Interseptimal interval: Difference between revisions

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In the theory of [[Margo_Schulter|Margo Schulter]], ''interseptimal'' is a category of intervals which occupy regions intermediate between two septimal ratios such as [[8/7|8/7]] and [[7/6|7/6]], or [[12/7|12/7]] and [[7/4|7/4]]. There are four interseptimal 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]], ''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 [http://www.bestii.com/%7Emschulter/IntervalSpectrumRegions.txt Regions of the Interval Spectrum]:


<ol><li>Maj2-min3 -- intermediate between 8/7 and 7/6 -- 240¢-260¢</li><li>Maj3-4 -- intermediate between [[9/7|9/7]] and [[21/16|21/16]] -- 440¢-468¢</li><li>5-min6 -- intermediate between [[32/21|32/21]] and [[14/9|14/9]] -- 732¢-760¢</li><li>Maj6-min7 -- intermediate between 12/7 and 7/4 -- 940¢-960¢</li></ol>
* 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|24edo]] at 250¢, 450¢, 750¢ and 950¢. They also appear in [[19edo|19edo]] and [[29edo|29edo]].
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:
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:


<ol><li>240¢-260¢ -- semifourth -- an interval of this size is around half the size of a perfect fourth.</li><li>440¢-468¢ -- semisixth -- an interval of this size is around half the size of a major sixth.</li><li>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).</li><li>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 [[edt|edts]] have a semitwelfth of approximately 951 cents, analogous to the 600 cent tritone shared by all even edos.</li></ol>
* 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 [[edt|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".
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).
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==
== Examples ==
Some interseptimal intervals in all four ranges, both just and tempered, are listed below.
Some interseptimal intervals in all four ranges, both just and tempered, are listed below.


===Maj2-min3 - 240¢-260¢===
=== Maj2-min3 - 240¢-260¢ ===


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


===Maj3-4 - 440-468===
=== Maj3-4 - 440-468 ===


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


===5-min6 - 732¢-760¢===
=== 5-min6 - 732¢-760¢ ===


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


===Maj6-min7 - 940-960===
=== Maj6-min7 - 940-960 ===


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


See: [[interval_category|Interval Category]], [[Gallery_of_Just_Intervals|Gallery of Just Intervals]]     [[Category:interseptimal]]
== See also ==
[[Category:interval_category]]
* [[Gentle region]]
* [[Gallery of Just Intervals]]
 
[[Category:interseptimal| ]] <!-- main article -->
[[Category:interval category]]

Revision as of 21:48, 18 October 2018

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\5edo 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\29edo 248.276 -
5\24edo 250.000 -
52/45 250.304 13
37/32 251.344 37
81/70 252.680 7
4\19edo 252.632 -
22/19 253.805 19
29/25 256.950 29
3\14edo 257.143 -
297/256 257.183 11
36/31 258.874 31
5\23edo 260.870 -

Maj3-4 - 440-468

Interval Cents Value Prime Limit (if applicable)
5\88cET or 11\30edo 440.000 -
40/31 441.278 31
7\19edo 442.015 -
31/24 443.081 31
10\27edo 444.444 -
22/17 446.363 17
35/27 449.275 7
3\8edo 450.000 -
48/37 450.611 37
13/10 454.214 13
11\29edo 455.172 -
125/96 456.986 5
8\21edo 457.143 -
56/43 457.308 43
43/33 458.245 43
30/23 459.994 23
5\13edo 461.538 -
47/36 461.597 47
64/49 462.348 7
98/75 463.069 7
17/13 464.428 17
12\31edo 464.516 -
7\18edo 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\18edo 733.333 -
19\31edo 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\21edo 742.857 -
182/125 743.014 5
18\29edo 744.828 -
20/13 745.786 13
37/24 749.389 37
5\8edo 750.000 -
54/35 750.725 7
17/11 753.637 17
17\27edo 755.556 -
48/31 756.919 31
12\19edo 757.895 -
31/20 758.722 31
19\30edo 760.000 -

Maj6-min7 - 940-960

Interval Cents Value Prime Limit (if applicable)
18\23edo 939.130 -
31/18 941.126 31
512/297 942.817 11
11\14edo 942.857 -
50/29 943.050 29
19/11 946.195 19
140/81 947.320 7
15\19edo 947.368 -
64/37 948.656 37
45/26 949.696 13
19\24edo 950.000 -
23\29edo 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\5edo 960.000 -
256/147 960.393 7

See also

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