Interseptimal interval: Difference between revisions

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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]]).

=== Dual "semichromatic" names ===

Since interseptimal intervals are typically well represented by any [[EDO]] or [[pergen]] that divides an instance of the prime 3 into 2''n'' parts, another option is to repurpose [[24edo#Quartertone Accidentals|quartertone accidentals]] to represent them, which is easy as long as we define our "half-sharps" or "half-flats" to be precisely half of a chromatic semitone. With this in mind, we get the following twinned identities for our interseptimals, with the simplest ones (assuming a half-fifth genchain) listed first:

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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]] or ~33¢.

* semifourth/chthonic = ultramajor second, inframinor third

* semisixth/naiadic = ultramajor third, infrafourth

* semitenth/cocytic = ultrafifth, inframinor sixth

* semitwelfth/ouranic = ultramajor sixth, inframinor seventh

Ultra- and infra- also work for intervals that are very close to 11/8's and 16/11's:

* ~11/8 or 550¢ = ultrafourth

* ~16/11 or 650¢ = infrafifth

=== 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 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:

@@ Line 23: / Line 23: @@
 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]]).
 === Dual "semichromatic" names ===
 Since interseptimal intervals are typically well represented by any [[EDO]] or [[pergen]] that divides an instance of the prime 3 into 2''n'' parts, another option is to repurpose [[24edo#Quartertone Accidentals|quartertone accidentals]] to represent them, which is easy as long as we define our "half-sharps" or "half-flats" to be precisely half of a chromatic semitone. With this in mind, we get the following twinned identities for our interseptimals, with the simplest ones (assuming a half-fifth genchain) listed first:
@@ Line 34: / Line 33: @@
 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]] or ~33¢.
+* semifourth/chthonic =  ultramajor second, inframinor third
+* semisixth/naiadic = ultramajor third, infrafourth
+* semitenth/cocytic = ultrafifth, inframinor sixth
+* semitwelfth/ouranic = ultramajor sixth, inframinor seventh
+Ultra- and infra- also work for intervals that are very close to 11/8's and 16/11's:
+* ~11/8 or 550¢ = ultrafourth
+* ~16/11 or 650¢ = infrafifth
 === 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 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: