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[[toc|flat]]
----
=Just Intonation explained= 
[[Just Intonation]] (JI) 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.<ref>Just Intonation is sometimes distinguished from //rational intonation,// by requiring that the ratios be lower than some arbitrary complexity (as for example measured by [[Tenney height]], [[Benedetti height]], etc.) 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 however, it casually suggests a rank two 7-limit [[Microtempering|microtempering]] system because of 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]].

</ref>

If you are used to speaking only in note names (e.g. the first 7 letters of the alphabet), 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]].

In languages other than English, the original conceptions of "Just Intonation" are more obviously retained in the terms used in those languages: German //Reine Stimmung// (pure, that is, beatless, tuning), Ukrainian //Натуральний стрій// and French //Gamme naturelle// (both referring to the "natural scale", that is, intervals derived from the harmonic series), Italian //Intonazione naturale// (natural intonation, once again intervals derived from harmonic series), and so on.

In the English language, the term "just" referred to "true, correct", and is still used today in this sense, in the crafts. In printing, to "justify" a line of type is to fit it precisely to a certain measure, for example. The original sense, then, was similar to that sense which is clearly retained in other languages as "natural".

Of course, a historical description of something as "natural" does not prove that something is "natural." Similarly labeling something "natural" without any ground, especially in the arts, is always very problematic. Nevertheless, the historical meanings of the terms for what we call "Just Intonation" do claim a "natural" status, and Just Intonation is indeed derived from genuine acoustic phenomena. How important, universal, etc., these phenomena are has been a matter of debate for thousands of years.

Specifying ratios of frequencies is another way of expressing the "natural scale", for it describes ratios between partials in the harmonic series (in their ideal form). So, contemporary usage of the term is in keeping with historical and international usages. However, just as harmonic vocabulary has expanded over the centuries, so has that which falls under "just intonation" expanded.

But, first things first. Let us take a look at why the idea of a "natural" or "just" tuning came about, and is still with us.

If we have a tone with a harmonic timbre and a fundamental frequency at 100 Hz (Hertz, or cycles per second), we will find the second harmonic component at 200 Hz, the third at 300 Hz, the fourth at 400 Hz...Yes, the harmonics are found at the fundamental frequency times 1, times 2, times 3...

The simplicity of it all can be difficult to believe at first. You can easily imagine people discovering this and getting carried away with ideas of "music of the spheres" and other mystical ideas. Yes, it IS amazing. Please keep in mind that not all sounds have a harmonic spectrum.<ref>[2] All manner of bells, gongs, percussion instruments, synthesizer sounds, have spectra that follow their own rules, usually very complex. Inharmonic tones can be found in otherwise harmonic spectra, and instruments with harmonic spectra may have inharmonic spectra during the attack portion of the sound. Loudly played brass instruments, for example, have a moment of extremely complex sound not unlike that of striking a piece of metal, followed by a moment in which the partials are "stretched" according to a more complex rule than simply multiplying by, 1, 2, 3, etc., before settling down into a harmonic series accompanied by various amounts of characteristic "noise". A breathily played flute has a large addition of inharmonic material, a "jinashi" shakuhachi flute is an excellent example of an instrument of varying harmonicity and inharmonicity.</ref>

Of course we are describing an ideal tone - in real life, tones waver, certain harmonics are missing, etc. Nevertheless this is the harmonic series, and measuring the spectra of violins (or any other stringed instruments), human voices, and woodwinds, for example, will reveal that this is indeed the pattern, and even in our "fuzzy" and "flawed" reality, spectra adhere to this pattern with impressive consistency.

In a tuning "according to the natural scale", we have for example a "perfect fifth" as simply the ratio between the third partial and the second partial: "3:2". In our example tone, that would be the ratio of 300 Hz to 200 Hz. Where we to want a Just Intonation perfect fifth above our original tone, its fundamental frequency would be found at 3/2 times the fundamental frequency of our original tone. So, 3/2 times 100 gives us 150. Our example perfect fifth has a fundamental frequency at 150 Hz.

Now, let us play our two example tones together, and we shall see why the German term is //Reine//, "pure", and why you'll hear "pure" used in English and many other languages as well. Let's call our first tone "Do" and our second tone, a perfect fifth higher, "Sol".

[[code]]
Tone   Frequencies of partials (Hz)
Do     100  200  300  400  500  600   700   800   900   ...
So     150  300  450  600  750  900  1050  1200  1350   ...
[[code]]

You see that the tones share the frequencies of some of the partials. These partials will "meld" when our Do and Sol are played together. This goes by the wonderful name of //Tonverschmelzung// in German. It is a very distinctive "blending" sound. If our Sol was tuned to, for example, 148 Hz, its second harmonic component would be at 296 Hz, and the two tones played together would not "meld together" at 300 Hz, but would "beat". That is, we would hear a throbbing sound, the "beat rate" of which is found by reckoning the distance in Hertz between the two near-coincident partials. In this case, 300 - 296 = 4 Hz, so we'd hear a beating of four times a second (this is like a rhythm of eighth notes at a metronome marking of 120 beats per minute).

One does not need to know of the harmonic series, nor even know how to read, or even count, to sing this.

There is more to it than this, of course, but the basic principles of Just Intonation are very simple. Hundreds of years ago, when the intonation of a few well-known intervals were the concern, understanding and defining "Just" was not difficult. These days, though, and going on from these basics, it can get a bit more complicated...


=Just Intonation in use= 

To start off your exploration of just intonation scales, the [[Gallery of 12-tone Just Intonation Scales]] is a good place to start.

