5\8 750.000 -
54/35 750.725 7
17/11 753.637 17
17\27 755.556 -
48/31 756.919 31
12\19 757.895 -
31/20 758.722 31
19\30 760.000 -

Maj6-min7 - 940-960¢

Interval Cents Value Prime Limit (if applicable)
18\23 939.130 -
31/18 941.126 31
512/297 942.817 11
11\14 942.857 -
50/29 943.050 29
19/11 946.195 19
140/81 947.320 7
15\19 947.368 -
64/37 948.656 37
45/26 949.696 13
19\24 950.000 -
23\29 951.724 -
26/15 952.259 13
125/72 955.031 5
33/19 955.760 19
1001/576 956.762 13
40/23 958.039 23
47/27 959.642 47
4\5 960.000 -
256/147 960.393 7

See also

Retrieved from "https://en.xen.wiki/index.php?title=Interseptimal_interval&oldid=46420"