|
|
| (108 intermediate revisions by 31 users not shown) |
| Line 1: |
Line 1: |
| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | {{interwiki |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| | | en = Just intonation |
| : This revision was by author [[User:guest|guest]] and made on <tt>2010-11-01 07:28:18 UTC</tt>.<br>
| | | de = Reine Stimmungen |
| : The original revision id was <tt>175265119</tt>.<br>
| | | es = Entonación Justa |
| : The revision comment was: <tt></tt><br>
| | | ja = 純正律 |
| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
| | | ko = 순정률 |
| <h4>Original Wikitext content:</h4>
| | | ro = Intervale raționale |
| <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]]
| | }} |
| ----
| | {{Wikipedia}} |
| =Just Intonation explained=
| |
| Just Intonation describes [[Gallery of Just Intervals|intervals]] between pitches by specifying ratios (of [[http://en.wikipedia.org/wiki/Rational_number|rational numbers]]) between the frequencies of pitches. This is sometimes distinguished from //rational intonation// by requiring that the ratios be ones of low complexity (as for example measured by [[Tenney height]]) but there is no clear dividing line. The matter is partially a question of intent. The rank two tuning system in which all intervals are given as combinations of the just perfect fourth, 4/3, and the just minor third, 6/5, would seem to be a nonoctave 5-limit just intonation system by definition. In practice, it can hardly be used except as a rank two 7-limit [[Microtempering|microtempering]] system because of certain very accurate approximations to the octave and to seven limit intervals: (6/5)^2/(4/3) = 27/25, the semitone maximus or just minor second; and (27/25)^9 is less than a cent short of an octave, while (27/25)^2 is almost precisely 7/6, the [[http://en.wikipedia.org/wiki/Septimal_minor_third|septimal minor third]].
| |
|
| |
|
| If you are used to speaking only in note names, you may need to study the relation between frequency and [[http://en.wikipedia.org/wiki/Pitch_%28music%29|pitch]]. Kyle Gann's //[[http://www.kylegann.com/tuning.html|Just Intonation Explained]]// is one good reference. A transparent illustration and one of just intonation's acoustic bases is the [[OverToneSeries|harmonic series]].
| | '''Just intonation''' ('''JI''') is an approach to [[musical tuning]] which uses tones whose frequencies are whole-number ratios of a given fundamental [[frequency]]. Just intonation includes the [[harmonic series]], which is the collection of tones found at integer multiples of a fundamental frequency; all just intervals can be found as the interval between two notes in the harmonic series. Just ratios of small numbers, called '''Low-complexity just intonation (LCJI)''' intervals, tend to be the most [[concordant|consonant]] in the sense that their sounds meld together. |
|
| |
|
| =Just Intonation used=
| | In the context of Western music theory prior to the 20th century, the term ''just intonation'' used alone usually refers to [[5-limit]] tuning--intervals where the numerators and denominators of any ratio used have no prime factors greater than 5. ''Extended just intonation'', a term coined by [[Ben Johnston]], refers to any tuning in the harmonic series regardless of [[prime limit]].<ref>From Ben Johnston "A Notation System for Extended Just Intonation." ''Maximum Clarity'', 2006, p. 77</ref> In current usage, just intonation typically refers to extended just intonation. The practice of just intonation without any particular constraint is sometimes referred to as '''rational intonation''' ('''RI''') or as [[free style JI]]. |
| The use of just intonation could be divided into these two flavors: | |
|
| |
|
| ==Free Style Just==
| | The structure of just intonation has several implications on music composition. [[Wolf interval|Wolf intervals]] and [[Comma|commas]], two kinds of dissonant intervals, may appear between distantly-related pitches. In addition, certain chord progressions are [[Comma pump|comma pumps]], which may cause the [[tonal center]] of a piece to drift up or down in pitch over time. These effects can be treated either as tools to use or as problems to be solved. Examples of approaches that try to solve these problems without greatly restricting the set of available ratios include pitch shifts, [[adaptive just intonation]] and [[temperament]]. Other approaches restrict the space of usable JI intervals in a way that makes these problems arise less frequently. |