The use of just intonation could be divided into these two flavors:

==Free Style Just== 

[[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== 
(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):

//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). See [[3-limit]], [[5-limit]], [[7-limit]], [[11-limit]], [[13-limit]].//

//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, incidentally, 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.//

//3. Other theorists who, in contrast to the above, advocate the use of [[http://en.wikipedia.org/wiki/Hexany|products sets]] of given arrays of prime numbers, such as [[http://en.wikipedia.org/wiki/Erv_Wilson|Ervin Wilson]], Robert Dussaut, and others.//

//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.//

//5. Restricting the denominator to one or very few values (the [[OverToneSeries|harmonic series]]).//

//6. Restricting the numerator to one or a very few values (the [[subharmonic series]] or [[aliquot scales]]).//

to this can be added
//7. 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.//

//8. While related to the above, the use of recurrent sequences is by some included under JI as it involves whole numbers. Wilson's [[http://anaphoria.com/wilsonintroMERU.html|Meru scales]] are a good example.//

=Variations on 'Just'= 
[[Regular Temperaments]] are just intonation systems of various [[harmonic limits]] with certain commas 'tempered out'
[[AdaptiveJI|Adaptive JI]]

=Links= 
[[Gallery of Just Intervals]]
[[Gallery of 12-tone Just Intonation Scales]]
[[Gallery of pentatonics]]
[[List of root-3rd-P5 triads in JI]]
[[Hahn distance]]
[[Arnold Dreyblatt]]
[[FiniteSubsetJI]]
[[hypergenesis58.scl|58 note 11 limit JI scale - hyper-Partchian!]]
[[boogiewoogiescale|Boogie woogie scale]]

=Articles= 
* [[http://en.wikipedia.org/wiki/Just_intonation|Wikipedia article on just intonation]]
* [[http://nowitzky.hostwebs.com/justint/|Just Intonation]] by Mark Nowitzky [[http://www.webcitation.org/5xeAm2lPL|Permalink]]
* [[http://www.kylegann.com/tuning.html|Just Intonation Explained]] by Kyle Gann [[http://www.webcitation.org/5xe2iC7Nq|Permalink]]
* [[http://www.kylegann.com/Octave.html|Anatomy of an Octave]] by Kyle Gann [[http://www.webcitation.org/5xe30LCev|Permalink]]
* [[http://www.dbdoty.com/Words/What-is-Just-Intonation.html|What is Just Intonation?]] by David B. Doty [[http://www.webcitation.org/5xe3MeWVq|Permalink]]
* [[http://lumma.org/tuning/faq/#whatisJI|What is "just intonation"?]] by Carl Lumma [[http://www.webcitation.org/65NwFAKLh|Permalink]]
* [[http://www.dbdoty.com/Words/werntz.html|A Response to Julia Werntz]] by David B. Doty [[http://www.webcitation.org/5xe38KWx4|Permalink]]
* [[http://lumma.org/tuning/gws/commaseq.htm|Comma Sequences]] by Gene Ward Smith [[http://www.webcitation.org/5xe4rPLZ0|Permalink]]

.

Original HTML content:

<html><head><title>Just intonation</title></head><body><!-- ws:start:WikiTextTocRule:26:&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:26 --><!-- ws:start:WikiTextTocRule:27: --><a href="#Just Intonation explained">Just Intonation explained</a><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --> | <a href="#Just Intonation in use">Just Intonation in use</a><!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --> | <a href="#Variations on 'Just'">Variations on 'Just'</a><!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --> | <a href="#Links">Links</a><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --> | <a href="#Articles">Articles</a><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: -->
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<!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc0"><a name="Just Intonation explained"></a><!-- ws:end:WikiTextHeadingRule:12 -->Just Intonation explained</h1>
 <a class="wiki_link" href="/Just%20Intonation">Just Intonation</a> (JI) 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.<!-- ws:start:WikiTextRefRule:10:&amp;lt;ref&amp;gt;Just Intonation is sometimes distinguished from &lt;em&gt;rational intonation,&lt;/em&gt; by requiring that the ratios be lower than some arbitrary complexity (as for example measured by &lt;a class=&quot;wiki_link&quot; href=&quot;/Tenney%20height&quot;&gt;Tenney height&lt;/a&gt;, &lt;a class=&quot;wiki_link&quot; href=&quot;/Benedetti%20height&quot;&gt;Benedetti height&lt;/a&gt;, etc.) but there is no clear dividing line. The matter is partially a question of intent.&lt;br /&gt;
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 however, it casually suggests a rank two 7-limit &lt;a class=&quot;wiki_link&quot; href=&quot;/Microtempering&quot;&gt;microtempering&lt;/a&gt; system because of 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 &lt;a class=&quot;wiki_link_ext&quot; href=&quot;http://en.wikipedia.org/wiki/Septimal_minor_third&quot; rel=&quot;nofollow&quot;&gt;septimal minor third&lt;/a&gt;.&lt;br /&gt;
&lt;br /&gt;
&amp;lt;/ref&amp;gt; --><sup id="cite_ref-1" class="reference"><a href="#cite_note-1">[1]</a></sup><!-- ws:end:WikiTextRefRule:10 --><br />
<br />
If you are used to speaking only in note names (e.g. the first 7 letters of the alphabet), 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 />
<br />
