EDT: Difference between revisions
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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 when using timbres whose overtones consist of primarily or only odd harmonics such as clarinets, square waves, or triangle waves. While is not known whether odd harmonics actually facilitate the ability to hear in tritave-equivalence, it is known 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 when using timbres whose overtones consist of primarily or only odd harmonics such as clarinets, square waves, or triangle waves. While is not known whether odd harmonics actually facilitate the ability to hear in tritave-equivalence, it is known 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 [[Bohlen–Pierce scale]], most commonly consisting of 13 equal divisions of the tritave (although a justly-intoned version exists as well), seems to have been the | The [[Bohlen–Pierce 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. | ||
== Rank two temperaments == | == Rank two temperaments == | ||
{{Todo|cleanup|inline=1|text=Rewrite for clarity}} | {{Todo|cleanup|improve readability|inline=1|text=Rewrite for clarity}} | ||
If factors of two are eliminated, the search for consonant intervals begins with the odd harmonic series, 1:3:5:7:9:.... We can take the second tritave of the series, 3:5:7:9, and find within it the two [[isoharmonic]] triads 3:5:7 and 5:7:9; the analogy here is with the third octave of the full harmonic series, 4:5:6:7:8, and the isoharmonic triad 4:5:6, the foundation of triadic harmony in [[5-limit]] theory. Hence, 3:5:7 or 5:7:9 can be viewed as the fundamental consonant triad of no-twos music, and if we then apply the 5-limit analogy one more time, these triads are bounded by the intervals [[7/3]] or [[9/5]] respectively, either of them filling the role of the "fifth" in diatonicism. | If factors of two are eliminated, the search for consonant intervals begins with the odd harmonic series, 1:3:5:7:9:.... We can take the second tritave of the series, 3:5:7:9, and find within it the two [[isoharmonic]] triads 3:5:7 and 5:7:9; the analogy here is with the third octave of the full harmonic series, 4:5:6:7:8, and the isoharmonic triad 4:5:6, the foundation of triadic harmony in [[5-limit]] theory. Hence, 3:5:7 or 5:7:9 can be viewed as the fundamental consonant triad of no-twos music, and if we then apply the 5-limit analogy one more time, these triads are bounded by the intervals [[7/3]] or [[9/5]] respectively, either of them filling the role of the "fifth" in diatonicism. | ||
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; 300 and beyond | ; 300 and beyond | ||
* [[316edt]], [[372edt]], [[415edt]], [[499edt]], [[527edt]], [[613edt]], [[729edt]], [[800edt]], [[953edt]], [[1213edt]], [[1342edt]], [[3401edt]], [[6181edt]], [[27208edt]] | * [[314edt|314]], [[316edt|316]], [[336edt|336]], [[372edt|372]], [[415edt|415]], [[428edt|428]], [[499edt|499]], [[527edt|527]], [[613edt|613]], [[729edt|729]], [[800edt|800]], [[953edt|953]], [[1213edt|1213]], [[1342edt|1342]], [[3401edt|3401]], [[6181edt|6181]], [[27208edt|27208]] | ||
* A [[list of tritave reduced harmonics]] for easy comparison of JI and temperaments in tritave-based systems. | * A [[list of tritave reduced harmonics]] for easy comparison of JI and temperaments in tritave-based systems. | ||
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* [[Consistency levels of small EDTs]] | * [[Consistency levels of small EDTs]] | ||
* [[Relative errors of small EDTs]] | * [[Relative errors of small EDTs]] | ||
* [[ | * [[List of tritave reduced harmonics]] | ||
* [[List of no-twos chords in JI]] | * [[List of no-twos chords in JI]] | ||
* Heinz Bohlen's work: [http://www.huygens-fokker.org/bpsite/otherscales.html ''The Bohlen-Pierce Site: Other Unusual Scales''] | * Heinz Bohlen's work: [http://www.huygens-fokker.org/bpsite/otherscales.html ''The Bohlen-Pierce Site: Other Unusual Scales''] | ||
[[Category:Edt| ]] <!-- main article --> | [[Category:Edt| ]] <!-- main article --> | ||
[[Category:Tritave]] | [[Category:Tritave]] | ||
[[Category:Acronyms]] | [[Category:Acronyms]] | ||