| = =
| |
| Lou Harrison used this term; it means that you choose just-intonation pitches, from the set of all possible just intervals (not from a mode or scale), as you use them in music. Dedicated page -> [[FreeStyleJI]]
| |
|
| |
|
| ==Constrained Just== | | == Consonance == |
| (In need of a better name maybe) Here are six ways that musicians and theorists have constrained the field of potential just ratios (from Jacques Dudon, "Differential Coherence", //1/1// vol. 11, no. 2: p.1):
| | LCJI intervals achieve consonance through alignment of [[Partial|partials]] if the interval has [[Harmonic timbre|harmonic timbre]]. In fact, alignment of partials is a stronger effect with harmonic timbre: if partials align at frequency n, they will also align at every multiple of n; and in addition, two notes whose partials align with the same root note will also have partials aligning with each other. This allows for the construction of just-intonation chords of more than two notes where every comprising interval is a consonance. |
|
| |
|
| //1. The principle of "[[Harmonic Limit|harmonic limits]]," which sets a threshold in order to place a limit on the largest prime number in any ratio (cf: Tanner's "psycharithmes" and his ordering by complexity; Gioseffe Zarlino's five-limit "senario," and the like; Helmholtz's theory of consonance with its "blending of partials," which, like the others, results in giving priority to the lowest prime numbers).//
| | Low-complexity JI intervals and chords also achieve consonance by being the ratios between harmonics of a (possibly unplayed) fundamental even if they do not have harmonic timbre. |
|
| |
|
| //2. Restrictions on the combinations of numbers that make up the numerator and denominator of the ratios under consideration, such as the "monophonic" system of [[http://en.wikipedia.org/wiki/Harry_Partch|Harry Partch]]'s [[http://en.wikipedia.org/wiki/Pitch_%28music%29|tonality diamond]]. This, incidentially, is an eleven-limit system that only makes use of ratios of the form n:d, where n and d are drawn only from harmonics 1,3 5 7 9, 11, or their octaves.//
| | ==Ways of using JI== |
| | Here are multiple ways in which musicians and theorists have used just intonation. |
|
| |
|
| //3. Other theorists who, in contrast to the above, advocate the use of the [[http://en.wikipedia.org/wiki/Hexany|products]] of a given set of prime numbers, such as Robert Dussaut, [[http://en.wikipedia.org/wiki/Erv_Wilson|Ervin Wilson]], and others.//
| | [[Free style JI|'''Free style JI''']] <br /> |
| | [[Lou Harrison]] used this term; it means that you choose just-intonation pitches from the set of all possible just intervals (not from a mode or scale) as you use them in music. |
|
| |
|
| //4. [[Just intonation subgroups|Restrictions on the variety of prime numbers]] used within a system, for example, 3 used with only one [sic, also included is 2] other prime 7, 11, or 13.... This is quite common practice with Ptolemy, Ibn-Sina, Al-Farabi, and Saf-al-Din, and with numerous contemporary composers working in Just Intonation.// | | '''Harmonic limits and subgroups'''<br /> |
| | [[Harmonic limit|Harmonic limits]] set a limit for the highest prime number in the factorization of any ratio used. [[Subgroup|Subgroups]] name a list of allowable prime numbers used. |
|
| |
|
| //5. Restricting the denominator to one or very few values (the [[OverToneSeries|harmonic series]]).// | | '''Restrictions on the denominator or numerator'''<br /> |
| | Some approaches restrict "the denominator to one or very few values"<ref name=":0">From Jacques Dudon, "Differential Coherence", ''1/1'' vol. 11, no. 2: p.1).</ref> (the [[harmonic series]], [[isoharmonic chord]]s, [[AFDO]]s/[[overtone scale]]s, [[primodality]], [[Ringer scale|ringer scales]]), the "numerator to one or a very few values" (the [[subharmonic series]], [[IFDO]]s/undertone scales), or both ([[Tonality diamond|tonality diamonds]]) |