In languages other than English, the original conceptions of &quot;Just Intonation&quot; are more obviously retained in the terms used in those languages: German <em>Reine Stimmung</em> (pure, that is, beatless, tuning), Ukrainian <em>Натуральний стрій</em> and French <em>Gamme naturelle</em> (both referring to the &quot;natural scale&quot;, that is, intervals derived from the harmonic series), Italian <em>Intonazione naturale</em> (natural intonation, once again intervals derived from harmonic series), and so on.<br />
<br />
In the English language, the term &quot;just&quot; referred to &quot;true, correct&quot;, and is still used today in this sense, in the crafts. In printing, to &quot;justify&quot; a line of type is to fit it precisely to a certain measure, for example. The original sense, then, was similar to that sense which is clearly retained in other languages as &quot;natural&quot;.<br />
<br />
Of course, a historical description of something as &quot;natural&quot; does not prove that something is &quot;natural.&quot; Similarly labeling something &quot;natural&quot; without any ground, especially in the arts, is always very problematic. Nevertheless, the historical meanings of the terms for what we call &quot;Just Intonation&quot; do claim a &quot;natural&quot; status, and Just Intonation is indeed derived from genuine acoustic phenomena. How important, universal, etc., these phenomena are has been a matter of debate for thousands of years.<br />
<br />
Specifying ratios of frequencies is another way of expressing the &quot;natural scale&quot;, for it describes ratios between partials in the harmonic series (in their ideal form). So, contemporary usage of the term is in keeping with historical and international usages. However, just as harmonic vocabulary has expanded over the centuries, so has that which falls under &quot;just intonation&quot; expanded.<br />
<br />
But, first things first. Let us take a look at why the idea of a &quot;natural&quot; or &quot;just&quot; tuning came about, and is still with us.<br />
<br />
If we have a tone with a harmonic timbre and a fundamental frequency at 100 Hz (Hertz, or cycles per second), we will find the second harmonic component at 200 Hz, the third at 300 Hz, the fourth at 400 Hz...Yes, the harmonics are found at the fundamental frequency times 1, times 2, times 3...<br />
<br />
The simplicity of it all can be difficult to believe at first. You can easily imagine people discovering this and getting carried away with ideas of &quot;music of the spheres&quot; and other mystical ideas. Yes, it IS amazing. Please keep in mind that not all sounds have a harmonic spectrum.<!-- ws:start:WikiTextRefRule:11:&amp;lt;ref&amp;gt;[2] All manner of bells, gongs, percussion instruments, synthesizer sounds, have spectra that follow their own rules, usually very complex. Inharmonic tones can be found in otherwise harmonic spectra, and instruments with harmonic spectra may have inharmonic spectra during the attack portion of the sound. Loudly played brass instruments, for example, have a moment of extremely complex sound not unlike that of striking a piece of metal, followed by a moment in which the partials are &amp;quot;stretched&amp;quot; according to a more complex rule than simply multiplying by, 1, 2, 3, etc., before settling down into a harmonic series accompanied by various amounts of characteristic &amp;quot;noise&amp;quot;. A breathily played flute has a large addition of inharmonic material, a &amp;quot;jinashi&amp;quot; shakuhachi flute is an excellent example of an instrument of varying harmonicity and inharmonicity.&amp;lt;/ref&amp;gt; --><sup id="cite_ref-2" class="reference"><a href="#cite_note-2">[2]</a></sup><!-- ws:end:WikiTextRefRule:11 --><br />
<br />
Of course we are describing an ideal tone - in real life, tones waver, certain harmonics are missing, etc. Nevertheless this is the harmonic series, and measuring the spectra of violins (or any other stringed instruments), human voices, and woodwinds, for example, will reveal that this is indeed the pattern, and even in our &quot;fuzzy&quot; and &quot;flawed&quot; reality, spectra adhere to this pattern with impressive consistency.<br />
<br />
In a tuning &quot;according to the natural scale&quot;, we have for example a &quot;perfect fifth&quot; as simply the ratio between the third partial and the second partial: &quot;3:2&quot;. In our example tone, that would be the ratio of 300 Hz to 200 Hz. Where we to want a Just Intonation perfect fifth above our original tone, its fundamental frequency would be found at 3/2 times the fundamental frequency of our original tone. So, 3/2 times 100 gives us 150. Our example perfect fifth has a fundamental frequency at 150 Hz.<br />
<br />
Now, let us play our two example tones together, and we shall see why the German term is <em>Reine</em>, &quot;pure&quot;, and why you'll hear &quot;pure&quot; used in English and many other languages as well. Let's call our first tone &quot;Do&quot; and our second tone, a perfect fifth higher, &quot;Sol&quot;.<br />
<br />
<!-- ws:start:WikiTextCodeRule:0:
&lt;pre class=&quot;text&quot;&gt;Tone   Frequencies of partials (Hz)&lt;br/&gt;Do     100  200  300  400  500  600   700   800   900   ...&lt;br/&gt;So     150  300  450  600  750  900  1050  1200  1350   ...&lt;/pre&gt;
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</style><pre class="text">Tone   Frequencies of partials (Hz)
Do     100  200  300  400  500  600   700   800   900   ...
So     150  300  450  600  750  900  1050  1200  1350   ...</pre>

