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'''Dynamic tonality''' is a [[paradigm]] for tuning and timbre which generalizes the special relationship between [[just intonation]] and the [[Harmonic series (music)|harmonic series]] to apply to a wider set of pseudo-just [[Musical tuning#Tuning systems|tunings]] and related<ref name="Relating">{{Cite journal |
'''Dynamic tonality''' is a [[paradigm]] for tuning and timbre which generalizes the special relationship between [[just intonation]], and the [[Harmonic series (music)|harmonic series]] to apply to a wider set of pseudo-just [[Musical tuning#Tuning systems|tunings]] and related<ref name="Relating">{{Cite journal |
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|title=Relating Tuning and Timbre |
|title=Relating Tuning and Timbre |
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|journal=Experimental Musical Instruments |
|journal=Experimental Musical Instruments |
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|date=29 Aug 2008 |
|date=29 Aug 2008 |
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|s2cid=1549755 |
|s2cid=1549755 |
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|url=http://oro.open.ac.uk/ |
|url=http://oro.open.ac.uk/21505/2/33.2.sethares.pdf |
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}} [http://sethares.engr.wisc.edu/paperspdf/tuningcontinua.pdf Alt URL] |
}} [http://sethares.engr.wisc.edu/paperspdf/tuningcontinua.pdf Alt URL] |
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</ref> |
</ref> |
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The main limitation of |
The main limitation of dynamic tonality is that it is best used with compatible [[isomorphic keyboard]] instruments and compatible synthesizers, or with voices and instruments whose sounds are transformed in real time via compatible digital tools.<ref name="Spectral_Tools"> |
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{{cite journal |
{{cite journal |
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| first = William | last = Sethares |
| first = William | last = Sethares |
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| doi = 10.1162/comj.2009.33.2.71 |
| doi = 10.1162/comj.2009.33.2.71 |
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| s2cid = 216636537 |
| s2cid = 216636537 |
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| quote = "Smooth changes of tuning and timbre are at the core of ''C2ShiningC'' … found on the Spectral Tools home page." |
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| quote-page = 13 |
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}}</ref> |
}}</ref> |
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==The static timbre paradigm== |
==The static timbre paradigm== |
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===Harmonic timbres=== |
===Harmonic timbres=== |
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A vibrating string, a column |
A vibrating string, a column of air, and the human voice all emit a specific pattern of [[Harmonic|partials]] corresponding to the harmonic series. The degree of correspondence varies, depending on the physical characteristics of the emitter. "Partials" are also called [[Harmonic#Partials, overtones, and harmonics|"harmonics"]] or "overtones." Each musical instrument's unique sound is called its [[timbre]], so an instrument's timbre can be called a "harmonic timbre" if its partials correspond closely to the harmonic series. |
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===Just tunings=== |
===Just tunings=== |
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[[Just intonation]] is a system of tuning that adjusts a tuning's |
[[Just intonation]] is a system of tuning that adjusts a tuning's notes to maximize their alignment with a harmonic timbre's partials. This alignment maximizes the [[Consonance and dissonance|consonance]] of music's [[Tonality|tonal]] [[Interval (music)|intervals]]. |
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===Temperament=== |
===Temperament=== |
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The harmonic series and just intonation share an [[Infinity|infinitely]] |
The harmonic series and just intonation share an [[Infinity|infinitely]] complicated – or infinite [[rank of an abelian group|rank]] – pattern that is determined by the infinite series of [[prime number]]s. A [[musical temperament|temperament]] is an attempt to reduce this complexity by [[map (mathematics)|mapping]] this rank-[[infinity symbol|∞]] pattern to a simpler, finite-rank pattern. |
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Throughout history, the pattern of notes in a tuning could be altered (that is, "tempered") by humans but the pattern of partials sounded by an [[Acoustic music|acoustic]] [[musical instrument]] was largely determined by the physics of their sound production. The resulting misalignment between "pseudo-just" tempered tunings and untempered timbres made temperament "a battleground for the great minds of Western civilization. |
Throughout history, the pattern of notes in a tuning could be altered (that is, "tempered") by humans but the pattern of partials sounded by an [[Acoustic music|acoustic]] [[musical instrument]] was largely determined by the physics of their sound production. The resulting misalignment between "pseudo-just" tempered tunings, and untempered timbres, made temperament "a battleground for the great minds of Western civilization".<ref name=Isacoff> |
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{{ |
{{cite book |
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|last=Isacoff |first=Stuart |
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|title=Temperament: How Music Became a Battleground for the Great Minds of Western Civilization |
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|last=Isacoff |
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|first=Stuart |
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|publisher=Knopf |
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|year=2003 |
|year=2003 |
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|publisher=Knopf |
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|title=Temperament: How music became a battleground for the great minds of western civilization |
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|isbn=978-0375403552 |
|isbn=978-0375403552 |
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|url=https://books.google.com/books?id=P-sAAAAACAAJ}} |
|url=https://books.google.com/books?id=P-sAAAAACAAJ |
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}} |
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</ref><ref name=Barbour> |
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</ref><ref name="Barbour">Barbour, J.M., 2004, [https://books.google.com/books?id=G-pG77pmlp4C ''Tuning and Temperament: A Historical Survey'']</ref><ref name="Duffin">Duffin, R.W., 2006, [https://books.google.com/books?id=gQWyHQAACAAJ ''How Equal Temperament Ruined Harmony (and Why You Should Care)'']</ref> This misalignment, in any tuning that is not fully Just (and hence infinitely complex), is the defining characteristic of the Static Timbre Paradigm. |
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{{cite book |
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|last=Barbour |first=J.M. |
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|year=2004 |
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|title=Tuning and Temperament: A historical survey |
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|publisher=Courier Corporation |
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|isbn=978-0-486-43406-3 |
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|url=https://books.google.com/books?id=G-pG77pmlp4C |
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|via=Google books |
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}} |
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</ref><ref name=Duffin> |
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{{cite book |
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|last=Duffin |first=R.W. |
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|year=2006 |
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|title=How Equal Temperament Ruined Harmony (and Why you should care) |
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|publisher=W. W. Norton & Company |
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|isbn=978-0-393-06227-4 |
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|url=https://books.google.com/books?id=gQWyHQAACAAJ |
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|via=Google books |
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}} |
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</ref> |
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This misalignment, in any tuning that is not fully Just (and hence infinitely complex), is the defining characteristic of any ''static timbre'' paradigm. |
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===Instruments=== |
===Instruments=== |
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Many of the pseudo-just temperaments proposed during this "temperament battle" were rank |
Many of the pseudo-just temperaments proposed during this "temperament battle" were rank 2 (two-dimensional) – such as [[quarter-comma meantone]] – that provided more than 12 notes per octave. However, the [[musical keyboard|standard piano-like keyboard]] is only rank 1 (one-dimensional), [[affordance|affording]] at most 12 notes per octave. Piano-like keyboards affording more than 12 notes per octave were developed by [[Nicola Vicentino|Vicentino]],{{r|Isacoff|p=127}} Colonna,{{r|Isacoff|p=131}} [[Marin Mersenne|Mersenne]],{{r|Isacoff|p=181}} [[Christiaan Huygens|Huygens]],{{r|Isacoff|p=185}} and [[Isaac Newton|Newton]],{{r|Isacoff|p=196}} but were all considered too cumbersome / too difficult to play.{{r|Isacoff|p=18}} |