|
| |
|
| //6. Restricting the numerator to one or a very few values (the [[subharmonic series]] or [[aliquot scales]]).// | | '''Mediants'''<br /> |
| | The use of harmonic and arithmetic [[Mediant (operation)|mediants]] as was common with the Ancient Greeks. This can also involve further divisions besides two parts as seen with Ptolemy sometimes using 3 parts. The Chinese have historically used as many as 10 parts. |
|
| |
|
| | '''Approximations/alterations of tempered tunings''' <br /> |
| | These are [[Detempering|detemperings]], including [[NEJI]] systems. |
|
| |
|
| =Variations on 'Just'=
| | '''Other approaches'''<br /> |
| [[Regular Temperaments]] are just intonation systems of various [[harmonic limits]] with certain commas 'tempered out' | | Other approaches include [http://anaphoria.com/wilsonintroMERU.html Meru scales], [[Tritriadic scale|titriadic scales]], and [[combination product sets|product sets]]. |
| [[AdaptiveJI|Adaptive JI]] | |
|
| |
|
| **Broken** links to JI theory pages on [[http://moinmoin.riters.com/microtonal|another microtonal wiki]], which await transfer to this wiki: | | ==Instruments== |
| [[http://moinmoin.riters.com/microtonal/index.cgi/58Note11LimitJI|58 note 11 limit JI]] - hyper-Partchian! | | {{todo|expand|comment=Expand the instruments section with more examples}} |
| [[http://moinmoin.riters.com/microtonal/index.cgi/Reduction|Reduction]] | | *The [[Kalimba#Array mbira|array mbira]] was designed by [[Bill Wesley]] as a versatile just intonation instrument, covering a 5 octave range. |
| [[http://moinmoin.riters.com/microtonal/index.cgi/Comma_20sequences|Comma sequences]]
| | *Most of [[Harry Partch]]'s instruments were designed to be for just intonation. |
| [[http://moinmoin.riters.com/microtonal/index.cgi/Hahn_20distance|Hahn distance]]
| | ==Music== |
| | {{Main|Music in just intonation}} |
|
| |
|
| =Scalesmith's gallery of Just Intonation scales= | | == Notation == |
| [[Gallery of 12-tone Just Intonation Scales]] | | There are various [[Musical notation|notation systems]] for just intonation. |
| [[boogiewoogiescale|Boogie woogie scale]] | | ==See also== |
| [[Arnold Dreyblatt]] | | {{todo|cleanup|inline=1}} |
| [[Gallery of pentatonics]]
| | *[[List of approaches to musical tuning]] |
| [[FiniteSubsetJI]]
| | *[[Gallery of just intervals]] |
| | *[[Gallery of 12-tone just intonation scales]] |
| | *[[Families of scales]] |
| | *[[boogiewoogiescale|Boogie woogie scale]] |
| | *[[:Category:Just intonation]] |
| | ==References== |
| | <references /> |
|
| |
|
| See also: [[Gallery of Just Intervals]]</pre></div>
| | ==Further reading== |
| <h4>Original HTML content:</h4>
| | *[http://www.tonalsoft.com/enc/j/just.aspx Just intonation] on the [[Tonalsoft Encyclopedia]] |
| <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Just intonation</title></head><body><!-- ws:start:WikiTextTocRule:14:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --><a href="#Just Intonation explained">Just Intonation explained</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> | <a href="#Just Intonation used">Just Intonation used</a><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --> | <a href="#toc3"> </a><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --> | <a href="#Variations on 'Just'">Variations on 'Just'</a><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --> | <a href="#Scalesmith's gallery of Just Intonation scales">Scalesmith's gallery of Just Intonation scales</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: -->
| | *[http://nowitzky.hostwebs.com/justint/ Just Intonation] by Mark Nowitzky |
| <!-- ws:end:WikiTextTocRule:22 --><hr />
| | *[http://www.kylegann.com/tuning.html Just Intonation Explained] by Kyle Gann |
| <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Just Intonation explained"></a><!-- ws:end:WikiTextHeadingRule:0 -->Just Intonation explained</h1>
| | *[http://www.kylegann.com/Octave.html Anatomy of an Octave] by Kyle Gann |
