EDT: Difference between revisions

Line 29:

'''One should bear in mind that, assuming tritave equivalence, when determining which harmonics are represented, the ratios of 3 in the denominator are fungible instead of those of 2.''' For example making the fifth harmonic 5:3 a "major sixth" by conventional (and arbitrarily silly for the purposes of xenharmony, even with octaves) pitch class terminology.

There are other uses, or conceptualizations, of tritave-based tunings. Purely intuitive use of these myriad, assuredly xenharmonic structures comes to mind (see "EDO" versus "equal temperament"). Another intent might be to find or define temperaments (such as Magic, Hanson, etc.), or to provide exact formulae for stretching/compressing what would musically be used as an "ordinary" octave of ~2:1. (And given the stable nature of octave-based systems, some aesthetic overlap even in the most tritave-equivalent of music, would be forseeable.) For instance, the Bernhard-Stopper (19edt) temperament, might for instance be found useful in tuning pianoforti, being equivalent to 12edo, except for a 1.2c sharp octave which is relevant to inharmonicity.

There are other uses, or conceptualizations, of tritave-based tunings. Purely intuitive use of these myriad, assuredly xenharmonic structures comes to mind (see "EDO" versus "equal temperament"). Another intent might be to find or define temperaments (such as Magic, Hanson, etc.), or to provide exact formulae for stretching/compressing what would musically be used as an "ordinary" octave of ~2:1. (And given the stable nature of octave-based systems, some aesthetic overlap even in the most tritave-equivalent of music, would be forseeable.) For instance, the Bernhard-Stopper (19edt) temperament, might for instance be found useful in tuning pianoforti, being equivalent to 12edo, except for a 1.2 cents sharp octave which is relevant to inharmonicity.

Below is a large list of EDTs; additionally, some equal divisions of the tritave are known by alternate names or have special interest:

*3edt (Liese generator)

*[[3edt]] (Liese generator)

*[[4edt]] (Vulture generator)

*[[~~5edt|~~5edt]] (Tritave counterpart of Magic)

*[[5edt]] (Tritave counterpart of Magic)

*[[~~6edt|~~6edt]] (Tritave counterpart of Hanson)

*[[6edt]] (Tritave counterpart of Hanson)

*[[~~7edt|~~7edt]] (Tritave counterpart of Orwell)

*[[7edt]] (Tritave counterpart of Orwell)

*[[~~8edt|~~8edt]] (Tritave counterpart of Vulture)

*[[8edt]] (Tritave counterpart of Vulture)

*[[~~11edt|~~11edt]] "Euler Temperament"

*[[11edt]] "Euler Temperament"

*[[BP|"Bohlen-Pierce" or "BP"]]

*[[15edt]] (Mowgli generator)

*[[19ED3|"Bernhard Stopper"]]

*[[~~39edt|~~39edt]] Triple Bohlen-Pierce (Erlich)

*[[39edt]] Triple Bohlen-Pierce (Erlich)

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| | [[24edt]]

| | [[15edo]]

| | This is only a rough correspondence, as the (5n)~~edo~~ ~ (8n)~~edt~~ sequence begins to break down. 24edt is only consistent through the 6-integer-limit, with discrepancy for the 7th harmonic.

| | This is only a rough correspondence, as the (8n)edt ~ (5n)edo sequence begins to break down. 24edt is only consistent through the 6-integer-limit, with discrepancy for the 7th harmonic.

|-

| | [[25edt]]

@@ Line 29: / Line 29: @@
 '''One should bear in mind that, assuming tritave equivalence, when determining which harmonics are represented, the ratios of 3 in the denominator are fungible instead of those of 2.''' For example making the fifth harmonic 5:3 a "major sixth" by conventional (and arbitrarily silly for the purposes of xenharmony, even with octaves) pitch class terminology.
-There are other uses, or conceptualizations, of tritave-based tunings. Purely intuitive use of these myriad, assuredly xenharmonic structures comes to mind (see "EDO" versus "equal temperament"). Another intent might be to find or define temperaments (such as Magic, Hanson, etc.), or to provide exact formulae for stretching/compressing what would musically be used as an "ordinary" octave of ~2:1. (And given the stable nature of octave-based systems, some aesthetic overlap even in the most tritave-equivalent of music, would be forseeable.) For instance, the Bernhard-Stopper (19edt) temperament, might for instance be found useful in tuning pianoforti, being equivalent to 12edo, except for a 1.2c sharp octave which is relevant to inharmonicity.
+There are other uses, or conceptualizations, of tritave-based tunings. Purely intuitive use of these myriad, assuredly xenharmonic structures comes to mind (see "EDO" versus "equal temperament"). Another intent might be to find or define temperaments (such as Magic, Hanson, etc.), or to provide exact formulae for stretching/compressing what would musically be used as an "ordinary" octave of ~2:1. (And given the stable nature of octave-based systems, some aesthetic overlap even in the most tritave-equivalent of music, would be forseeable.) For instance, the Bernhard-Stopper (19edt) temperament, might for instance be found useful in tuning pianoforti, being equivalent to 12edo, except for a 1.2 cents sharp octave which is relevant to inharmonicity.
 Below is a large list of EDTs; additionally, some equal divisions of the tritave are known by alternate names or have special interest:
-*3edt (Liese generator)
+*[[3edt]] (Liese generator)
 *[[4edt]] (Vulture generator)
-*[[5edt|5edt]] (Tritave counterpart of Magic)
+*[[5edt]] (Tritave counterpart of Magic)
-*[[6edt|6edt]] (Tritave counterpart of Hanson)
+*[[6edt]] (Tritave counterpart of Hanson)
-*[[7edt|7edt]] (Tritave counterpart of Orwell)
+*[[7edt]] (Tritave counterpart of Orwell)
-*[[8edt|8edt]] (Tritave counterpart of Vulture)
+*[[8edt]] (Tritave counterpart of Vulture)
-*[[11edt|11edt]] "Euler Temperament"
+*[[11edt]] "Euler Temperament"
 *[[BP|"Bohlen-Pierce" or "BP"]]
+*[[15edt]] (Mowgli generator)
 *[[19ED3|"Bernhard Stopper"]]
-*[[39edt|39edt]] Triple Bohlen-Pierce (Erlich)
+*[[39edt]] Triple Bohlen-Pierce (Erlich)
@@ Line 473: / Line 474: @@
 | | [[24edt]]
 | | [[15edo]]
-| | This is only a rough correspondence, as the (5n)edo ~ (8n)edt sequence begins to break down. 24edt is only consistent through the 6-integer-limit, with discrepancy for the 7th harmonic.
+| | This is only a rough correspondence, as the (8n)edt ~ (5n)edo sequence begins to break down. 24edt is only consistent through the 6-integer-limit, with discrepancy for the 7th harmonic.
 |-
 | | [[25edt]]