Interseptimal interval: Difference between revisions

In the theory of [[Margo Schulter]], '''interseptimal''' is a category of [[interval region]]s for [[interval]]s 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¢

* 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 both [[5L 2s|diatonic]] and [[chromatic]] categories, they are inevitably xenharmonic.

** 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.

* 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.

* 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

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" | unison

| perfect

| perfect

| half-augmented

| (interseptimal)

! colspan="2" |

| augmented

| minor

| rowspan="3" |2nd

! colspan="3" |

| (interpental)

| neutral

| rowspan="3" | penta-2nd

| rowspan="3" | subthird

| minor

| major

| neutral

| (interseptimal)

! colspan="2" |

| major

| minor

! colspan="3" |

| (interpental)

| neutral

| rowspan="5" | penta-3rd

| rowspan="5" | fourthoid

| diminished

| major

| half-diminished

| (interseptimal)

! colspan="2" |

| perfect

| half-augmented

| (interpental)

| half-augmented

| augmented

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

| augmented

| rowspan="5" | penta-4th

| rowspan="5" | fifthoid

| diminished

| diminished

| half-diminished

| (interpental)

| half-diminished

| perfect

| perfect

| half-augmented

| (interseptimal)

! colspan="2" |

| augmented

| minor

! colspan="3" |

| (interpental)

| neutral

| rowspan="3" |penta-5th

| rowspan="3" |subseventh

| minor

| major

| neutral

| (interseptimal)

! colspan="2" |

| major

| minor

| rowspan="3" |7th

! colspan="3" |

| (interpental)

| neutral

| rowspan="3" | hexave

| rowspan="3" | octoid

| diminished

| major

| half-diminished

| (interseptimal)

! colspan="2" |

| perfect

| perfect

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]].

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

! Cents Value

! Prime Limit (if applicable)

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! Cents Value

! Prime Limit (if applicable)

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! Cents Value

! Prime Limit (if applicable)

| 5\[[Bohlen-Pierce]]

| -

| -

| -

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| -

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! Cents Value

! Prime Limit (if applicable)

| -

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