| Just Intonation describes <a class="wiki_link" href="/Gallery%20of%20Just%20Intervals">intervals</a> between pitches by specifying ratios (of <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Rational_number" rel="nofollow">rational numbers</a>) between the frequencies of pitches. This is sometimes distinguished from <em>rational intonation</em> by requiring that the ratios be ones of low complexity (as for example measured by <a class="wiki_link" href="/Tenney%20height">Tenney height</a>) but there is no clear dividing line. The matter is partially a question of intent. The rank two tuning system in which all intervals are given as combinations of the just perfect fourth, 4/3, and the just minor third, 6/5, would seem to be a nonoctave 5-limit just intonation system by definition. In practice, it can hardly be used except as a rank two 7-limit <a class="wiki_link" href="/Microtempering">microtempering</a> system because of certain very accurate approximations to the octave and to seven limit intervals: (6/5)^2/(4/3) = 27/25, the semitone maximus or just minor second; and (27/25)^9 is less than a cent short of an octave, while (27/25)^2 is almost precisely 7/6, the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_minor_third" rel="nofollow">septimal minor third</a>.<br />
| | *[http://www.dbdoty.com/Words/What-is-Just-Intonation.html What is Just Intonation?] by David B. Doty |
| <br />
| | *[http://lumma.org/tuning/faq/#whatisJI What is "just intonation"?] by Carl Lumma |
| If you are used to speaking only in note names, you may need to study the relation between frequency and <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Pitch_%28music%29" rel="nofollow">pitch</a>. Kyle Gann's <em><a class="wiki_link_ext" href="http://www.kylegann.com/tuning.html" rel="nofollow">Just Intonation Explained</a></em> is one good reference. A transparent illustration and one of just intonation's acoustic bases is the <a class="wiki_link" href="/OverToneSeries">harmonic series</a>.<br />
| | *[http://www.dbdoty.com/Words/werntz.html A Response to Julia Werntz] by David B. Doty |
| <br />
| | *[http://lumma.org/tuning/gws/commaseq.htm Comma Sequences] by Gene Ward Smith |
| <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Just Intonation used"></a><!-- ws:end:WikiTextHeadingRule:2 -->Just Intonation used</h1>
| | *[https://casfaculty.case.edu/ross-duffin/just-intonation-in-renaissance-theory-practice/ Just Intonation in Renaissance Theory & Practice] by Ross W. Duffin |
| The use of just intonation could be divided into these two flavors:<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Just Intonation used-Free Style Just"></a><!-- ws:end:WikiTextHeadingRule:4 -->Free Style Just</h2>
| |
| <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><!-- ws:end:WikiTextHeadingRule:6 --> </h1>
| |
| Lou Harrison used this term; it means that you choose just-intonation pitches, from the set of all possible just intervals (not from a mode or scale), as you use them in music. Dedicated page -&gt; <a class="wiki_link" href="/FreeStyleJI">FreeStyleJI</a><br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Just Intonation used-Constrained Just"></a><!-- ws:end:WikiTextHeadingRule:8 -->Constrained Just</h2>
| |
| (In need of a better name maybe) Here are six ways that musicians and theorists have constrained the field of potential just ratios (from Jacques Dudon, &quot;Differential Coherence&quot;, <em>1/1</em> vol. 11, no. 2: p.1):<br />
| |
| <br />
| |
| <em>1. The principle of &quot;<a class="wiki_link" href="/Harmonic%20Limit">harmonic limits</a>,&quot; which sets a threshold in order to place a limit on the largest prime number in any ratio (cf: Tanner's &quot;psycharithmes&quot; and his ordering by complexity; Gioseffe Zarlino's five-limit &quot;senario,&quot; and the like; Helmholtz's theory of consonance with its &quot;blending of partials,&quot; which, like the others, results in giving priority to the lowest prime numbers).</em><br />
| |
| <br />
| |