<!-- ws:end:WikiTextCodeRule:0 --><br />
You see that the tones share the frequencies of some of the partials. These partials will &quot;meld&quot; when our Do and Sol are played together. This goes by the wonderful name of <em>Tonverschmelzung</em> in German. It is a very distinctive &quot;blending&quot; sound. If our Sol was tuned to, for example, 148 Hz, its second harmonic component would be at 296 Hz, and the two tones played together would not &quot;meld together&quot; at 300 Hz, but would &quot;beat&quot;. That is, we would hear a throbbing sound, the &quot;beat rate&quot; of which is found by reckoning the distance in Hertz between the two near-coincident partials. In this case, 300 - 296 = 4 Hz, so we'd hear a beating of four times a second (this is like a rhythm of eighth notes at a metronome marking of 120 beats per minute).<br />
<br />
One does not need to know of the harmonic series, nor even know how to read, or even count, to sing this.<br />
<br />
There is more to it than this, of course, but the basic principles of Just Intonation are very simple. Hundreds of years ago, when the intonation of a few well-known intervals were the concern, understanding and defining &quot;Just&quot; was not difficult. These days, though, and going on from these basics, it can get a bit more complicated...<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc1"><a name="Just Intonation in use"></a><!-- ws:end:WikiTextHeadingRule:14 -->Just Intonation in use</h1>
 <br />
To start off your exploration of just intonation scales, the <a class="wiki_link" href="/Gallery%20of%2012-tone%20Just%20Intonation%20Scales">Gallery of 12-tone Just Intonation Scales</a> is a good place to start.<br />
<br />
The use of just intonation could be divided into these two flavors:<br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc2"><a name="Just Intonation in use-Free Style Just"></a><!-- ws:end:WikiTextHeadingRule:16 -->Free Style Just</h2>
 <br />
<a class="wiki_link" href="/Lou%20Harrison">Lou Harrison</a> 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:18:&lt;h2&gt; --><h2 id="toc3"><a name="Just Intonation in use-Constrained Just"></a><!-- ws:end:WikiTextHeadingRule:18 -->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). See <a class="wiki_link" href="/3-limit">3-limit</a>, <a class="wiki_link" href="/5-limit">5-limit</a>, <a class="wiki_link" href="/7-limit">7-limit</a>, <a class="wiki_link" href="/11-limit">11-limit</a>, <a class="wiki_link" href="/13-limit">13-limit</a>.</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, incidentally, 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 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Hexany" rel="nofollow">products sets</a> of given arrays of prime numbers, such as <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Erv_Wilson" rel="nofollow">Ervin Wilson</a>, Robert Dussaut, 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 />
to this can be added<br />
<em>7. 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.</em><br />
<br />
<em>8. While related to the above, the use of recurrent sequences is by some included under JI as it involves whole numbers. Wilson's <a class="wiki_link_ext" href="http://anaphoria.com/wilsonintroMERU.html" rel="nofollow">Meru scales</a> are a good example.</em><br />
<br />
<!-- ws:start:WikiTextHeadingRule:20:&lt;h1&gt; --><h1 id="toc4"><a name="Variations on 'Just'"></a><!-- ws:end:WikiTextHeadingRule:20 -->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 />
<!-- ws:start:WikiTextHeadingRule:22:&lt;h1&gt; --><h1 id="toc5"><a name="Links"></a><!-- ws:end:WikiTextHeadingRule:22 -->Links</h1>
 <a class="wiki_link" href="/Gallery%20of%20Just%20Intervals">Gallery of Just Intervals</a><br />
<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="/Gallery%20of%20pentatonics">Gallery of pentatonics</a><br />
<a class="wiki_link" href="/List%20of%20root-3rd-P5%20triads%20in%20JI">List of root-3rd-P5 triads in JI</a><br />
<a class="wiki_link" href="/Hahn%20distance">Hahn distance</a><br />
<a class="wiki_link" href="/Arnold%20Dreyblatt">Arnold Dreyblatt</a><br />
<a class="wiki_link" href="/FiniteSubsetJI">FiniteSubsetJI</a><br />
<a class="wiki_link" href="/hypergenesis58.scl">58 note 11 limit JI scale - hyper-Partchian!</a><br />
<a class="wiki_link" href="/boogiewoogiescale">Boogie woogie scale</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:24:&lt;h1&gt; --><h1 id="toc6"><a name="Articles"></a><!-- ws:end:WikiTextHeadingRule:24 -->Articles</h1>
 <ul><li><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Just_intonation" rel="nofollow">Wikipedia article on just intonation</a></li><li><a class="wiki_link_ext" href="http://nowitzky.hostwebs.com/justint/" rel="nofollow">Just Intonation</a> by Mark Nowitzky <a class="wiki_link_ext" href="http://www.webcitation.org/5xeAm2lPL" rel="nofollow">Permalink</a></li><li><a class="wiki_link_ext" href="http://www.kylegann.com/tuning.html" rel="nofollow">Just Intonation Explained</a> by Kyle Gann <a class="wiki_link_ext" href="http://www.webcitation.org/5xe2iC7Nq" rel="nofollow">Permalink</a></li><li><a class="wiki_link_ext" href="http://www.kylegann.com/Octave.html" rel="nofollow">Anatomy of an Octave</a> by Kyle Gann <a class="wiki_link_ext" href="http://www.webcitation.org/5xe30LCev" rel="nofollow">Permalink</a></li><li><a class="wiki_link_ext" href="http://www.dbdoty.com/Words/What-is-Just-Intonation.html" rel="nofollow">What is Just Intonation?</a> by David B. Doty <a class="wiki_link_ext" href="http://www.webcitation.org/5xe3MeWVq" rel="nofollow">Permalink</a></li><li><a class="wiki_link_ext" href="http://lumma.org/tuning/faq/#whatisJI" rel="nofollow">What is &quot;just intonation&quot;?</a> by Carl Lumma <a class="wiki_link_ext" href="http://www.webcitation.org/65NwFAKLh" rel="nofollow">Permalink</a></li><li><a class="wiki_link_ext" href="http://www.dbdoty.com/Words/werntz.html" rel="nofollow">A Response to Julia Werntz</a> by David B. Doty <a class="wiki_link_ext" href="http://www.webcitation.org/5xe38KWx4" rel="nofollow">Permalink</a></li><li><a class="wiki_link_ext" href="http://lumma.org/tuning/gws/commaseq.htm" rel="nofollow">Comma Sequences</a> by Gene Ward Smith <a class="wiki_link_ext" href="http://www.webcitation.org/5xe4rPLZ0" rel="nofollow">Permalink</a></li></ul><br />
.<!-- ws:start:WikiTextReferencesRule:210: --><hr class="references" /><ol class="references">
<li id="cite_note-1"><a href="#cite_ref-1">^</a> Just Intonation is sometimes distinguished from <em>rational intonation,</em> by requiring that the ratios be lower than some arbitrary complexity (as for example measured by <a class="wiki_link" href="/Tenney%20height">Tenney height</a>, <a class="wiki_link" href="/Benedetti%20height">Benedetti height</a>, etc.) but there is no clear dividing line. The matter is partially a question of intent.<br />
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 however, it casually suggests a rank two 7-limit <a class="wiki_link" href="/Microtempering">microtempering</a> system because of 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 />