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==The dynamic tonality paradigm== |
==The dynamic tonality paradigm== |
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The goal of |
The goal of dynamic tonality is to enable [[consonance and dissonance|consonance]] beyond the range of tunings and temperaments in which harmonic timbres have traditionally been played. Dynamic tonality delivers consonance by tempering the intervals between notes (into "pseudo-just tunings") and also tempering the intervals between partials (into "pseudo-harmonic timbres") through digital synthesis and/or processing. Aligning the notes of a pseudo-just tuning's notes and the partials of a pseudo-harmonic timbre (or ''vice versa'') enables consonance. |
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The defining characteristic of |
The defining characteristic of dynamic tonality is that a given rank-2 temperament (as defined by a period {{mvar|α}}, a generator {{mvar|β}}, and a [[Comma (music)#Comma sequence|comma sequence]])<ref name=Fingering> |
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{{cite journal |
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|first1=A. |last1=Milne |
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|first2=W.A. |last2=Sethares |author2-link=William Sethares |
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|journal=Computer Music Journal |
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|first3=J. |last3=Plamondon |
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|first1=A.|last1=Milne |
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|first2=W.A. |last2=Sethares |
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|first3=J.|last3=Plamondon |
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|volume=31 |
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|issue=4 |
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|pages=15–32 |
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|doi=10.1162/comj.2007.31.4.15 |
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|date=Winter 2007 |
|date=Winter 2007 |
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|title=Isomorphic controllers and dynamic tuning: Invariant fingering over a tuning continuum |
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|journal=[[Computer Music Journal]] |
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|volume=31 |issue=4 |pages=15–32 |
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|doi=10.1162/comj.2007.31.4.15 |doi-access=free |
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|s2cid=27906745 |
|s2cid=27906745 |
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|url=http://oro.open.ac.uk/21503/1/comj.2007.31.4.15 |
|url=http://oro.open.ac.uk/21503/1/comj.2007.31.4.15 |
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}} |
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|doi-access=free |
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}} |
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</ref> is used to generate, ''in real time during performance'', the same set of intervals{{r|Continua}} among: |
</ref> is used to generate, ''in real time during performance'', the same set of intervals{{r|Continua}} among: |
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# A pseudo-just tuning's notes; |
# A pseudo-just tuning's notes; |
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Generating all three from the same temperament solves two problems and creates (at least) three opportunities. |
Generating all three from the same temperament solves two problems and creates (at least) three opportunities. |
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# Dynamic |
# Dynamic tonality solves the problem{{r|Isacoff}}{{r|Barbour}}{{r|Duffin}} of maximizing the consonance<ref name=TTSS> |
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{{Cite book |
{{Cite book |
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|last=Sethares |first=W.A. |author-link=William Sethares |
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|year=2004 |
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|title=Tuning, Timbre, Spectrum, Scale |
|title=Tuning, Timbre, Spectrum, Scale |
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|last=Sethares |
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|first=W.A. |
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|publisher=Springer |
|publisher=Springer |
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|year=2004 |
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|isbn=978-1852337971 |
|isbn=978-1852337971 |
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|url=https://books.google.com/books?id=KChoKKhjOb0C |
|url=https://books.google.com/books?id=KChoKKhjOb0C |
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|via=Google books |
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}} |
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</ref> of tempered tunings, and extends that solution across a wider range of tunings than were previously considered to be consonant.{{r|Fingering}}{{r|Continua}} |
</ref> of tempered tunings, and extends that solution across a wider range of tunings than were previously considered to be consonant.{{r|Fingering}}{{r|Continua}} |
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# Dynamic Tonality solves<ref>{{cite AV media |
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# Dynamic Tonality [https://www.youtube.com/watch?v=lWK1d9fzlVQ%5D solves] the "cumbersome" problem cited by Isacoff{{r|Isacoff|p=18,104,196}} by generating a keyboard that is (a) [[Isomorphic keyboard|isomorphic]] with its temperament{{r|Fingering}} (in every octave, key, and tuning), and yet is (b) tiny (the size of the keyboards on [[Chromatic button accordion|squeezeboxes]] such as [[concertina]]s, [[bandoneon]]s, and [[Bayan (accordion)|bayans]]). The creators of Dynamic Tonality could find no evidence that any of Isacoff's Great Minds knew about isomorphic keyboards or recognized the connection between the rank of a temperament and the dimensions of a keyboard (as described in Milne ''et al.'' 2007).{{r|Fingering}} |
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|people=Jim Plamondon (upload) |
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# Dynamic Tonality gives musicians the opportunity to explore new musical effects (see "[[Dynamic tonality#New musical effects|New musical effects]]," below). |
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|title=Motion sensing 1 |
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# Dynamic Tonality creates the opportunity for musicians to explore rank-2 temperaments other than the syntonic temperament (such as [[Schismatic temperament|schismatic]], [[magic temperament|Magic]], and [[George Secor#Miracle temperament|miracle]]) easily and with maximum consonance. |
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|medium=video |
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# Dynamic Tonality creates the opportunity for a significant increase in the efficiency of music education.<ref name="Sight_reading">{{Cite journal |
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|publisher=Thrumtronics |
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|title=Sight-reading music theory: A thought experiment on improving pedagogical efficiency |
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|url=https://www.youtube.com/watch?v=lWK1d9fzlVQ |
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|journal=Technical Report, Thumtronics Pty Ltd |
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|via=[[YouTube]] |
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|url=https://www.academia.edu/2654225 |
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|url-status=dead |
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|first1=Jim |last1=Plamondon |
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|access-date=2024-01-20 |
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|archive-date=2024-01-13 |
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|archive-url=https://web.archive.org/web/20240113193717/https://www.youtube.com/watch?v=lWK1d9fzlVQ&%3Bfeature=youtu.be |
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}} <!-- "archive.org" was unable to backup the video, only the page --></ref> the "cumbersome" problem cited by Isacoff{{r|Isacoff|p=18,104,196}} by generating a keyboard that is (a) [[Isomorphic keyboard|isomorphic]] with its temperament{{r|Fingering}} (in every octave, key, and tuning), and yet is (b) tiny (the size of the keyboards on [[Chromatic button accordion|squeezeboxes]] such as [[concertina]]s, [[bandoneon]]s, and [[Bayan (accordion)|bayans]]). The creators of dynamic tonality could find no evidence that any of Isacoff's Great Minds knew about isomorphic keyboards or recognized the connection between the rank of a temperament and the dimensions of a keyboard.{{r|Fingering}} |
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# Dynamic tonality gives musicians the opportunity to explore new musical effects (see "[[#New musical effects|New musical effects]]," below). |
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# Dynamic tonality creates the opportunity for musicians to explore rank-2 temperaments other than the syntonic temperament (such as [[Schismatic temperament|schismatic]], [[magic temperament|Magic]], and [[George Secor#Miracle temperament|miracle]]) easily and with maximum consonance. |
