EDT: Difference between revisions

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'''EDT'''~~: Equal Division of~~ the ~~Tritave (~~3/1, 3rd harmonic~~, just perfect twelfth). Also sometimes written as ED3~~.

The '''equal division of the tritave''' or '''twelfth''' ('''EDT''') or '''3rd harmonic''' ('''ED3''') is a [[tuning]] obtained by dividing the [[3/1|3rd harmonic]] in a certain number of [[equal]] steps.

== Introduction ==

Western music generally revolves around the principle of ~~'''~~octave equivalence~~'''~~: notes an octave apart are often perceived in western music as being the same ''chroma'' but differing in pitch height. As the octave corresponds to a 2/1 frequency ratio, it has been proposed that the next-simplest after the octave, the 3/1, can also be used to evoke a sense of chroma equivalence. This interval corresponds to a perfect twelfth in the diatonic scale, but when used to refer to an equivalence interval it is often called the "[[tritave]]".

Western music generally revolves around the principle of [[octave equivalence]]: notes an octave apart are often perceived in western music as being the same ''chroma'' but differing in pitch height. As the octave corresponds to a 2/1 frequency ratio, it has been proposed that the next-simplest after the octave, the 3/1, can also be used to evoke a sense of chroma equivalence. This interval corresponds to a perfect twelfth in the diatonic scale, but when used to refer to an equivalence interval it is often called the "[[tritave]]".

It has been argued that pitches a tritave apart can never truly be heard as equivalent in all of the ways that octaves are, with some claiming that the [http://www.mmk.ei.tum.de/persons/ter/top/octequiv.html tonotopic representation of the mammalian auditory system] is inherently biased towards octave-equivalence. With proper context, experience, and training, however, at least some people find that they can experience some degree of tritave equivalence especially using timbres restricted to odd harmonics such as clarinets. It is not known whether odd harmonics actually facilitate the ability to hear in tritave-equivalence. Either way, it is certain that musically valuable organizations of pitch can arise through the equal division of non-octave intervals, regardless of whether the period is perceived as being truly chroma-equivalent, and as such the multitude of equal divisions of the tritave are rich and ripe for exploration.

It has been argued that pitches a tritave apart can never truly be heard as equivalent in all of the ways that octaves are, with some claiming that the [http://www.mmk.ei.tum.de/persons/ter/top/octequiv.html tonotopic representation of the mammalian auditory system]{{dead link}} is inherently biased towards octave-equivalence. With proper context, experience, and training, however, at least some people find that they can experience some degree of tritave equivalence especially using timbres restricted to odd harmonics such as clarinets. It is not known whether odd harmonics actually facilitate the ability to hear in tritave-equivalence. Either way, it is certain that musically valuable organizations of pitch can arise through the equal division of non-octave intervals, regardless of whether the period is perceived as being truly chroma-equivalent, and as such the multitude of equal divisions of the tritave are rich and ripe for exploration.

The [[BP|Bohlen-Pierce (BP) scale]], most commonly consisting of 13 equal divisions of the tritave (although a justly-intoned version exists as well), seems to have been the second such arrangement to be seriously studied and made into music, the first being the [[Obikhod]] pitch set of the Russian Orthodox Church which seems to have been by extension of the diatonic scale. The BP scale was independently discovered by Heinz Bohlen, John Pierce and Kees Van Prooijen. Bohlen found it while looking for triads with equal-difference tones, Prooijen uncovered it while searching for equally-tempered scales with accurate higher harmonics, and Pierce stumbled upon it trying to find consonant chords other than 4:5:6. Though they all started with different goals in mind, each of them amazingly ended up at the same destination.

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* A list of [[Tritave_Reduced_Harmonics|tritave reduced harmonics]] for easy comparison of JI and temperaments in tritave-based systems.

* Also may be found convenient: http://www.nonoctave.com/tuning/twelfth.html

* Also may be found convenient: [http://www.nonoctave.com/tuning/twelfth.html Nonoctave.com | Tuning | Equal Divisions of the Twelfth]

== EDT-EDO correspondences ==

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* [[Tritave Reduced Harmonics]]

* [[No-twos 31-limit]]

* Heinz Bohlen's work: http://www.huygens-fokker.org/bpsite/otherscales.html ~~[[Category~~:~~Edt|~~ ]~~] ~~

* Heinz Bohlen's work: [http://www.huygens-fokker.org/bpsite/otherscales.html ''The Bohlen-Pierce Site: Other Unusual Scales'']

[[Category:Edt| ]]

[[Category:Equal-step tuning]]

[[Category:Edonoi]]