<br />
</li>
<li id="cite_note-2"><a href="#cite_ref-2">^</a> [2] All manner of bells, gongs, percussion instruments, synthesizer sounds, have spectra that follow their own rules, usually very complex. Inharmonic tones can be found in otherwise harmonic spectra, and instruments with harmonic spectra may have inharmonic spectra during the attack portion of the sound. Loudly played brass instruments, for example, have a moment of extremely complex sound not unlike that of striking a piece of metal, followed by a moment in which the partials are &quot;stretched&quot; according to a more complex rule than simply multiplying by, 1, 2, 3, etc., before settling down into a harmonic series accompanied by various amounts of characteristic &quot;noise&quot;. A breathily played flute has a large addition of inharmonic material, a &quot;jinashi&quot; shakuhachi flute is an excellent example of an instrument of varying harmonicity and inharmonicity.</li>
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+: This revision was by author [[User:spt3125|spt3125]] and made on <tt>2014-05-25 20:40:15 UTC</tt>.<br>
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 ----
 =Just Intonation explained=
-JI, or [[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 [[Benedetti 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]].
+[[Just Intonation]] (JI) 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.&lt;ref&gt;Just Intonation is sometimes distinguished from //rational intonation,// by requiring that the ratios be lower than some arbitrary complexity (as for example measured by [[Tenney height]], [[Benedetti height]], etc.) 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 however, it casually suggests a rank two 7-limit [[Microtempering|microtempering]] system because of 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]].
+&lt;/ref&gt;
-"Just Intonation", as we find it commonly used today, describes [[Gallery of Just Intervals|intervals]] between pitches by specifying ratios ([[http://en.wikipedia.org/wiki/Rational_number|rational numbers]]) of pitch frequencies.
+If you are used to speaking only in note names (e.g. the first 7 letters of the alphabet), 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]].
-In languages other than English, the original conceptions of "Just Intonation" are more obviously retained in the terms used in those languages: Reine Stimmung (pure, that is, beatless, tuning) in German, Натуральний стрій in Ukrainian and Gamme naturelle in French, (both referring to the "natural scale", that is, intervals derived from the harmonic series), Intonazione naturale (natural intonation, once again intervals derived from harmonic series) in Italian, and so on.
+In languages other than English, the original conceptions of "Just Intonation" are more obviously retained in the terms used in those languages: German //Reine Stimmung// (pure, that is, beatless, tuning), Ukrainian //Натуральний стрій// and French //Gamme naturelle// (both referring to the "natural scale", that is, intervals derived from the harmonic series), Italian //Intonazione naturale// (natural intonation, once again intervals derived from harmonic series), and so on.
 In the English language, the term "just" referred to "true, correct", and is still used today in this sense, in the crafts. In printing, to "justify" a line of type is to fit it precisely to a certain measure, for example. The original sense, then, was similar to that sense which is clearly retained in other languages as "natural".
@@ Line 25: / Line 26: @@
 But, first things first. Let us take a look at why the idea of a "natural" or "just" tuning came about, and is still with us.
-If we have a tone with a harmonic timbre and a fundamental frequency at 100 Hz, we will find the second harmonic component at 200 Hz, the third at 300 Hz, the fourth at 400 Hz...Yes, the harmonics are found at the fundamental frequency times 1, times 2, times 3...
+If we have a tone with a harmonic timbre and a fundamental frequency at 100 Hz (Hertz, or cycles per second), we will find the second harmonic component at 200 Hz, the third at 300 Hz, the fourth at 400 Hz...Yes, the harmonics are found at the fundamental frequency times 1, times 2, times 3...
-The simplicity of it all can be difficult to believe at first. You can easily imagine people discovering this and getting carried away with ideas of "music of the spheres" and other mystical ideas. Yes, it IS amazing. Please keep in mind that not all sounds have a harmonic spectrum.*
+The simplicity of it all can be difficult to believe at first. You can easily imagine people discovering this and getting carried away with ideas of "music of the spheres" and other mystical ideas. Yes, it IS amazing. Please keep in mind that not all sounds have a harmonic spectrum.&lt;ref&gt;[2] All manner of bells, gongs, percussion instruments, synthesizer sounds, have spectra that follow their own rules, usually very complex. Inharmonic tones can be found in otherwise harmonic spectra, and instruments with harmonic spectra may have inharmonic spectra during the attack portion of the sound. Loudly played brass instruments, for example, have a moment of extremely complex sound not unlike that of striking a piece of metal, followed by a moment in which the partials are "stretched" according to a more complex rule than simply multiplying by, 1, 2, 3, etc., before settling down into a harmonic series accompanied by various amounts of characteristic "noise". A breathily played flute has a large addition of inharmonic material, a "jinashi" shakuhachi flute is an excellent example of an instrument of varying harmonicity and inharmonicity.&lt;/ref&gt;
-Of course we are describing an ideal tone- in real life, tones waver, certain harmonics are missing, etc. Nevertheless this is the harmonic series, and measuring the spectra of violins (or any other stringed instruments), human voices, and woodwinds, for example, will reveal that this is indeed the pattern, and even in our "fuzzy" and "flawed" reality, spectra adhere to this pattern with impressive consistency.
+Of course we are describing an ideal tone - in real life, tones waver, certain harmonics are missing, etc. Nevertheless this is the harmonic series, and measuring the spectra of violins (or any other stringed instruments), human voices, and woodwinds, for example, will reveal that this is indeed the pattern, and even in our "fuzzy" and "flawed" reality, spectra adhere to this pattern with impressive consistency.
-In a tuning "according to the natural scale", we have for example a "perfect fifth" as simply the ratio between the third partial and the second partial: " 3:2". In our example tone, that would be the ratio of 300 Hz to 200 Hz. Where we to want a Just Intonation perfect fifth above our original tone, its fundamental frequency would be found at 3/2 times the fundamental frequency of our original tone. So, 3/2 times 100 gives us 150. Our example perfect fifth has a fundamental frequency at 150 Hz.
