EDT: Difference between revisions

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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. 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] 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 ~~first~~ such arrangement to be seriously studied and made into music. 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.

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.

== Rank two temperaments ==

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The named but not necessarily no twos rank two temperament which 13EDT "supports" is [[Sirius]], which takes a generator between ~7:6 and ~6:5. Like Arcturus, I speak advisedly of 13EDT supporting it because the most proper small MOS of it is triskaidecatonic. Unlike Arcturus, there is a smaller MOS of it than this which is technically proper. However, this MOS is the Grumpy heptatonic scale the use of which is made problematic by the uniqueness of the step of the second size. It is problematic to have the step of the second size be unique in a subscale of an edx because it creates a strong sense of a second equal division of a y strictly less than x, and this sense of two different equal divisions trying to happen in the same scale causes ordinary concepts of equivalence to break down in spectacular ways. If this "problem" has not been named yet, "cross-equivalence artifacting" would be a perfect name for it.

The final interval which 13EDT can reasonably use to generate a rank two temperament is its false 3/2 of 5 degrees. By a weird coincidence, it will generate the [[5L 3s]] unfair father octatonic scale just as if it were an interval of an edo, except that the scale will not always contain a false 4/3 as it must in an EDO. This means, most importantly, that 16/15 cannot be assumed to be a "comma" tempered out by this false Father temperament when it is taken as a temperament of full just intonation. By a second, and totally separate, weird coincidence, the well-known Bohlen-Pierce temperament is its index-2 subtemperament.

The final interval which 13EDT can reasonably use to generate a rank two temperament is its false 3/2 of 5 degrees. By a weird coincidence, it will generate the [[5L 3s (tritave-equivalent)|5L 3s]] unfair father octatonic scale just as if it were an interval of an edo, except that the scale will not always contain a false 4/3 as it must in an EDO. This means, most importantly, that 16/15 cannot be assumed to be a "comma" tempered out by this false Father temperament when it is taken as a temperament of full just intonation. By a second, and totally separate, weird coincidence, the well-known Bohlen-Pierce temperament is its index-2 subtemperament.

Due to the fact of its 9/7 generator, the temperament which is to BP what neutral temperaments are to syntonic temperaments does not become intelligibly a division of the tritave until extended to 17 tones whereas EDOs supporting various neutral temperaments have an "ordinary" heptatonic scale which is intelligibly a division of the octave. Additionally, 7 and 9 being consecutive odd numbers means that trying to force this temperament into a no-twos subgroup induces very poor "approximations" of less intelligible higher harmonics. To avoid this, this temperament should be assumed to be a temperment of the 3.5.7.8 subgroup tempering out 245/243 and 64/63, the familiar comma from EDOs supporting the Superpythagorean or Parapythagorean diatonic scale.

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 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. 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] 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 first such arrangement to be seriously studied and made into music. 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.
+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.
 == Rank two temperaments ==
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 The named but not necessarily no twos rank two temperament which 13EDT "supports" is [[Sirius]], which takes a generator between ~7:6 and ~6:5. Like Arcturus, I speak advisedly of 13EDT supporting it because the most proper small MOS of it is triskaidecatonic. Unlike Arcturus, there is a smaller MOS of it than this which is technically proper. However, this MOS is the Grumpy heptatonic scale the use of which is made problematic by the uniqueness of the step of the second size. It is problematic to have the step of the second size be unique in a subscale of an edx because it creates a strong sense of a second equal division of a y strictly less than x, and this sense of two different equal divisions trying to happen in the same scale causes ordinary concepts of equivalence to break down in spectacular ways. If this "problem" has not been named yet, "cross-equivalence artifacting" would be a perfect name for it.
-The final interval which 13EDT can reasonably use to generate a rank two temperament is its false 3/2 of 5 degrees. By a weird coincidence, it will generate the [[5L 3s]] unfair father octatonic scale just as if it were an interval of an edo, except that the scale will not always contain a false 4/3 as it must in an EDO. This means, most importantly, that 16/15 cannot be assumed to be a "comma" tempered out by this false Father temperament when it is taken as a temperament of full just intonation. By a second, and totally separate, weird coincidence, the well-known Bohlen-Pierce temperament is its index-2 subtemperament.
+The final interval which 13EDT can reasonably use to generate a rank two temperament is its false 3/2 of 5 degrees. By a weird coincidence, it will generate the [[5L 3s (tritave-equivalent)|5L 3s]] unfair father octatonic scale just as if it were an interval of an edo, except that the scale will not always contain a false 4/3 as it must in an EDO. This means, most importantly, that 16/15 cannot be assumed to be a "comma" tempered out by this false Father temperament when it is taken as a temperament of full just intonation. By a second, and totally separate, weird coincidence, the well-known Bohlen-Pierce temperament is its index-2 subtemperament.
 Due to the fact of its 9/7 generator, the temperament which is to BP what neutral temperaments are to syntonic temperaments does not become intelligibly a division of the tritave until extended to 17 tones whereas EDOs supporting various neutral temperaments have an "ordinary" heptatonic scale which is intelligibly a division of the octave. Additionally, 7 and 9 being consecutive odd numbers means that trying to force this temperament into a no-twos subgroup induces very poor "approximations" of less intelligible higher harmonics. To avoid this, this temperament should be assumed to be a temperment of the 3.5.7.8 subgroup tempering out 245/243 and 64/63, the familiar comma from EDOs supporting the Superpythagorean or Parapythagorean diatonic scale.