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.

== 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]], 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". 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

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 that the traditional Western heptatonic framework. In a pentatonic framework, there are 5 categories of intervals in the octave rather than 7, into which interseptimals fit much more cleanly:

* the subthird or penta-2nd (imperfect)

** <- semifourth/chthonic region

* the fourthoid or penta-3rd (perfect)

** <- semisixth/naiadic region

* the fifthoid or penta-4th (perfect)

** <- semitenth/cocytic region

* the subseventh or penta-5th (imperfect)

** <- semitwelfth/ouranic region

* the octoid or hexave (perfect, reducing to a unison)

Of especial interest are the chthonic and ouranic regions, being very close to intervals of [[5edo|5EDO]], and in central positions within the two imperfect interval categories. In this framework, they can easily function as "neutral/middle" intervals to which other intervals in their pentatonic category are compared. 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)

| -

| -

| -

| -

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=== Maj3–4 – 440-468¢ ===

 

! Cents Value

! Prime Limit (if applicable)

| -

| -

| -

| -

| -

| -

| -

| -

| -

=== 5–min6 – 732-760¢ ===

 

! Cents Value

! Prime Limit (if applicable)

| 5\[[Bohlen-Pierce]]

| -

| -

| -

| -

| -

| -

| -

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=== Maj6–min7 – 940-960¢ ===

 

! Cents Value

! Prime Limit (if applicable)

| -

| -

| -

| -

| -

| -

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

[[Category:Interval category]]