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# Dynamic tonality creates the opportunity for a significant increase in the efficiency of music education.<ref name=Sight_reading> |
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{{cite report |
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|first1=Jim |last1=Plamondon |
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|first2=Andrew J. |last2=Milne |
|first2=Andrew J. |last2=Milne |
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|first3=William |last3=Sethares |
|first3=William |last3=Sethares |author3-link=William Sethares |
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|date=2009 |
|date=2009 |
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|title=Sight-reading music theory: A thought experiment on improving pedagogical efficiency |
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|access-date=11 May 2020}} |
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|type=Technical Report |
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|publisher=Thumtronics Pty Ltd |
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|url=https://www.academia.edu/2654225 |
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|access-date=11 May 2020 |
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}} |
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</ref> |
</ref> |
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A rank-2 temperament defines a rank-2 ( |
A rank-2 temperament defines a rank-2 (two-dimensional) note space, as shown in video 1 (note space). |
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[[File:Note-space.webm|thumb|Video 1: |
[[File:Note-space.webm|thumb|Video 1: generating a rank-2 note space]] |
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The syntonic temperament is a rank-2 temperament defined by its period (just perfect octave, 1 |
The syntonic temperament is a rank-2 temperament defined by its period (just perfect octave, {{sfrac|1|2}}), its generator (just perfect fifth, {{sfrac|3|2}}) and its comma sequence (which starts with the syntonic comma, {{sfrac|81|80}}, which names the temperament). The construction of the syntonic temperament's note-space is shown in video 2 (Syntonic note-space). |
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[[File:Syntonic_space_%281080%29.webm|thumb|Video |
[[File:Syntonic_space_%281080%29.webm|thumb|Video 2: generating the syntonic temperament's note space]] |
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The valid tuning range of the syntonic temperament is show in Figure |
The valid tuning range of the syntonic temperament is show in Figure 1. |
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[[File:Rank-2_temperaments_with_the_generator_close_to_a_fifth_and_period_an_octave.jpg |thumb|Figure 1: The valid tuning range of the syntonic temperament, noting its valid tuning ranges at different {{mvar|p}}-limits and some notable tunings within those ranges.]] |
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A keyboard that is generated by a temperament is said to be [[Isomorphic keyboard|isomorphic]] with that temperament (from the Greek "iso" meaning "same," and "morph" meaning "shape"). Isomorphic keyboards are also known as [[generalized keyboard]]s. [[Isomorphic keyboards]] have the unique properties of transpositional invariance<ref> |
A keyboard that is generated by a temperament is said to be [[Isomorphic keyboard|isomorphic]] with that temperament (from the Greek "iso" meaning "same," and "morph" meaning "shape"). Isomorphic keyboards are also known as [[generalized keyboard]]s. [[Isomorphic keyboards]] have the unique properties of transpositional invariance<ref> |
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{{cite report |
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|last=Keislar |first=D. |
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|date=April 1988 |
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|title=History and Principles of Microtonal Keyboard Design |
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|id=Report No. STAN-M-45 |
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|series=Center for Computer Research in Music and Acoustics |
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|publisher=[[Stanford University]] |
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|place=Paolo Alto, CA |
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|url=http://ccrma.stanford.edu/STANM/stanms/stanm45/stanm45.pdf |
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|via=ccrma.stanford.edu |
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}} |
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</ref> |
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and tuning invariance{{r|Fingering|p=4}} when used with [[Rank of an abelian group|rank-2]] [[regular temperament|temperaments]] of [[just intonation]]. That is, such keyboards expose a given [[interval (music)|musical interval]] with "the same shape" in every octave of every key of every tuning of such a temperament. |
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Of the various isomorphic keyboards now known (e.g., the [[Robert Holford Macdowell Bosanquet|Bosanquet]], [[Jankó keyboard|Janko]], [[Adriaan Fokker|Fokker]], and [[Array system|Wesley]]), the [[Wicki-Hayden note layout|Wicki-Hayden]] keyboard is optimal for dynamic tonality across the entire valid 5-limit tuning range of the syntonic temperament.{{r|Continua|p=7-10}} The isomorphic keyboard shown in this article's videos is the Wicki-Hayden keyboard, for that reason. It also has symmetries related to [[Diatonic set theory|Diatonic Set Theory]], as shown in Video 3 (Same shape). |
Of the various isomorphic keyboards now known (e.g., the [[Robert Holford Macdowell Bosanquet|Bosanquet]], [[Jankó keyboard|Janko]], [[Adriaan Fokker|Fokker]], and [[Array system|Wesley]]), the [[Wicki-Hayden note layout|Wicki-Hayden]] keyboard is optimal for dynamic tonality across the entire valid 5-limit tuning range of the syntonic temperament.{{r|Continua|p=7-10}} The isomorphic keyboard shown in this article's videos is the Wicki-Hayden keyboard, for that reason. It also has symmetries related to [[Diatonic set theory|Diatonic Set Theory]], as shown in Video 3 (Same shape). |
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[[File:Same_shape_V2_%281080%29.webm|thumb|Video |
[[File:Same_shape_V2_%281080%29.webm|thumb|Video 3: Same shape in every octave, key, and tuning]] |
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The Wicki-Hayden keyboard embodies a [[tonnetz]], as shown in |
The Wicki-Hayden keyboard embodies a [[tonnetz]], as shown in video 4 (tonnetz). The tonnetz is a lattice diagram representing tonal space first described by [[Leonhard Euler|Euler]] (1739),<ref> |
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{{cite book |
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| last = Euler |
| last = Euler | first = Leonhard | author-link = Leonhard Euler |
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| first = Leonhard |
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|author-link=Leonhard Euler |
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| year = 1739 |
| year = 1739 |
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| |
| lang = la |
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| title = Tentamen novae theoriae musicae ex certissismis harmoniae principiis dilucide expositae |
| title = Tentamen novae theoriae musicae ex certissismis harmoniae principiis dilucide expositae |
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| pages = 147 |
| pages = 147 |
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| publisher = Saint Petersburg Academy |
| publisher = Saint Petersburg Academy |
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}} |
}} |
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</ref> |
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</ref> which is a central feature of [[Neo-Riemannian theory|Neo-Riemannian music theory]]. |
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which is a central feature of [[Neo-Riemannian theory|Neo-Riemannian music theory]]. |
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[[File:Tonnetz_(1080).webm|thumb|Video 4: The keyboard generated by the syntonic temperament embodies a tonnetz.]] |
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[[File:Tonnetz_(1080).webm|thumb|Video 4: the keyboard generated by the syntonic temperament embodies a tonnetz.]] |
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===Non-Western tunings=== |
===Non-Western tunings=== |
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The endpoints of the valid 5 |
The endpoints of the valid 5 limit tuning range of the syntonic temperament, shown in Figure 1, are: |
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* |
* Perfect 5 = 686 cents {{nobr|(7 {{sc|TET}}):}} The minor second is as wide as the major second, so the diatonic scale is a seven-note [[whole tone scale]]. This is the traditional tuning of the traditional Thai [[Ranat ek#Tuning|''ranat ek'']], in which the ''ranat's'' inharmonic timbre is maximally consonant.{{r|TTSS|p=303}} Other non-Western musical cultures are also [[Equal temperament#7-tone equal temperament|reported to tune their instruments in 7 TET]], including the Mandinka ''[[balafon]]''.<ref> |
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{{ |
{{cite book |
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|last=Jessup |first=L. |
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|title=The Mandinka Balafon: An Introduction with Notation for Teaching |