| <em>2. Restrictions on the combinations of numbers that make up the numerator and denominator of the ratios under consideration, such as the &quot;monophonic&quot; system of <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Harry_Partch" rel="nofollow">Harry Partch</a>'s <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Pitch_%28music%29" rel="nofollow">tonality diamond</a>. This, incidentially, is an eleven-limit system that only makes use of ratios of the form n:d, where n and d are drawn only from harmonics 1,3 5 7 9, 11, or their octaves.</em><br />
| |
| <br />
| |
| <em>3. Other theorists who, in contrast to the above, advocate the use of the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Hexany" rel="nofollow">products</a> of a given set of prime numbers, such as Robert Dussaut, <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Erv_Wilson" rel="nofollow">Ervin Wilson</a>, and others.</em><br />
| |
| <br />
| |
| <em>4. <a class="wiki_link" href="/Just%20intonation%20subgroups">Restrictions on the variety of prime numbers</a> used within a system, for example, 3 used with only one [sic, also included is 2] other prime 7, 11, or 13.... This is quite common practice with Ptolemy, Ibn-Sina, Al-Farabi, and Saf-al-Din, and with numerous contemporary composers working in Just Intonation.</em><br />
| |
| <br />
| |
| <em>5. Restricting the denominator to one or very few values (the <a class="wiki_link" href="/OverToneSeries">harmonic series</a>).</em><br />
| |
| <br />
| |
| <em>6. Restricting the numerator to one or a very few values (the <a class="wiki_link" href="/subharmonic%20series">subharmonic series</a> or <a class="wiki_link" href="/aliquot%20scales">aliquot scales</a>).</em><br />
| |
| <br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Variations on 'Just'"></a><!-- ws:end:WikiTextHeadingRule:10 -->Variations on 'Just'</h1>
| |
| <a class="wiki_link" href="/Regular%20Temperaments">Regular Temperaments</a> are just intonation systems of various <a class="wiki_link" href="/harmonic%20limits">harmonic limits</a> with certain commas 'tempered out'<br />
| |
| <a class="wiki_link" href="/AdaptiveJI">Adaptive JI</a><br />
| |
| <br />
| |
| <strong>Broken</strong> links to JI theory pages on <a class="wiki_link_ext" href="http://moinmoin.riters.com/microtonal" rel="nofollow">another microtonal wiki</a>, which await transfer to this wiki:<br />
| |
| <a class="wiki_link_ext" href="http://moinmoin.riters.com/microtonal/index.cgi/58Note11LimitJI" rel="nofollow">58 note 11 limit JI</a> - hyper-Partchian!<br />
| |
| <a class="wiki_link_ext" href="http://moinmoin.riters.com/microtonal/index.cgi/Reduction" rel="nofollow">Reduction</a><br />
| |
| <a class="wiki_link_ext" href="http://moinmoin.riters.com/microtonal/index.cgi/Comma_20sequences" rel="nofollow">Comma sequences</a><br />
| |
| <a class="wiki_link_ext" href="http://moinmoin.riters.com/microtonal/index.cgi/Hahn_20distance" rel="nofollow">Hahn distance</a><br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Scalesmith's gallery of Just Intonation scales"></a><!-- ws:end:WikiTextHeadingRule:12 -->Scalesmith's gallery of Just Intonation scales</h1>
| |
| <a class="wiki_link" href="/Gallery%20of%2012-tone%20Just%20Intonation%20Scales">Gallery of 12-tone Just Intonation Scales</a><br />
| |
| <a class="wiki_link" href="/boogiewoogiescale">Boogie woogie scale</a><br />
| |
| <a class="wiki_link" href="/Arnold%20Dreyblatt">Arnold Dreyblatt</a><br />
| |
| <a class="wiki_link" href="/Gallery%20of%20pentatonics">Gallery of pentatonics</a><br />
| |
| <a class="wiki_link" href="/FiniteSubsetJI">FiniteSubsetJI</a><br />
| |
| <br />
| |
| See also: <a class="wiki_link" href="/Gallery%20of%20Just%20Intervals">Gallery of Just Intervals</a></body></html></pre></div>
| |
Just intonation (JI) is an approach to musical tuning which uses tones whose frequencies are whole-number ratios of a given fundamental frequency. Just intonation includes the harmonic series, which is the collection of tones found at integer multiples of a fundamental frequency; all just intervals can be found as the interval between two notes in the harmonic series. Just ratios of small numbers, called Low-complexity just intonation (LCJI) intervals, tend to be the most consonant in the sense that their sounds meld together.