+In a tuning "according to the natural scale", we have for example a "perfect fifth" as simply the ratio between the third partial and the second partial: "3:2". In our example tone, that would be the ratio of 300 Hz to 200 Hz. Where we to want a Just Intonation perfect fifth above our original tone, its fundamental frequency would be found at 3/2 times the fundamental frequency of our original tone. So, 3/2 times 100 gives us 150. Our example perfect fifth has a fundamental frequency at 150 Hz.
-Now, let us play our two example tones together, and we shall see why the German term is "Reine", "pure", and why you'll hear "pure" used in English as well (and in Czech, Slovak, etc.) Let's call our first tone "Do" and our second tone, a perfect fifth higher, "Sol".
+Now, let us play our two example tones together, and we shall see why the German term is //Reine//, "pure", and why you'll hear "pure" used in English and many other languages as well. Let's call our first tone "Do" and our second tone, a perfect fifth higher, "Sol".
-Tone.... Frequencies of Partial Number
+[[code]]
-"Do" 100 200 300 400 500 600 700 800 900
+Tone   Frequencies of partials (Hz)
-"Sol" 150 300 450 600 750 900
+Do     100  200  300  400  500  600   700   800   900   ...
+So     150  300  450  600  750  900  1050  1200  1350   ...
+[[code]]
-You see that the tones share the frequencies of some of the partials. These partials will "meld" when our Do and Sol are played together. This goes by the wonderful name of "Tonverschmelzung" in German. It is a very distinctinctive "blending" sound. If our Sol was tuned to, for example, 148 Hertz, its second harmonic component would be at 296 Hertz, and the two tones played together would not "meld together" at 300 Hertz, but would "beat". That is, we would hear a throbbing sound, the "beat rate" of which is found by reckoning the distance in Hertz between the two near-coincident partials. In this case, 300-296=4 Hertz, so we'd hear a beating of four times a second (this is like a rhythm of eighth notes at a metronome marking of 120 beats per minute).
+You see that the tones share the frequencies of some of the partials. These partials will "meld" when our Do and Sol are played together. This goes by the wonderful name of //Tonverschmelzung// in German. It is a very distinctive "blending" sound. If our Sol was tuned to, for example, 148 Hz, its second harmonic component would be at 296 Hz, and the two tones played together would not "meld together" at 300 Hz, but would "beat". That is, we would hear a throbbing sound, the "beat rate" of which is found by reckoning the distance in Hertz between the two near-coincident partials. In this case, 300 - 296 = 4 Hz, so we'd hear a beating of four times a second (this is like a rhythm of eighth notes at a metronome marking of 120 beats per minute).
 One does not need to know of the harmonic series, nor even know how to read, or even count, to sing this.
@@ Line 45: / Line 48: @@
 There is more to it than this, of course, but the basic principles of Just Intonation are very simple. Hundreds of years ago, when the intonation of a few well-known intervals were the concern, understanding and defining "Just" was not difficult. These days, though, and going on from these basics, it can get a bit more complicated...
-If you are used to speaking only in note names (e.g. the first 7 letters of the alphabet), 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]].
-*All manner of bells, gongs, percussion instruments, synthesizer sounds, have spectra that follow their own rules, usually very complex. Inharmonic tones can be found in otherwise harmonic spectra, and instruments with harmonic spectra may have inharmonic spectra during the attack portion of the sound. Loudly played brass instruments, for example, have a moment of extremely complex sound not unlike that of striking a piece of metal, followed by a moment in which the partials are "stretched" according to a more complex rule than simply multiplying by, 1, 2, 3, etc., before settling down into a harmonic series accompanied by various amounts of characteristic "noise". A breathily played flute has a large addition of inharmonic material, a "jinashi" shakuhachi flute is an excellent example of an instrument of varying harmonicity and inharmonicity.
-="Pure"? "Rational"? Various shades of "Just" Intonation=
-Just Intonation is sometimes distinguished from //rational intonation,// by requiring that the ratios be lower than some arbitrary complexity (as for example measured by [[Tenney height]]).
-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 however, it casually suggests a rank two 7-limit [[Microtempering|microtempering]] system because of 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]].
 =Just Intonation in use=
-To start off your exploration of just intonation scales, [[Gallery of 12-tone Just Intonation Scales|this]] is a good place to start.
+To start off your exploration of just intonation scales, the [[Gallery of 12-tone Just Intonation Scales]] is a good place to start.
 The use of just intonation could be divided into these two flavors:
@@ Line 65: / Line 57: @@
 ==Free Style Just==
-[[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; [[FreeStyleJI]]
+[[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; [[FreeStyleJI]]
 ==Constrained Just==
@@ Line 72: / Line 64: @@
 //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). See [[3-limit]], [[5-limit]], [[7-limit]], [[11-limit]], [[13-limit]].//
-//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.//
+//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, incidentally, 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.//
-//3. Other theorists who, in contrast to the above, advocate the use of [[http://en.wikipedia.org/wiki/Hexany|products sets]] of given arrays of prime numbers, such as [[http://en.wikipedia.org/wiki/Erv_Wilson|Ervin Wilson]],////Robert Dussaut,// //and others.//
+//3. Other theorists who, in contrast to the above, advocate the use of [[http://en.wikipedia.org/wiki/Hexany|products sets]] of given arrays of prime numbers, such as [[http://en.wikipedia.org/wiki/Erv_Wilson|Ervin Wilson]], Robert Dussaut, and others.//
 //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.//
@@ Line 83: / Line 75: @@
 to this can be added
-//7. 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.//
+//7. 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.//
-//8. While related to the above, the use of recurrent sequences is by some included under JI as it involves whole numbers. Wilson's [[http://anaphoria.com/wilsonintroMERU.html|Meru scales]] are a good example as well as Jacques Dudon//
+//8. While related to the above, the use of recurrent sequences is by some included under JI as it involves whole numbers. Wilson's [[http://anaphoria.com/wilsonintroMERU.html|Meru scales]] are a good example.//
 =Variations on 'Just'=
@@ Line 92: / Line 84: @@
 =Links=
-[[hypergenesis58.scl|58 note 11 limit JI]] - hyper-Partchian!
-[[Hahn distance]]
 [[Gallery of Just Intervals]]
 [[Gallery of 12-tone Just Intonation Scales]]
-[[boogiewoogiescale|Boogie woogie scale]]