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|last=Jessup |
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|first=L. |
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|publisher=Xylo Publications |
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|year=1983 |
|year=1983 |
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|title=The Mandinka Balafon: An introduction with notation for teaching |
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}} |
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|publisher=Xylo Publications |
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}} |
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</ref> |
</ref> |
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* |
* Perfect 5 = 720 cents {{nobr|(5 {{sc|TET}}):}} The minor second has zero width, so the diatonic scale is a five-note [[whole tone scale]]. This is [[Equal temperament#5-tone and 9-tone equal temperament|arguably]] the [[Gamelan#Tuning|''slendro'']] scale of Java's ''[[gamelan]]'' orchestras, with which the ''gamelan's'' inharmonic timbres are maximally consonant.{{r|TTSS|p=73}} |
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===Dynamic timbres=== |
===Dynamic timbres=== |
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The partials of a pseudo-harmonic timbre are digitally mapped, as defined by a temperament, to specific notes of a pseudo |
The partials of a pseudo-harmonic timbre are digitally mapped, as defined by a temperament, to specific notes of a pseudo – just tuning. When the temperament's generator changes in width, the tuning of the temperament's notes changes, and the partials change along with those notes – yet their relative position remains invariant on the temperament-generated isomorphic keyboard. The frequencies of notes and partials change with the generator's width, but the relationships among the notes, partials, and note-controlling buttons remain the same: as defined by the temperament. The mapping of partials to the notes of the syntonic temperament is animated in video 5. |
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[[File:Mapping Partials 1080.webm|thumb|Video |
[[File:Mapping Partials 1080.webm|thumb|Video 5: Animates the mapping of partials to notes in accordance with the syntonic temperament.]] |
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===Dynamic tuning=== |
===Dynamic tuning=== |
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On an isomorphic keyboard, any given musical structure—a [[Musical scale|scale]], a [[ |
On an isomorphic keyboard, any given musical structure—a [[Musical scale|scale]], a [[chord (music)|chord]], a [[chord progression]], or an entire [[song]]—has exactly the same fingering in every tuning of a given temperament. This allows a performer to learn to play a song in one tuning of a given temperament and then to play it with exactly the same finger-movements, on exactly the same note-controlling buttons, in every other tuning of that temperament. See video 3 (Same shape). |
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For example, one could learn to play [[Rodgers and Hammerstein|Rodgers and Hammerstein's]] [[Do-Re-Mi]] in its original [[equal temperament|12 |
For example, one could learn to play [[Rodgers and Hammerstein|Rodgers and Hammerstein's]] "[[Do-Re-Mi]]" song in its original [[equal temperament|12 tone equal temperament]] {{nobr|(12 {{sc|tet}})}} and then play it with exactly the same finger-movements, on exactly the same note-controlling buttons, while smoothly changing the tuning in real time across the [[syntonic temperament]]'s tuning continuum. |
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The process of digitally tempering a pseudo-harmonic timbre's partials to align with a tempered pseudo-just tuning's notes is shown in |
The process of digitally tempering a pseudo-harmonic timbre's partials to align with a tempered pseudo-just tuning's notes is shown in video 6 (Dynamic tuning & timbre).{{r|Spectral_Tools}} |
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[[File:Dynamic-tuning-and-timbre.webm|thumb|Video |
[[File:Dynamic-tuning-and-timbre.webm|thumb|Video 6: Dynamic tuning & timbre.]] |
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===New musical effects=== |
===New musical effects=== |
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====Tuning-based effects==== |
====Tuning-based effects==== |
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Dynamic Tonality's novel tuning-based effects<ref name= |
Dynamic Tonality's novel tuning-based effects<ref name=Dynamic> |
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{{cite conference |
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|first1=Jim |last1=Plamondon |
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|title=Dynamic Tonality: Extending the Framework of Tonality into the 21st Century |
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|first2=Andrew J. |last2=Milne |
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|first3=William |last3=Sethares |author3-link=William Sethares |
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|year=2009 |
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|title=Dynamic tonality: Extending the framework of tonality into the 21st century |
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|conference=Proceedings of the Annual Conference of the South Central Chapter of the College Music Society |
|conference=Proceedings of the Annual Conference of the South Central Chapter of the College Music Society |
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|url=http://sethares.engr.wisc.edu/paperspdf/CMS2009.pdf |
|url=http://sethares.engr.wisc.edu/paperspdf/CMS2009.pdf |
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}} |
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|first1=Jim |last1=Plamondon |
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|first2=Andrew J. |last2=Milne |
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|first3=William |last3=Sethares |
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|date=2009}} |
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</ref> include: |
</ref> include: |
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* ''Polyphonic tuning bends'', in which the pitch of the tonic remains fixed while the pitches of all other notes change to reflect changes in the tuning, with notes that are close to the tonic in [[pitch space|tonal space]] changing pitch only slightly and those that are distant changing considerably; |
* ''Polyphonic tuning bends'', in which the pitch of the tonic remains fixed while the pitches of all other notes change to reflect changes in the tuning, with notes that are close to the tonic in [[pitch space|tonal space]] changing pitch only slightly and those that are distant changing considerably; |
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====Timbre-based effects==== |
====Timbre-based effects==== |
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The developers of dynamic tonality have invented novel vocabulary to describe the effects on timbre by raising or lowering the relative amplitude of partials.<ref name=X_System> |
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Dynamic Tonality's novel timbre effects<ref name="X_System">{{cite journal |author1=Milne, A. |author2=Sethares, W.|author3=Plamondon, J. | year = 2006 | title=X System |journal=Technical Report, Thumtronics Inc. |url=http://oro.open.ac.uk/21510/1/X_System.pdf |access-date=2020-05-02}} [[File:CC BY-SA icon.svg|50px]] The definitions of ''primeness'', ''conicality'', and ''richness'' were copied from this source, which is available under a [https://creativecommons.org/licenses/by-sa/3.0/ Creative Commons Attribution-ShareAlike 3.0 Unported] license and the [[Wikipedia:Text of the GNU Free Documentation License|GNU Free Documentation License]].</ref> include: |
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{{cite report |
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* ''Primeness'': Partials 2, 4, 8, 16, ..., 2<sup>n</sup> are factorised only by prime 2, and so these partials can be said to embody ''twoness''. Partials 3, 9, 27, ..., 3<sup>n</sup> are factorised only by prime 3, and so can be said to embody ''threeness''. Partials 5, 25, 125, ..., 5<sup>n</sup> are factorised only by prime 5, and so can be said to embody ''fiveness''. Other partials are factorised by two, or more, different primes. Partials 12 is factorised by both 2 and 3, and so embodies both twoness and threeness; partial 15 is factorised by 3 and 5, and so embodies both threeness and fiveness. ''Primeness'' empowers the musician to manipulate any given timbre such that its twoness, threeness, fiveness, ..., ''primeness'' can be enhanced or reduced. Adding a second comma to the syntonic temperament's comma sequence defines the 7th partial (see Video 5), thus similarly enabling ''sevenness''. |
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|first1=A. |last1=Milne |
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* ''Conicality'': Turning down twoness will lead to an odd-partial-only timbre – a “hollow or nasal” sound<ref name="Helmholtz1885"> |
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|first2=W. |last2=Sethares |author2-link=William Sethares |
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|first3=J. |last3=Plamondon |
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|year = 2006 |
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|title=X System |
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|type=Technical Report |
