Interseptimal interval: Difference between revisions

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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]:
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]:


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


* 240¢-260¢ – '''semifourth''' – an interval of this size is around half the size of a perfect fourth.
* 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.
* 440¢–468¢ – '''semisixth''' – an interval of this size is around half the size of a major sixth.
** [[Zhea Erose]] calls the 440c-464c region '''naiadic''' (from ''naiad'', a kind of ancient Greek water spirit) and uses it frequently.
** The term '''naiadic''' (from ''naiad'', a kind of ancient Greek water spirit) refers to the 440c–464c region by [[Zhea Erose]], who uses it frequently.
* 732¢-760¢ – '''semitenth''' – an interval of this size is around half the size of a minor tenth (i. e., an octave plus a minor third). Another possible name is sesquifourth (since this is also about one and a half times the size of a perfect fourth).
* 732¢–760¢ – '''semitenth''' – an interval of this size is around half the size of a minor tenth (i. e., an octave plus a minor 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]]s have a semitwelfth of approximately 951 cents, analogous to the 600 cent tritone shared by all even edos.
** The term '''cocytic''' was proposed by [[Inthar]], and is carried on with by [[Flora Canou]]<ref>Flora Canou criticizes ''semisixth'' and ''semitenth'' as they fail to make clear whether the interval to be split is major or minor, and prefers ''naiadic'' and ''cocytic''.</ref>.
* 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]]s 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".
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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 center-1 right-2"
{| class="wikitable center-1 right-2"
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|}
|}


=== Maj3-4 – 440-468¢ ===
=== Maj3–4 – 440-468¢ ===


{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
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|}
|}


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


{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
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|}
|}


=== Maj6-min7 – 940-960¢ ===
=== Maj6–min7 – 940-960¢ ===


{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
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* [[Gentle region]]
* [[Gentle region]]
* [[Gallery of just intervals]]
* [[Gallery of just intervals]]
== Notes ==


[[Category:Interseptimal| ]] <!-- main article -->
[[Category:Interseptimal| ]] <!-- main article -->
[[Category:Interval category]]
[[Category:Interval category]]

Revision as of 19:09, 28 October 2021

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.
    • The term naiadic (from naiad, a kind of ancient Greek water spirit) refers to the 440c–464c region by Zhea Erose, who uses it frequently.
  • 732¢–760¢ – semitenth – an interval of this size is around half the size of a minor tenth (i. e., an octave plus a minor 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

Notes

  1. ↑ Flora Canou criticizes semisixth and semitenth as they fail to make clear whether the interval to be split is major or minor, and prefers naiadic and cocytic.
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