+[[Gallery of pentatonics]]
+[[List of root-3rd-P5 triads in JI]]
+[[Hahn distance]]
 [[Arnold Dreyblatt]]
-[[Gallery of pentatonics]]
 [[FiniteSubsetJI]]
-[[List of root-3rd-P5 triads in JI]]
+[[hypergenesis58.scl|58 note 11 limit JI scale - hyper-Partchian!]]
+[[boogiewoogiescale|Boogie woogie scale]]
 =Articles=
@@ Line 110: / Line 102: @@
 * [[http://lumma.org/tuning/faq/#whatisJI|What is "just intonation"?]] by Carl Lumma [[http://www.webcitation.org/65NwFAKLh|Permalink]]
 * [[http://www.dbdoty.com/Words/werntz.html|A Response to Julia Werntz]] by David B. Doty [[http://www.webcitation.org/5xe38KWx4|Permalink]]
-* [[http://lumma.org/tuning/gws/commaseq.htm|Comma Sequences]] by Gene Ward Smith [[http://www.webcitation.org/5xe4rPLZ0|Permalink]]</pre></div>
+* [[http://lumma.org/tuning/gws/commaseq.htm|Comma Sequences]] by Gene Ward Smith [[http://www.webcitation.org/5xe4rPLZ0|Permalink]]
+.</pre></div>
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-<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Just intonation&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:16:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt;&lt;a href="#Just Intonation explained"&gt;Just Intonation explained&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt; | &lt;a href="#x&amp;quot;Pure&amp;quot;? &amp;quot;Rational&amp;quot;? Various shades of &amp;quot;Just&amp;quot; Intonation"&gt;&amp;quot;Pure&amp;quot;? &amp;quot;Rational&amp;quot;? Various shades of &amp;quot;Just&amp;quot; Intonation&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt; | &lt;a href="#Just Intonation in use"&gt;Just Intonation in use&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;!-- ws:start:WikiTextTocRule:20: --&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt;&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt; | &lt;a href="#Variations on 'Just'"&gt;Variations on 'Just'&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt; | &lt;a href="#Links"&gt;Links&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt; | &lt;a href="#Articles"&gt;Articles&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt;
+<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Just intonation&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:26:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&gt;&lt;a href="#Just Intonation explained"&gt;Just Intonation explained&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&gt; | &lt;a href="#Just Intonation in use"&gt;Just Intonation in use&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt;&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt;&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt; | &lt;a href="#Variations on 'Just'"&gt;Variations on 'Just'&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:31 --&gt;&lt;!-- ws:start:WikiTextTocRule:32: --&gt; | &lt;a href="#Links"&gt;Links&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:32 --&gt;&lt;!-- ws:start:WikiTextTocRule:33: --&gt; | &lt;a href="#Articles"&gt;Articles&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:33 --&gt;&lt;!-- ws:start:WikiTextTocRule:34: --&gt;
-&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;hr /&gt;
+&lt;!-- ws:end:WikiTextTocRule:34 --&gt;&lt;hr /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Just Intonation explained"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Just Intonation explained&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Just Intonation explained"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Just Intonation explained&lt;/h1&gt;
-  JI, or &lt;a class="wiki_link" href="/Just%20Intonation"&gt;Just Intonation&lt;/a&gt; describes &lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&gt;intervals&lt;/a&gt; between pitches by specifying ratios (of &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Rational_number" rel="nofollow"&gt;rational numbers&lt;/a&gt;) between the frequencies of pitches. This is sometimes distinguished from &lt;em&gt;rational intonation&lt;/em&gt; by requiring that the ratios be ones of low complexity (as for example measured by &lt;a class="wiki_link" href="/Benedetti%20height"&gt;Benedetti height&lt;/a&gt;) 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 &lt;a class="wiki_link" href="/Microtempering"&gt;microtempering&lt;/a&gt; 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 &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_minor_third" rel="nofollow"&gt;septimal minor third&lt;/a&gt;.&lt;br /&gt;
+  &lt;a class="wiki_link" href="/Just%20Intonation"&gt;Just Intonation&lt;/a&gt; (JI) describes &lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&gt;intervals&lt;/a&gt; between pitches by specifying ratios (of &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Rational_number" rel="nofollow"&gt;rational numbers&lt;/a&gt;) between the frequencies of pitches.&lt;!-- ws:start:WikiTextRefRule:10:&amp;amp;lt;ref&amp;amp;gt;Just Intonation is sometimes distinguished from &amp;lt;em&amp;gt;rational intonation,&amp;lt;/em&amp;gt; by requiring that the ratios be lower than some arbitrary complexity (as for example measured by &amp;lt;a class=&amp;quot;wiki_link&amp;quot; href=&amp;quot;/Tenney%20height&amp;quot;&amp;gt;Tenney height&amp;lt;/a&amp;gt;, &amp;lt;a class=&amp;quot;wiki_link&amp;quot; href=&amp;quot;/Benedetti%20height&amp;quot;&amp;gt;Benedetti height&amp;lt;/a&amp;gt;, etc.) but there is no clear dividing line. The matter is partially a question of intent.&amp;lt;br /&amp;gt;
+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 however, it casually suggests a rank two 7-limit &amp;lt;a class=&amp;quot;wiki_link&amp;quot; href=&amp;quot;/Microtempering&amp;quot;&amp;gt;microtempering&amp;lt;/a&amp;gt; system because of 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 &amp;lt;a class=&amp;quot;wiki_link_ext&amp;quot; href=&amp;quot;http://en.wikipedia.org/wiki/Septimal_minor_third&amp;quot; rel=&amp;quot;nofollow&amp;quot;&amp;gt;septimal minor third&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
+&amp;lt;br /&amp;gt;
+&amp;amp;lt;/ref&amp;amp;gt; --&gt;&lt;sup id="cite_ref-1" class="reference"&gt;&lt;a href="#cite_note-1"&gt;[1]&lt;/a&gt;&lt;/sup&gt;&lt;!-- ws:end:WikiTextRefRule:10 --&gt;&lt;br /&gt;
 &lt;br /&gt;
-If you are used to speaking only in note names, you may need to study the relation between frequency and &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Pitch_%28music%29" rel="nofollow"&gt;pitch&lt;/a&gt;. Kyle Gann's &lt;em&gt;&lt;a class="wiki_link_ext" href="http://www.kylegann.com/tuning.html" rel="nofollow"&gt;Just Intonation Explained&lt;/a&gt;&lt;/em&gt; is one good reference. A transparent illustration and one of just intonation's acoustic bases is the &lt;a class="wiki_link" href="/OverToneSeries"&gt;harmonic series&lt;/a&gt;.&lt;br /&gt;