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|publisher = Thumtronics Inc. |
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|url=http://oro.open.ac.uk/21510/1/X_System.pdf |
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|access-date=2020-05-02 |
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}} [[File:CC BY-SA icon.svg|50px]] The descriptions of ''primeness'', ''conicality'', and ''richness'' were copied from this source, which is available under a [https://creativecommons.org/licenses/by-sa/3.0/ Creative Commons Attribution-ShareAlike 3.0 Unported] license and the [[Wikipedia:Text of the GNU Free Documentation License|GNU Free Documentation License]]. |
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</ref> |
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Their new terms include ''primeness'', ''conicality'', and ''richness'', with ''primeness'' being further subdivided into ''twoness'', ''threeness'', ''fiveness'' etc.: |
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; ''Primeness'': The overall term ''primeness'' refers to the level to which overtones or partials of the fundamental tone whose harmonic order is a multiple of some prime factor; for example: |
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:* The order of partials 2, 4, 8, 16, ..., 2{{sup|{{mvar|n}}}} (for {{mvar|n}} = 1, 2, 3 ...) only contain the [[prime number|prime]] [[Factor (arithmetic)|factor]] 2, so this particular set of partials is described as having ''twoness'', only. |
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:* The of partials numbered 3, 9, 27, ..., 3{{sup|{{mvar|n}}}} can only have their order divided evenly by the prime number 3, and so can be said to only demonstrate ''threeness''. |
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:* Partials of order 5, 25, 125, ..., 5{{sup|{{mvar|n}}}} can only be factored by prime 5, and so those are said to have ''fiveness''. |
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: Other partials' orders may be factorised by several primes: Partial 12 can be factored by both 2 and 3, and so shows both ''twoness'' and ''threeness''; partial 15 can be factored by both 3 and 5, and so shows both ''threeness'' and ''fiveness''. If yet another appropriately-sized comma is introduced into the syntonic temperament's sequence of commas and semitones it can provide for a [[seventh harmonic|7th order/ partial]] (see video 5), and thus enable ''sevenness''. |
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: Consideration of ''primeness'' of a sound is meant to enable a musician to thoughtfully manipulate a timber by enhancing or reducing its ''twoness'', ''threeness'', ''fiveness'', ..., ''primeness''. |
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; ''Conicality'': Specifically turning down ''twoness'' produces timbre whose partials are predominantly odd order – a “hollow or nasal” sound<ref name=Helmholtz1885> |
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{{cite book |
{{cite book |
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|last1=Helmholtz |first1=H. |author1-link=Hermann von Helmholtz |
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|url=https://archive.org/details/onsensationston00unkngoog |
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|last2=Ellis |first2=A.J. |author2-link=Alexander J. Ellis |
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|last=von Helmholtz |
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|first=Hermann |
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|year=1885 |
|year=1885 |
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|title=On the |
|title=On the SensationsofTone as a Physiological Basis for a TheoryofMusic |
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|edition= |
|edition=2nd English |
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|translator- |
|translator-first=A.J. |translator-last=Ellis |translator-link=Alexander J. Ellis |
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|location=London, UK |
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|translator-last1= Ellis |
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|location=London |
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|publisher=Longmans, Green, and Co. |
|publisher=Longmans, Green, and Co. |
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|page=[https://archive.org/details/onsensationston00unkngoog/page/n77 52] |
|page=[https://archive.org/details/onsensationston00unkngoog/page/n77 52] |
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|url=https://archive.org/details/onsensationston00unkngoog |
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|access-date=2020-05-13 }} |
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|access-date=2020-05-13 |via=archive.org |
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</ref> reminiscent of cylindrical closed bore instruments (''e.g.'' clarinet). As the twoness is turned up, the even partials are gradually introduced creating a sound more reminiscent of open cylindrical bore instruments (''e.g.'' flute, shakuhachi), or conical bore instruments (''e.g.'' bassoon, oboe, saxophone). This perceptual feature is called conicality. |
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}} <!-- Translator A.J. Ellis, himself a physicist, authored an extensive "Appendix XX" to the English translation that is extremely helpful for understanding microtonality and just intonation. --> |
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* ''Richness'': When richness is at minimum, only the fundamental sounds; as it is increased, the twoness gain is increased, then the threeness gain, then the fiveness gain, etc.. |
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</ref> reminiscent of cylindrical closed bore instruments (an [[vessel flute|ocarina]], for example, or a few types of [[organ pipe]]s). As the ''twoness'' increases, the even partials increase, creating a sound more reminiscent of open cylindrical bore instruments ([[concert flute]]s, for example, or [[shakuhachi]]), or conical bore instruments ([[bassoon]]s, [[oboe]]s, [[saxophone]]s). This perceptual feature is called ''conicality''. |
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; ''Richness'': The term ''richness'' is close to common use for describing sound; in this context, it means the extent to which a timbre's spectrum contains partials whose orders include many different prime factors: The more prime factors are present in the orders of a timber's loud partials, the more ''rich'' the sound is. When richness is at minimum, only the fundamental sound is present; as it is increased, the ''twoness'' is increased, then the ''threeness'', then the ''fiveness'', etc. |
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===Superset of static timbre paradigm=== |
===Superset of static timbre paradigm=== |
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One can use Dynamic Tonality to temper only the tuning of notes, without tempering timbres, thus embracing the Static Timbre Paradigm. |
One can use Dynamic Tonality to temper only the tuning of notes, without tempering timbres, thus embracing the Static Timbre Paradigm. |
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Similarly, using a synthesizer control such as the Tone Diamond,<ref> |
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Similarly, using a synthesizer control such as the Tone Diamond,<ref>Milne, A., [https://www.academia.edu/589140/The_Tone_Diamond ''The Tone Diamond''], Technical report, The MARCS Institute for Brain, Behaviour and Development, University of Western Sydney, April 2002.</ref> a musician can opt to maximize regularity, harmonicity, or consonance—or trade off among them in real time (with some of the jammer's 10 degrees of freedom mapped to the Tone Diamond's variables), with consistent fingering. This enables musicians to choose tunings that are regular or irregular, equal or non-equal, major-biased or minor-biased—and enables the musician to slide smoothly among these tuning options in real time, exploring the [[Affect (psychology)|emotional affect]] of each variation and the changes among them. |
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{{cite report |
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|last=Milne |first=A. |
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|date=April 2002 |
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|title=The Tone Diamond |
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|type=Technical report |
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|series=[[MARCS Institute for Brain, Behaviour, and Development]] |
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|publisher=[[University of Western Sydney]] |
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|url=https://www.academia.edu/589140 |
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|via=academia.edu |
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}} |
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</ref> |
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a musician can opt to maximize regularity, harmonicity, or consonance – or trade off among them in real time (with some of the jammer's 10 degrees of freedom mapped to the tone diamond's variables), with consistent fingering. This enables musicians to choose tunings that are regular or irregular, equal or non-equal, major-biased or minor-biased – and enables the musician to slide smoothly among these tuning options in real time, exploring the [[affect (psychology)|emotional affect]] of each variation and the changes among them. |
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===Compared to microtonality=== |
