Interseptimal interval: Difference between revisions

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¢

* 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. In other words, interseptimals are the [[interordinal]] intervals with respect to the diatonic [[mos]] [[5L 2s]].

A recent, more JI-agnostic synonym is '''interordinal'''.

== Categorical and Notational Approaches ==

** 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 '''naiadic''' (from ''naiad'', a kind of ancient Greek water spirit) refers to the 440–464¢ region by [[Zhea Erose]], who uses it frequently.

** 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>

* 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". 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]]).

* semifourth/chthonic = semi-augmented second (+11/2), semi-diminished third (-13/2)

* semisixth/naiadic = semi-diminished fourth (-9/2), semi-augmented third (+15/2)

* semitenth/cocytic = semi-augmented fifth (+9/2), semi-diminished sixth (-15/2)

* semitwelfth/ouranic = semi-diminished seventh (-11/2), semi-augmented sixth (+13/2)

While this does not give the interseptimals a single distinct ''notational'' name, it does reflect their ambiguity and flexibility with regards to the surrounding interval categories that many are so fond of. Furthermore, as both identities are exactly 12 notational fifths apart (i.e a direct analogue of the [[Pythagorean comma]]), composers can use a mechanism similar to the [[Color notation|"po and qu" of Color Notation]], or the plus and minus accidentals (+/-) proposed in [[Rational Comma Notation (RCN)|Rational Comma Notation]], to freely switch between the two identities.

Alternatively, one can use the ''ultra-'' prefix for sharpening by ~50¢ and ''infra-'' for flattening by ~50¢, analogous to ''super-'' and ''sub-'' for modifications by [[64/63]] (in a [[12edo]]-related context such as [[36edo]], 33¢).

* semifourth/chthonic = ultramajor second, inframinor third

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

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

These names and this notation are inspired by analogous names for [[interordinal]]s in other mosses, but they refer to interval regions rather than exact intervals implied by a concrete mos tuning.

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.

|+The pentatonic framework

! colspan="2" |names

!quality

!boundaries

! colspan="2" |heptatonic equivalent

| rowspan="3" |1sn

| rowspan="3" |unison

|perfect

|1/1 to 64/63

|perfect

|1sn

|half-augmented

|(interseptimal)

! colspan="2" |

|augmented

|28/27 to 16/15

|minor

| rowspan="3" |2nd

! colspan="3" |

|(interpental)

|neutral

| rowspan="3" |penta-2nd

| rowspan="3" |subthird

|minor

|10/9 to 8/7

|major

|neutral

|(interseptimal)

! colspan="2" |

|major

|7/6 to 6/5

|minor

| rowspan="3" |3rd

! colspan="3" |

|(interpental)

|neutral

| rowspan="5" |penta-3rd

| rowspan="5" |fourthoid

|diminished

|5/4 to 9/7

|major

|half-diminished

|(interseptimal)

! colspan="2" |

|perfect

|21/16 to 27/20

|perfect

| rowspan="3" |4th

|half-augmented

|(interpental)

|half-augmented

|augmented

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

|augmented

| rowspan="5" |penta-4th

| rowspan="5" |fifthoid

|diminished

|diminished

| rowspan="3" |5th

|half-diminished

|(interpental)

|half-diminished

|perfect

|40/27 to 32/21

|perfect

|half-augmented

|(interseptimal)

! colspan="2" |

|augmented

|14/9 to 8/5

|minor

| rowspan="3" |6th

! colspan="3" |

|(interpental)

|neutral

| rowspan="3" |penta-5th

| rowspan="3" |subseventh

|minor

|5/3 to 12/7

|major

|neutral

|(interseptimal)

! colspan="2" |

|major

|7/4 to 9/5

|minor

| rowspan="3" |7th

! colspan="3" |

|(interpental)

|neutral

| rowspan="3" |hexave

| rowspan="3" |octoid

|diminished

|15/8 to 27/14

|major

|half-diminished

|(interseptimal)

! colspan="2" |

|perfect

|63/32 to 2/1

|perfect

|8ve

Interseptimal intervals are now easily named. However there are now hard-to-name "interpental" intervals which would be neutral intervals in the heptatonic framework, containing such ratios as 12/11, 11/9, etc. This is because interseptimal intervals are the neutral intervals with respect to the parent [[mos]] [[2L 3s]] of the diatonic mos [[5L 2s]]. See [[Neutral and interordinal k-mossteps]] for a partial generalization of this behavior to other mosses.

So composing in a pentatonic framework may allow interseptimal intervals to play much more pivotal roles than usual.

=== Maj2–min3 – 240-260¢ ===

 

! Cents Value

! Prime Limit (if applicable)

| -

| -

| -

| -

| -

| -

=== Maj3–4 – 440-468¢ ===

 

! Cents Value

! Prime Limit (if applicable)

| -

| -

| -

| -

| -

| -

| -

| -

| -

=== 5–min6 – 732-760¢ ===

 

! Cents Value

! Prime Limit (if applicable)

| 5\[[Bohlen-Pierce]]

| -

| -

| -

| -

| -

| -

| -

| -

| -

=== Maj6–min7 – 940-960¢ ===

 

! Cents Value

! Prime Limit (if applicable)

| -

| -

| -

| -

| -

| -

[[Category:Interval category]]