In the context of Western music theory prior to the 20th century, the term just intonation used alone usually refers to 5-limit tuning--intervals where the numerators and denominators of any ratio used have no prime factors greater than 5. Extended just intonation, a term coined by Ben Johnston, refers to any tuning in the harmonic series regardless of prime limit.[1] In current usage, just intonation typically refers to extended just intonation. The practice of just intonation without any particular constraint is sometimes referred to as rational intonation (RI) or as free style JI.
The structure of just intonation has several implications on music composition. Wolf intervals and commas, two kinds of dissonant intervals, may appear between distantly-related pitches. In addition, certain chord progressions are comma pumps, which may cause the tonal center of a piece to drift up or down in pitch over time. These effects can be treated either as tools to use or as problems to be solved. Examples of approaches that try to solve these problems without greatly restricting the set of available ratios include pitch shifts, adaptive just intonation and temperament. Other approaches restrict the space of usable JI intervals in a way that makes these problems arise less frequently.
Consonance
LCJI intervals achieve consonance through alignment of partials if the interval has harmonic timbre. In fact, alignment of partials is a stronger effect with harmonic timbre: if partials align at frequency n, they will also align at every multiple of n; and in addition, two notes whose partials align with the same root note will also have partials aligning with each other. This allows for the construction of just-intonation chords of more than two notes where every comprising interval is a consonance.
Low-complexity JI intervals and chords also achieve consonance by being the ratios between harmonics of a (possibly unplayed) fundamental even if they do not have harmonic timbre.
Ways of using JI
Here are multiple ways in which musicians and theorists have used just intonation.
Free style JI
Lou Harrison used this term; it means that you choose just-intonation pitches from the set of all possible just intervals (not from a mode or scale) as you use them in music.
Harmonic limits and subgroups
Harmonic limits set a limit for the highest prime number in the factorization of any ratio used. Subgroups name a list of allowable prime numbers used.
Restrictions on the denominator or numerator
Some approaches restrict "the denominator to one or very few values"[2] (the harmonic series, isoharmonic chords, AFDOs/overtone scales, primodality, ringer scales), the "numerator to one or a very few values" (the subharmonic series, IFDOs/undertone scales), or both (tonality diamonds)
Mediants
The use of harmonic and arithmetic mediants as was common with the Ancient Greeks. This can also involve further divisions besides two parts as seen with Ptolemy sometimes using 3 parts. The Chinese have historically used as many as 10 parts.
Approximations/alterations of tempered tunings
These are detemperings, including NEJI systems.
Other approaches
Other approaches include Meru scales, titriadic scales, and product sets.
Instruments
- The array mbira was designed by Bill Wesley as a versatile just intonation instrument, covering a 5 octave range.
- Most of Harry Partch's instruments were designed to be for just intonation.
Music
Notation
There are various notation systems for just intonation.
See also
|
Todo: cleanup
|
References
- ↑ From Ben Johnston "A Notation System for Extended Just Intonation." Maximum Clarity, 2006, p. 77
- ↑ From Jacques Dudon, "Differential Coherence", 1/1 vol. 11, no. 2: p.1).
Further reading