@@ Line 4: / Line 4: @@
 |de = Edt
 }}
-'''EDT''': Equal Division of the Tritave (3/1, 3rd harmonic, just perfect twelfth).  Also sometimes written as ED3.
+The '''equal division of the tritave''' or '''twelfth''' ('''EDT''') or '''3rd harmonic''' ('''ED3''') is a [[tuning]] obtained by dividing the [[3/1|3rd harmonic]] in a certain number of [[equal]] steps.
 == Introduction ==
-Western music generally revolves around the principle of '''octave equivalence''': notes an octave apart are often perceived in western music as being the same ''chroma'' but differing in pitch height. As the octave corresponds to a 2/1 frequency ratio, it has been proposed that the next-simplest after the octave, the 3/1, can also be used to evoke a sense of chroma equivalence. This interval corresponds to a perfect twelfth in the diatonic scale, but when used to refer to an equivalence interval it is often called the "[[tritave]]".
+Western music generally revolves around the principle of [[octave equivalence]]: notes an octave apart are often perceived in western music as being the same ''chroma'' but differing in pitch height. As the octave corresponds to a 2/1 frequency ratio, it has been proposed that the next-simplest after the octave, the 3/1, can also be used to evoke a sense of chroma equivalence. This interval corresponds to a perfect twelfth in the diatonic scale, but when used to refer to an equivalence interval it is often called the "[[tritave]]".
-It has been argued that pitches a tritave apart can never truly be heard as equivalent in all of the ways that octaves are, with some claiming that the [http://www.mmk.ei.tum.de/persons/ter/top/octequiv.html tonotopic representation of the mammalian auditory system] is inherently biased towards octave-equivalence. With proper context, experience, and training, however, at least some people find that they can experience some degree of tritave equivalence especially using timbres restricted to odd harmonics such as clarinets. It is not known whether odd harmonics actually facilitate the ability to hear in tritave-equivalence. Either way, it is certain that musically valuable organizations of pitch can arise through the equal division of non-octave intervals, regardless of whether the period is perceived as being truly chroma-equivalent, and as such the multitude of equal divisions of the tritave are rich and ripe for exploration.
+It has been argued that pitches a tritave apart can never truly be heard as equivalent in all of the ways that octaves are, with some claiming that the [http://www.mmk.ei.tum.de/persons/ter/top/octequiv.html tonotopic representation of the mammalian auditory system]{{dead link}} is inherently biased towards octave-equivalence. With proper context, experience, and training, however, at least some people find that they can experience some degree of tritave equivalence especially using timbres restricted to odd harmonics such as clarinets. It is not known whether odd harmonics actually facilitate the ability to hear in tritave-equivalence. Either way, it is certain that musically valuable organizations of pitch can arise through the equal division of non-octave intervals, regardless of whether the period is perceived as being truly chroma-equivalent, and as such the multitude of equal divisions of the tritave are rich and ripe for exploration.
 The [[BP|Bohlen-Pierce (BP) scale]], most commonly consisting of 13 equal divisions of the tritave (although a justly-intoned version exists as well), seems to have been the second such arrangement to be seriously studied and made into music, the first being the [[Obikhod]] pitch set of the Russian Orthodox Church which seems to have been by extension of the diatonic scale. The BP scale was independently discovered by Heinz Bohlen, John Pierce and Kees Van Prooijen. Bohlen found it while looking for triads with equal-difference tones, Prooijen uncovered it while searching for equally-tempered scales with accurate higher harmonics, and Pierce stumbled upon it trying to find consonant chords other than 4:5:6. Though they all started with different goals in mind, each of them amazingly ended up at the same destination.
@@ Line 169: / Line 169: @@
 * A list of [[Tritave_Reduced_Harmonics|tritave reduced harmonics]] for easy comparison of JI and temperaments in tritave-based systems.
-* Also may be found convenient: http://www.nonoctave.com/tuning/twelfth.html
+* Also may be found convenient: [http://www.nonoctave.com/tuning/twelfth.html Nonoctave.com | Tuning | Equal Divisions of the Twelfth]
 == EDT-EDO correspondences ==
@@ Line 1,153: / Line 1,153: @@
 * [[Tritave Reduced Harmonics]]
 * [[No-twos 31-limit]]
-* Heinz Bohlen's work: http://www.huygens-fokker.org/bpsite/otherscales.html    [[Category:Edt| ]] <!-- main article -->
+* Heinz Bohlen's work: [http://www.huygens-fokker.org/bpsite/otherscales.html ''The Bohlen-Pierce Site: Other Unusual Scales'']
+[[Category:Edt| ]] <!-- main article -->
 [[Category:Equal-step tuning]]
 [[Category:Edonoi]]