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 essay [http://www.bestii.com/%7Emschulter/IntervalSpectrumRegions.txt ''Regions of the Interval Spectrum'']:

* Maj2–min3 – intermediate between [[8/7]] and [[7/6]] – 240¢–260¢

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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.

** The term '''chthonic''' (from ''khthon'', an ancient Greek word referring to spirits of the underworld) refers to the 240-260¢ region by [[Zhea Erose]].<ref>as per [[Primodal Archive]]</ref>

** The term '''chthonic''' (from ''khthon'', an ancient Greek word referring to spirits of the underworld) refers to the 240-260¢ region by [[Zhea Erose]].<ref>As per [[Primodal Archive]].</ref>

* 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 440–464¢ 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).

** The term '''cocytic''' was proposed by [[Inthar]], by analogy with ''naiadic''.~~<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>~~

** The term '''cocytic''' was proposed by [[Inthar]], by analogy with ''naiadic''.

* 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 '''ouranic''' (by analogy with chthonic, and to match with the other terms) is proposed by [[User:Kaiveran|Kaiveran]].

~~This~~ makes notating these intervals very easy as long as we have an agreed-upon symbol for "semi". By analogy with the "semi" names, the tritone could also be called a semioctave, although the term tritone is so well-established (and so well represented by an unsplit 3-limit) that there seems 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]]).

One might want to use a mixture of above terms. [[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''. However, ''semifourth'' and ''semitwelfth'' are clear enough, so the Greek terms seems practically redundant.

The terminology makes notating these intervals very easy as long as we have an agreed-upon symbol for "semi". By analogy with the "semi" names, the tritone could also be called a semioctave, although the term tritone is so well-established (and so well represented by an unsplit 3-limit) that there seems 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]]).

=== Dual "semichromatic" names ===

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* ~11/8 or ~550¢ = ultrafourth, infratritone, infrasemioctave

* ~16/11 or ~650¢ = infrafifth, ultratritone, ultrasemioctave

=== "Inter" names ===

Both the "semi-nth" names and the Greek-derived names above are less intuitive than they could be and require some amount of memorization. For this reason, Inthar has proposed the following terms that explicitly name the diatonic interval categories that the interseptimals fall between:

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=== Within a pentatonic framework ===

A pentatonic framework, as elucidated in Kite Giedraitis's [http://www.tallkite.com/AlternativeTunings.html Alternative Tuning guide], is far more amenable to interseptimal intervals than the traditional Western heptatonic framework.

{| class="wikitable"

|+The pentatonic framework

! colspan="2" |names

! colspan="2" | names

!quality

! quality

!boundaries

! boundaries

! colspan="2" |heptatonic equivalent

! colspan="2" | heptatonic equivalent

|-

| rowspan="3" |1sn

| rowspan="3" | 1sn

| rowspan="3" |unison

| rowspan="3" | unison

|perfect

| perfect

|1/1 to 64/63

| 1/1 to 64/63

|perfect

| perfect

|1sn

| 1sn

|-

|half-augmented

| half-augmented

|(interseptimal)

| (interseptimal)

! colspan="2" |

|-

|augmented

| augmented

|28/27 to 16/15

| 28/27 to 16/15

|minor

| minor

| rowspan="3" |2nd

|-

! colspan="3" |

|(interpental)

| (interpental)

|neutral

| neutral

|-

| rowspan="3" |penta-2nd

| rowspan="3" | penta-2nd

| rowspan="3" |subthird

| rowspan="3" | subthird

|minor

| minor

|10/9 to 8/7

| 10/9 to 8/7

|major

| major

|-

|neutral

| neutral

|(interseptimal)

| (interseptimal)

! colspan="2" |

|-

|major

| major

|7/6 to 6/5

| 7/6 to 6/5

|minor

| minor

| rowspan="3" |3rd

| rowspan="3" | 3rd

|-

! colspan="3" |

|(interpental)

| (interpental)

|neutral

| neutral

|-

| rowspan="5" |penta-3rd

| rowspan="5" | penta-3rd

| rowspan="5" |fourthoid

| rowspan="5" | fourthoid

|diminished

| diminished

|5/4 to 9/7

| 5/4 to 9/7

|major

| major

|-

|half-diminished

| half-diminished

|(interseptimal)

| (interseptimal)

! colspan="2" |

|-

|perfect

| perfect

|21/16 to 27/20

| 21/16 to 27/20

|perfect

| perfect

| rowspan="3" |4th

| rowspan="3" | 4th

|-

|half-augmented

| half-augmented

|(interpental)

| (interpental)

|half-augmented

| half-augmented

|-

|augmented

| augmented

| rowspan="2" |7/5 to 10/7

|augmented

| augmented

|-

| rowspan="5" |penta-4th

| rowspan="5" | penta-4th

| rowspan="5" |fifthoid

| rowspan="5" | fifthoid

|diminished

| diminished

|diminished

| diminished

| rowspan="3" |5th

| rowspan="3" | 5th

|-

|half-diminished

| half-diminished

|(interpental)

| (interpental)

|half-diminished

| half-diminished

|-

|perfect

| perfect

|40/27 to 32/21

| 40/27 to 32/21

|perfect

| perfect

|-

|half-augmented

| half-augmented

|(interseptimal)

| (interseptimal)