===Compared to microtonality=== |
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===Example: C2ShiningC=== |
===Example: C2ShiningC=== |
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An early example of |
An early example of dynamic tonality can be heard in the song "C2ShiningC".<ref> |
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{{cite AV media |
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|title=C2ShiningC |
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|medium=music recording |
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|people=[[William Sethares|W.A. Sethares]] (provider) |
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|series=personal academic website |
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|publisher=[[University of Wisconsin]] |
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|url=https://sethares.engr.wisc.edu/mp3s/C2ShiningC.mp3 |
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|via=wisc.edu |
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}} |
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</ref>{{r|Spectral_Tools}} |
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This sound example contains only one chord, [[Major chord| |
This sound example contains only one chord, [[Major chord|C{{sub|maj}}]], played throughout, yet a sense of [[Tension (music)|harmonic tension]] is imparted by a tuning progression and a timbre progression, as follows: |
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::{| style="text-align:center;vertical-align:center;" |
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<blockquote> |
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|- |
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| {{nobr| {{big|C}}{{sub|maj}} 19 {{sc|tet}} }} <br/> ''harmonic'' ||  {{math|⟶}}  |
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| {{nobr| {{big|C}}{{sub|maj}} 5 {{sc|tet}} }} <br/> ''harmonic'' ||  {{math|⟶}}  |
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| {{nobr| {{big|C}}{{sub|maj}} 19 {{sc|tet}} }} <br/> ''consonant'' ||  {{math|⟶}}  |
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| {{nobr| {{big|C}}{{sub|maj}} 5 {{sc|tet}} }} <br/> ''consonant'' |
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|} |
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* The ''timbre'' progresses from a harmonic timbre (with partials following the [[harmonic series (music)|harmonic series]]) to a 'pseudo-harmonic' timbre (with partials adjusted to align with the notes of the current tuning) and back again. |
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'''Cmaj 19-tet/harmonic -> Cmaj 5-tet/harmonic -> Cmaj 19-tet/consonant -> Cmaj 5-tet/consonant''' |
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* Twice as rapidly, the ''tuning'' progresses (via polyphonic tuning bends), within the [[syntonic temperament]], from an initial tuning in which the tempered [[perfect fifth]] (p5) is 695 [[musical cent|cents]] wide (19 tone equal temperament, {{nobr|19 {{sc|tet}})}} to a second tuning in which the p5 is 720 cents wide {{nobr|(5 {{sc|tet}}),}} and back again. |
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</blockquote> |
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As the tuning changes, the pitches of all notes except the [[tonic (music)|tonic]] change, and the widths of all [[interval (music)|intervals]] except the [[octave]] change; however, the relationships among the intervals (as defined by the [[syntonic temperament]]'s period, generator, and [[comma sequence]]) remain invariant (that is, constant; not varying) throughout. This invariance among a temperament's interval relationships is what makes invariant fingering (on an isomorphic keyboard) possible, even while the tuning is changing. In the [[syntonic temperament]], the tempered [[major third]] (M3) is as wide as four tempered [[perfect fifth]]s (p5‑s) minus two [[octaves]] – so the M3's width changes across the tuning progression |
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* The ''timbre'' progresses from a harmonic timbre (with partials following the [[harmonic series (music)|harmonic series]]) to a 'pseudo-harmonic' timbre (with partials adjusted to align with the notes of the current tuning) and back again. |
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* from 380 cents in {{nobr|19 {{sc|tet}}}} (p5 = 695 cents), where the C{{sub|maj}} triad's M3 is very close in width to its [[just intonation|just]] width of 386.3 cents, |
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* Twice as rapidly, the ''tuning'' progresses (via polyphonic tuning bends), within the [[syntonic temperament]], from an initial tuning in which the tempered perfect fifth (P5) is 695 cents wide (19-tone equal temperament, 19-tet) to a second tuning in which the P5 is 720 cents wide (5-tet), and back again. |
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The piece is recorded in (or transposed in the recording to) D major, despite its name. |
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As the tuning changes, the pitches of all notes except the [[Tonic (music)|tonic]] change, and the widths of all [[Interval (music)|intervals]] except the [[octave]] change; however, the relationships among the intervals (as defined by the [[syntonic temperament]]'s period, generator, and [[comma sequence]]) remain invariant (''i.e.'', consistent) throughout. This invariance among a temperament's interval relationships is what makes invariant fingering (on an isomorphic keyboard) possible, even while the tuning is changing. |
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In the [[syntonic temperament]], the tempered [[major third]] (M3) is as wide as four tempered [[perfect fifth]]s (P5's) minus two [[octaves]]—so the M3's width changes across the tuning progression |
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* |
* to 480 cents in {{nobr|5 {{sc|tet}}}} (p5 = 720 cents), where the C{{sub|maj}} triad's M3 is close in width to a slightly flat [[perfect fourth]] of 498 cents, making the [[major chord|C{{sub|maj}}]] chord sound rather like a {{nobr|[[suspended chord|C{{sup|sus 4}}]].}} |
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Thus, the tuning progression's widening of the C{{sub|maj}}'s M3 from a nearly just [[major third]] in {{nobr|19 {{sc|tet}}}} to a slightly flat [[perfect fourth]] in {{nobr|5 {{sc|tet}}}} creates the [[tension (music)|harmonic tension]] of a {{nobr|[[suspended chord|C{{sup|sus 4}}]]}} within a [[major chord|C{{sub|maj}}]] chord, which is relieved by the return to {{nobr|19 {{sc|tet}}}}. This example proves that dynamic tonality offers new means of creating and then releasing [[Tension (music)|harmonic tension]], ''even within a single chord''. |
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* to 480 cents in 5-tet (P5 = 720), where the Cmaj triad's M3 is close in width to a slightly flat [[perfect fourth]] of 498 cents, making the [[Major chord|Cmaj]] chord sound rather like a [[Suspended chord|Csus4]]. |
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Thus, the tuning progression's widening of the Cmaj's M3 from a nearly-just [[major third]] in 19-tet to a slightly flat [[perfect fourth]] in 5-tet creates [[Tension (music)|harmonic tension]], which is relieved by the return to 19-tet. |
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This analysis is presented in [[Major chord|C{{sub|maj}}]] as originally intended, despite the recording actually being in [[Major chord|D{{sub|maj}}]]. |
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This is an example of Dynamic Tonality's ability to expand the frontiers of tonality by offering new means of creating tension and release, ''even within a single chord''. |
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===History=== |
===History=== |
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Dynamic |
Dynamic tonality was developed primarily by a collaboration between [[William Sethares]], Andrew Milne, and James ("Jim") Plamondon. |
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[[File:Thummer_prototype.png|thumb|A prototype of the Thummer |
[[File:Thummer_prototype.png|thumb|A prototype of the Thummer]] |
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The latter formed Thumtronics Pty Ltd to develop an expressive, tiny, electronic Wicki-Hayden keyboard instrument: Thumtronics' "Thummer."<ref> |
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The latter formed Thumtronics Pty Ltd. to develop an expressive, tiny, electronic Wicki-Hayden keyboard instrument: Thumtronics' "Thummer."<ref> |
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{{cite news |
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{{cite news |
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|last1=Jurgensen |
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|last = Jurgensen |first = John |
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|first1=John |
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|date=7 December 2007 |
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|title=The Soul of a New Instrument |
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|title=The soul of a new instrument |
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|url=https://www.wsj.com/articles/SB119698832376116538 |
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|newspaper = [[The Wall Street Journal]] |
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|access-date=26 July 2021 |
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|url=https://www.wsj.com/articles/SB119698832376116538 |
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|publisher=Wall Street Journal |
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|access-date=26 July 2021 |
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|date=7 December 2007}}</ref><ref> |
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}} |
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{{cite magazine |
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|last1=Beschizza |
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|first1=Rob |
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|title=The Thummer: A Musical Instrument for the 21st Century? |
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|url=https://www.wired.com/2007/01/the-thummer-a-m/ |
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|magazine=Wired |
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|access-date=26 July 2021 |
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|date=1 March 2007}} |
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</ref><ref> |
</ref><ref> |
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{{cite magazine |
{{cite magazine |
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|last = Beschizza |first = Rob |
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|last1=Van Buskirk |
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|date=March 2007 |
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|first1=Eliot |
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|title=Thummer musical instrument |
|title=The ''Thummer'': A musical instrument for the 21st century? |
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|magazine=[[Wired (magazine)|Wired]] |
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|url=https://www.wired.com/2007/09/thummer-musical/ |
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|url=https://www.wired.com/2007/01/the-thummer-a-m/ |
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|magazine=Wired |
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|access-date=26 July 2021 |
|access-date=26 July 2021 |
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}} |
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|date=25 September 2007}} |
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</ref><ref> |
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{{cite magazine |
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|last = van Buskirk |first = Eliot |
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|date=25 September 2007 |
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|title=Thummer musical instrument combines buttons, Wii-style motion detection |
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|magazine=[[Wired (magazine)|Wired]] |
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|url=https://www.wired.com/2007/09/thummer-musical/ |
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|access-date=26 July 2021 |
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}} |
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</ref><ref> |
</ref><ref> |
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{{cite web |
{{cite web |
||
| |
|last = Merrett |first = Andy |
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|date=26 September 2007 |
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|first1=Andy |
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|title=Thummer: |
|title=''Thummer'': New concept musical instrument based on QWERTY keyboard and motion detection |
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|website= Tech Digest |
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|url=https://www.techdigest.tv/2007/09/thummer_new_con.html |
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|access-date=26 July 2021 |
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|website=Tech Digest |
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}} |
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|access-date=26 July 2021 |
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|date=26 September 2007}} |
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</ref><ref> |
</ref><ref> |
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{{cite web |
{{cite web |
||
| |
|last = Strauss |first = Paul |
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|date=25 September 2007 |
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|first1=Paul |
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|title=Thummer: This |
|title=''Thummer'': This synthesizerisall about expression |
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|website=TechnaBob |
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|url=https://technabob.com/blog/2007/09/25/thummer-this-synthesizer-is-all-about-expression/ |
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|url=https://technabob.com/blog/2007/09/25/thummer-this-synthesizer-is-all-about-expression/ |
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|website=TechnaBob |
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|access-date=26 July 2021 |
|access-date=26 July 2021 |
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}} |
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|date=25 September 2007}} |
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</ref> The generic name for a Thummer-like instrument is "jammer." With [[ |
</ref> The generic name for a Thummer-like instrument is "jammer." With [[analog stick#Dual analog sticks|two thumb-sticks]] and internal motion sensors, a jammer would [[affordance#As perceived action possibilities|afford]] 10 [[degrees of freedom (mechanics)|degrees of freedom]], which would make it the most expressive polyphonic instrument available. Without the [[expressive potential (electronic music)|expressive potential]] of a jammer, musicians lack the expressive power needed to exploit ''dynamic tonality'' in real time, so dynamic tonality's new tonal frontiers remain largely unexplored. |
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{{-}} |
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==References== |
==References== |
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{{reflist|25em}} |
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<references/> |
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== External links == |
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* [https://www.dynamictonality.com/spectools.htm Spectral Tools home page], with examples of and tools for creating music that takes advantage of dynamic tonality |
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[[Category:Musical temperaments]] |
[[Category:Musical temperaments]] |
![]() |
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Dynamic tonality is a paradigm for tuning and timbre which generalizes the special relationship between just intonation, and the harmonic series to apply to a wider set of pseudo-just tunings and related[1] pseudo-harmonic timbres.[2]
The main limitation of dynamic tonality is that it is best used with compatible isomorphic keyboard instruments and compatible synthesizers, or with voices and instruments whose sounds are transformed in real time via compatible digital tools.[3]
A vibrating string, a column of air, and the human voice all emit a specific pattern of partials corresponding to the harmonic series. The degree of correspondence varies, depending on the physical characteristics of the emitter. "Partials" are also called "harmonics" or "overtones." Each musical instrument's unique sound is called its timbre, so an instrument's timbre can be called a "harmonic timbre" if its partials correspond closely to the harmonic series.
Just intonation is a system of tuning that adjusts a tuning's notes to maximize their alignment with a harmonic timbre's partials. This alignment maximizes the consonance of music's tonal intervals.
The harmonic series and just intonation share an infinitely complicated – or infinite rank – pattern that is determined by the infinite series of prime numbers. A temperament is an attempt to reduce this complexity by mapping this rank-∞ pattern to a simpler, finite-rank pattern.
Throughout history, the pattern of notes in a tuning could be altered (that is, "tempered") by humans but the pattern of partials sounded by an acoustic musical instrument was largely determined by the physics of their sound production. The resulting misalignment between "pseudo-just" tempered tunings, and untempered timbres, made temperament "a battleground for the great minds of Western civilization".[4][5][6] This misalignment, in any tuning that is not fully Just (and hence infinitely complex), is the defining characteristic of any static timbre paradigm.
Many of the pseudo-just temperaments proposed during this "temperament battle" were rank 2 (two-dimensional) – such as quarter-comma meantone – that provided more than 12 notes per octave. However, the standard piano-like keyboard is only rank 1 (one-dimensional), affording at most 12 notes per octave. Piano-like keyboards affording more than 12 notes per octave were developed by Vicentino,[4]: 127 Colonna,[4]: 131 Mersenne,[4]: 181 Huygens,[4]: 185 and Newton,[4]: 196 but were all considered too cumbersome / too difficult to play.[4]: 18
The goal of dynamic tonality is to enable consonance beyond the range of tunings and temperaments in which harmonic timbres have traditionally been played. Dynamic tonality delivers consonance by tempering the intervals between notes (into "pseudo-just tunings") and also tempering the intervals between partials (into "pseudo-harmonic timbres") through digital synthesis and/or processing. Aligning the notes of a pseudo-just tuning's notes and the partials of a pseudo-harmonic timbre (orvice versa) enables consonance.
