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:

Since interseptimal intervals are typically well represented by any [[EDO]] or [[pergen]] that divides its approximate 3/1 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:

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

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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:
+Since interseptimal intervals are typically well represented by any [[EDO]] or [[pergen]] that divides its approximate 3/1 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:
 * semifourth/chthonic = semi-augmented second (+11/2), semi-diminished third (-13/2)