! colspan="2" |

|-

|augmented

| augmented

|14/9 to 8/5

| 14/9 to 8/5

|minor

| minor

| rowspan="3" |6th

| rowspan="3" | 6th

|-

! colspan="3" |

|(interpental)

| (interpental)

|neutral

| neutral

|-

| rowspan="3" |penta-5th

| rowspan="3" |subseventh

|minor

| minor

|5/3 to 12/7

| 5/3 to 12/7

|major

| major

|-

|neutral

| neutral

|(interseptimal)

| (interseptimal)

! colspan="2" |

|-

|major

| major

|7/4 to 9/5

| 7/4 to 9/5

|minor

| minor

| rowspan="3" |7th

|-

! colspan="3" |

|(interpental)

| (interpental)

|neutral

| neutral

|-

| rowspan="3" |hexave

| rowspan="3" | hexave

| rowspan="3" |octoid

| rowspan="3" | octoid

|diminished

| diminished

|15/8 to 27/14

| 15/8 to 27/14

|major

| major

|-

|half-diminished

| half-diminished

|(interseptimal)

| (interseptimal)

! colspan="2" |

|-

|perfect

| perfect

|63/32 to 2/1

| 63/32 to 2/1

|perfect

| perfect

|8ve

| 8ve

|}

Note the two additional interseptimal regions. The boundary ratios are mostly either 81/80 or 64/63 away from a 3-limit interval. The exceptions are 7/5 and 10/7, which are only a [[5120/5103|Saruyo]] comma away from the 3-limit diminished 5th and augmented 4th respectively.

@@ Line 1: / Line 1: @@
-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 essay [http://www.bestii.com/%7Emschulter/IntervalSpectrumRegions.txt ''Regions of the Interval Spectrum'']:
 * Maj2–min3 – intermediate between [[8/7]] and [[7/6]] – 240¢–260¢
@@ Line 18: / Line 18: @@
 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.
-** The term '''chthonic''' (from ''khthon'', an ancient Greek word referring to spirits of the underworld) refers to the 240-260¢ region by [[Zhea Erose]].<ref>as per [[Primodal Archive]]</ref>
+** The term '''chthonic''' (from ''khthon'', an ancient Greek word referring to spirits of the underworld) refers to the 240-260¢ region by [[Zhea Erose]].<ref>As per [[Primodal Archive]].</ref>
 * 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 440–464¢ 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).
-** The term '''cocytic''' was proposed by [[Inthar]], by analogy with ''naiadic''.<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>
+** The term '''cocytic''' was proposed by [[Inthar]], by analogy with ''naiadic''.
 * 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 '''ouranic''' (by analogy with chthonic, and to match with the other terms) is proposed by [[User:Kaiveran|Kaiveran]].
-This makes notating these intervals very easy as long as we have an agreed-upon symbol for "semi". By analogy with the "semi" names, the tritone could also be called a semioctave, although the term tritone is so well-established (and so well represented by an unsplit 3-limit) that there seems 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]]).
+One might want to use a mixture of above terms. [[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''. However, ''semifourth'' and ''semitwelfth'' are clear enough, so the Greek terms seems practically redundant.
+The terminology makes notating these intervals very easy as long as we have an agreed-upon symbol for "semi". By analogy with the "semi" names, the tritone could also be called a semioctave, although the term tritone is so well-established (and so well represented by an unsplit 3-limit) that there seems 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]]).
 === Dual "semichromatic" names ===
@@ Line 46: / Line 48: @@
 * ~11/8 or ~550¢ = ultrafourth, infratritone, infrasemioctave
 * ~16/11 or ~650¢ = infrafifth, ultratritone, ultrasemioctave
 === "Inter" names ===
 Both the "semi-nth" names and the Greek-derived names above are less intuitive than they could be and require some amount of memorization. For this reason, Inthar has proposed the following terms that explicitly name the diatonic interval categories that the interseptimals fall between:
@@ Line 55: / Line 58: @@
 === Within a pentatonic framework ===
 A pentatonic framework, as elucidated in Kite Giedraitis's [http://www.tallkite.com/AlternativeTunings.html Alternative Tuning guide], is far more amenable to interseptimal intervals than the traditional Western heptatonic framework.
 {| class="wikitable"
 |+The pentatonic framework
-! colspan="2" |names
+! colspan="2" | names
-!quality
+! quality
-!boundaries
+! boundaries
-! colspan="2" |heptatonic equivalent
+! colspan="2" | heptatonic equivalent
 |-
-| rowspan="3" |1sn
+| rowspan="3" | 1sn
-| rowspan="3" |unison
+| rowspan="3" | unison
-|perfect
+| perfect
-|1/1 to 64/63
+| 1/1 to 64/63
-|perfect
+| perfect
-|1sn
+| 1sn
 |-
-|half-augmented
+| half-augmented
-|(interseptimal)
+| (interseptimal)
 ! colspan="2" |
 |-
-|augmented
+| augmented
-|28/27 to 16/15
+| 28/27 to 16/15
-|minor
+| minor
 | rowspan="3" |2nd
 |-
 ! colspan="3" |
-|(interpental)
+| (interpental)
-|neutral
+| neutral
 |-
-| rowspan="3" |penta-2nd
+| rowspan="3" | penta-2nd
-| rowspan="3" |subthird
+| rowspan="3" | subthird
-|minor
+| minor
-|10/9 to 8/7
+| 10/9 to 8/7
-|major
+| major
 |-
-|neutral
+| neutral
-|(interseptimal)
+| (interseptimal)
 ! colspan="2" |
 |-
-|major
+| major
-|7/6 to 6/5
+| 7/6 to 6/5
-|minor
+| minor
-| rowspan="3" |3rd
+| rowspan="3" | 3rd
 |-
 ! colspan="3" |
-|(interpental)
+| (interpental)
-|neutral
+| neutral
 |-
-| rowspan="5" |penta-3rd
+| rowspan="5" | penta-3rd
-| rowspan="5" |fourthoid
+| rowspan="5" | fourthoid
-|diminished
+| diminished
-|5/4 to 9/7
+| 5/4 to 9/7
-|major
+| major
 |-
-|half-diminished
+| half-diminished
-|(interseptimal)
+| (interseptimal)
 ! colspan="2" |
 |-
-|perfect
+| perfect
-|21/16 to 27/20
+| 21/16 to 27/20
-|perfect
+| perfect
-| rowspan="3" |4th
+| rowspan="3" | 4th
 |-
-|half-augmented
+| half-augmented
-|(interpental)
+| (interpental)
-|half-augmented
+| half-augmented
 |-
-|augmented
+| augmented
 | rowspan="2" |7/5 to 10/7
-|augmented
+| augmented
 |-
-| rowspan="5" |penta-4th
+| rowspan="5" | penta-4th
-| rowspan="5" |fifthoid
+| rowspan="5" | fifthoid
-|diminished
+| diminished
-|diminished
+| diminished
-| rowspan="3" |5th
+| rowspan="3" | 5th
 |-
-|half-diminished
+| half-diminished
-|(interpental)
+| (interpental)
-|half-diminished
+| half-diminished
 |-
-|perfect
+| perfect
-|40/27 to 32/21
+| 40/27 to 32/21
-|perfect
+| perfect
 |-
-|half-augmented
+| half-augmented
-|(interseptimal)
+| (interseptimal)
 ! colspan="2" |
 |-
-|augmented
+| augmented
-|14/9 to 8/5
+| 14/9 to 8/5
-|minor
+| minor
-| rowspan="3" |6th
+| rowspan="3" | 6th
 |-
 ! colspan="3" |
-|(interpental)
+| (interpental)
-|neutral
+| neutral
 |-
 | rowspan="3" |penta-5th
 | rowspan="3" |subseventh
-|minor
+| minor
-|5/3 to 12/7
+| 5/3 to 12/7
-|major
+| major
 |-
-|neutral
+| neutral
-|(interseptimal)
+| (interseptimal)
 ! colspan="2" |
 |-
-|major
+| major
-|7/4 to 9/5
+| 7/4 to 9/5
-|minor
+| minor
 | rowspan="3" |7th
 |-
 ! colspan="3" |
-|(interpental)
+| (interpental)
-|neutral
+| neutral
 |-
-| rowspan="3" |hexave
+| rowspan="3" | hexave
-| rowspan="3" |octoid
+| rowspan="3" | octoid
-|diminished
+| diminished
-|15/8 to 27/14
+| 15/8 to 27/14
-|major
+| major
 |-
-|half-diminished
+| half-diminished
-|(interseptimal)
+| (interseptimal)
 ! colspan="2" |
 |-
-|perfect
+| perfect
-|63/32 to 2/1
+| 63/32 to 2/1
-|perfect
+| perfect
-|8ve
+| 8ve
 |}
 Note the two additional interseptimal regions. The boundary ratios are mostly either 81/80 or 64/63 away from a 3-limit interval. The exceptions are 7/5 and 10/7, which are only a [[5120/5103|Saruyo]] comma away from the 3-limit diminished 5th and augmented 4th respectively.