The defining characteristic of dynamic tonality is that a given rank-2 temperament (as defined by a period α, a generator β, and a comma sequence)[7] is used to generate, in real time during performance, the same set of intervals[2] among:
Generating all three from the same temperament solves two problems and creates (at least) three opportunities.
A rank-2 temperament defines a rank-2 (two-dimensional) note space, as shown in video 1 (note space).
The syntonic temperament is a rank-2 temperament defined by its period (just perfect octave, 1/2), its generator (just perfect fifth, 3/2) and its comma sequence (which starts with the syntonic comma, 81/80, which names the temperament). The construction of the syntonic temperament's note-space is shown in video 2 (Syntonic note-space).
The valid tuning range of the syntonic temperament is show in Figure 1.
A keyboard that is generated by a temperament is said to be isomorphic with that temperament (from the Greek "iso" meaning "same," and "morph" meaning "shape"). Isomorphic keyboards are also known as generalized keyboards. Isomorphic keyboards have the unique properties of transpositional invariance[11] and tuning invariance[7]: 4 when used with rank-2 temperamentsofjust intonation. That is, such keyboards expose a given musical interval with "the same shape" in every octave of every key of every tuning of such a temperament.
Of the various isomorphic keyboards now known (e.g., the Bosanquet, Janko, Fokker, and Wesley), the Wicki-Hayden keyboard is optimal for dynamic tonality across the entire valid 5-limit tuning range of the syntonic temperament.[2]: 7-10 The isomorphic keyboard shown in this article's videos is the Wicki-Hayden keyboard, for that reason. It also has symmetries related to Diatonic Set Theory, as shown in Video 3 (Same shape).
The Wicki-Hayden keyboard embodies a tonnetz, as shown in video 4 (tonnetz). The tonnetz is a lattice diagram representing tonal space first described by Euler (1739),[12] which is a central feature of Neo-Riemannian music theory.
The endpoints of the valid 5 limit tuning range of the syntonic temperament, shown in Figure 1, are:
The partials of a pseudo-harmonic timbre are digitally mapped, as defined by a temperament, to specific notes of a pseudo – just tuning. When the temperament's generator changes in width, the tuning of the temperament's notes changes, and the partials change along with those notes – yet their relative position remains invariant on the temperament-generated isomorphic keyboard. The frequencies of notes and partials change with the generator's width, but the relationships among the notes, partials, and note-controlling buttons remain the same: as defined by the temperament. The mapping of partials to the notes of the syntonic temperament is animated in video 5.
On an isomorphic keyboard, any given musical structure—a scale, a chord, a chord progression, or an entire song—has exactly the same fingering in every tuning of a given temperament. This allows a performer to learn to play a song in one tuning of a given temperament and then to play it with exactly the same finger-movements, on exactly the same note-controlling buttons, in every other tuning of that temperament. See video 3 (Same shape).
For example, one could learn to play Rodgers and Hammerstein's "Do-Re-Mi" song in its original 12 tone equal temperament (12TET) and then play it with exactly the same finger-movements, on exactly the same note-controlling buttons, while smoothly changing the tuning in real time across the syntonic temperament's tuning continuum.
The process of digitally tempering a pseudo-harmonic timbre's partials to align with a tempered pseudo-just tuning's notes is shown in video 6 (Dynamic tuning & timbre).[3]
Dynamic Tonality enables two new kinds of real-time musical effects:
Dynamic Tonality's novel tuning-based effects[14] include:
The developers of dynamic tonality have invented novel vocabulary to describe the effects on timbre by raising or lowering the relative amplitude of partials.[15] Their new terms include primeness, conicality, and richness, with primeness being further subdivided into twoness, threeness, fiveness etc.:
One can use Dynamic Tonality to temper only the tuning of notes, without tempering timbres, thus embracing the Static Timbre Paradigm.
Similarly, using a synthesizer control such as the Tone Diamond,[17] a musician can opt to maximize regularity, harmonicity, or consonance – or trade off among them in real time (with some of the jammer's 10 degrees of freedom mapped to the tone diamond's variables), with consistent fingering. This enables musicians to choose tunings that are regular or irregular, equal or non-equal, major-biased or minor-biased – and enables the musician to slide smoothly among these tuning options in real time, exploring the emotional affect of each variation and the changes among them.
Imagine that the valid tuning range of a temperament (as defined in Dynamic Tonality) is a string, and that individual tunings are beads on that string. The microtonal community has typically focused primarily on the beads, whereas Dynamic Tonality is focused primarily on the string. Both communities care about both beads and strings; only their focus and emphasis differ.
An early example of dynamic tonality can be heard in the song "C2ShiningC".[18][3]
This sound example contains only one chord, Cmaj, played throughout, yet a sense of harmonic tension is imparted by a tuning progression and a timbre progression, as follows:
Cmaj19TET harmonic |
⟶ | Cmaj5TET harmonic |
⟶ | Cmaj19TET consonant |
⟶ | Cmaj5TET consonant |
As the tuning changes, the pitches of all notes except the tonic change, and the widths of all intervals except the octave change; however, the relationships among the intervals (as defined by the syntonic temperament's period, generator, and comma sequence) remain invariant (that is, constant; not varying) throughout. This invariance among a temperament's interval relationships is what makes invariant fingering (on an isomorphic keyboard) possible, even while the tuning is changing. In the syntonic temperament, the tempered major third (M3) is as wide as four tempered perfect fifths (p5‑s) minus two octaves – so the M3's width changes across the tuning progression
Thus, the tuning progression's widening of the Cmaj's M3 from a nearly just major thirdin19TET to a slightly flat perfect fourthin5TET creates the harmonic tension of a Csus 4 within a Cmaj chord, which is relieved by the return to 19TET. This example proves that dynamic tonality offers new means of creating and then releasing harmonic tension, even within a single chord.
This analysis is presented in Cmaj as originally intended, despite the recording actually being in Dmaj.
Dynamic tonality was developed primarily by a collaboration between William Sethares, Andrew Milne, and James ("Jim") Plamondon.
The latter formed Thumtronics Pty Ltd. to develop an expressive, tiny, electronic Wicki-Hayden keyboard instrument: Thumtronics' "Thummer."[19][20][21][22][23] The generic name for a Thummer-like instrument is "jammer." With two thumb-sticks and internal motion sensors, a jammer would afford 10 degrees of freedom, which would make it the most expressive polyphonic instrument available. Without the expressive potential of a jammer, musicians lack the expressive power needed to exploit dynamic tonality in real time, so dynamic tonality's new tonal frontiers remain largely unexplored.
Smooth changes of tuning and timbre are at the core of C2ShiningC … found on the Spectral Tools home page.