+If you are used to speaking only in note names (e.g. the first 7 letters of the alphabet), you may need to study the relation between frequency and &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Pitch_%28music%29" rel="nofollow"&gt;pitch&lt;/a&gt;. Kyle Gann's &lt;em&gt;&lt;a class="wiki_link_ext" href="http://www.kylegann.com/tuning.html" rel="nofollow"&gt;Just Intonation Explained&lt;/a&gt;&lt;/em&gt; is one good reference. A transparent illustration and one of just intonation's acoustic bases is the &lt;a class="wiki_link" href="/OverToneSeries"&gt;harmonic series&lt;/a&gt;.&lt;br /&gt;
-&lt;br /&gt;
-&amp;quot;Just Intonation&amp;quot;, as we find it commonly used today, describes &lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&gt;intervals&lt;/a&gt; between pitches by specifying ratios (&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Rational_number" rel="nofollow"&gt;rational numbers&lt;/a&gt;) of pitch frequencies.&lt;br /&gt;
 &lt;br /&gt;
-In languages other than English, the original conceptions of &amp;quot;Just Intonation&amp;quot; are more obviously retained in the terms used in those languages: Reine Stimmung (pure, that is, beatless, tuning) in German, Натуральний стрій in Ukrainian and Gamme naturelle in French, (both referring to the &amp;quot;natural scale&amp;quot;, that is, intervals derived from the harmonic series), Intonazione naturale (natural intonation, once again intervals derived from harmonic series) in Italian, and so on.&lt;br /&gt;
+In languages other than English, the original conceptions of &amp;quot;Just Intonation&amp;quot; are more obviously retained in the terms used in those languages: German &lt;em&gt;Reine Stimmung&lt;/em&gt; (pure, that is, beatless, tuning), Ukrainian &lt;em&gt;Натуральний стрій&lt;/em&gt; and French &lt;em&gt;Gamme naturelle&lt;/em&gt; (both referring to the &amp;quot;natural scale&amp;quot;, that is, intervals derived from the harmonic series), Italian &lt;em&gt;Intonazione naturale&lt;/em&gt; (natural intonation, once again intervals derived from harmonic series), and so on.&lt;br /&gt;
 &lt;br /&gt;
 In the English language, the term &amp;quot;just&amp;quot; referred to &amp;quot;true, correct&amp;quot;, and is still used today in this sense, in the crafts. In printing, to &amp;quot;justify&amp;quot; a line of type is to fit it precisely to a certain measure, for example. The original sense, then, was similar to that sense which is clearly retained in other languages as &amp;quot;natural&amp;quot;.&lt;br /&gt;
@@ Line 131: / Line 126: @@
 But, first things first. Let us take a look at why the idea of a &amp;quot;natural&amp;quot; or &amp;quot;just&amp;quot; tuning came about, and is still with us.&lt;br /&gt;
 &lt;br /&gt;
-If we have a tone with a harmonic timbre and a fundamental frequency at 100 Hz, we will find the second harmonic component at 200 Hz, the third at 300 Hz, the fourth at 400 Hz...Yes, the harmonics are found at the fundamental frequency times 1, times 2, times 3...&lt;br /&gt;
+If we have a tone with a harmonic timbre and a fundamental frequency at 100 Hz (Hertz, or cycles per second), we will find the second harmonic component at 200 Hz, the third at 300 Hz, the fourth at 400 Hz...Yes, the harmonics are found at the fundamental frequency times 1, times 2, times 3...&lt;br /&gt;
 &lt;br /&gt;
-The simplicity of it all can be difficult to believe at first. You can easily imagine people discovering this and getting carried away with ideas of &amp;quot;music of the spheres&amp;quot; and other mystical ideas. Yes, it IS amazing. Please keep in mind that not all sounds have a harmonic spectrum.*&lt;br /&gt;
+The simplicity of it all can be difficult to believe at first. You can easily imagine people discovering this and getting carried away with ideas of &amp;quot;music of the spheres&amp;quot; and other mystical ideas. Yes, it IS amazing. Please keep in mind that not all sounds have a harmonic spectrum.&lt;!-- ws:start:WikiTextRefRule:11:&amp;amp;lt;ref&amp;amp;gt;[2] All manner of bells, gongs, percussion instruments, synthesizer sounds, have spectra that follow their own rules, usually very complex. Inharmonic tones can be found in otherwise harmonic spectra, and instruments with harmonic spectra may have inharmonic spectra during the attack portion of the sound. Loudly played brass instruments, for example, have a moment of extremely complex sound not unlike that of striking a piece of metal, followed by a moment in which the partials are &amp;amp;quot;stretched&amp;amp;quot; according to a more complex rule than simply multiplying by, 1, 2, 3, etc., before settling down into a harmonic series accompanied by various amounts of characteristic &amp;amp;quot;noise&amp;amp;quot;. A breathily played flute has a large addition of inharmonic material, a &amp;amp;quot;jinashi&amp;amp;quot; shakuhachi flute is an excellent example of an instrument of varying harmonicity and inharmonicity.&amp;amp;lt;/ref&amp;amp;gt; --&gt;&lt;sup id="cite_ref-2" class="reference"&gt;&lt;a href="#cite_note-2"&gt;[2]&lt;/a&gt;&lt;/sup&gt;&lt;!-- ws:end:WikiTextRefRule:11 --&gt;&lt;br /&gt;
 &lt;br /&gt;
-Of course we are describing an ideal tone- in real life, tones waver, certain harmonics are missing, etc. Nevertheless this is the harmonic series, and measuring the spectra of violins (or any other stringed instruments), human voices, and woodwinds, for example, will reveal that this is indeed the pattern, and even in our &amp;quot;fuzzy&amp;quot; and &amp;quot;flawed&amp;quot; reality, spectra adhere to this pattern with impressive consistency.&lt;br /&gt;
+Of course we are describing an ideal tone - in real life, tones waver, certain harmonics are missing, etc. Nevertheless this is the harmonic series, and measuring the spectra of violins (or any other stringed instruments), human voices, and woodwinds, for example, will reveal that this is indeed the pattern, and even in our &amp;quot;fuzzy&amp;quot; and &amp;quot;flawed&amp;quot; reality, spectra adhere to this pattern with impressive consistency.&lt;br /&gt;
 &lt;br /&gt;
-In a tuning &amp;quot;according to the natural scale&amp;quot;, we have for example a &amp;quot;perfect fifth&amp;quot; as simply the ratio between the third partial and the second partial: &amp;quot; 3:2&amp;quot;. In our example tone, that would be the ratio of 300 Hz to 200 Hz. Where we to want a Just Intonation perfect fifth above our original tone, its fundamental frequency would be found at 3/2 times the fundamental frequency of our original tone. So, 3/2 times 100 gives us 150. Our example perfect fifth has a fundamental frequency at 150 Hz.&lt;br /&gt;
+In a tuning &amp;quot;according to the natural scale&amp;quot;, we have for example a &amp;quot;perfect fifth&amp;quot; as simply the ratio between the third partial and the second partial: &amp;quot;3:2&amp;quot;. In our example tone, that would be the ratio of 300 Hz to 200 Hz. Where we to want a Just Intonation perfect fifth above our original tone, its fundamental frequency would be found at 3/2 times the fundamental frequency of our original tone. So, 3/2 times 100 gives us 150. Our example perfect fifth has a fundamental frequency at 150 Hz.&lt;br /&gt;
 &lt;br /&gt;
-Now, let us play our two example tones together, and we shall see why the German term is &amp;quot;Reine&amp;quot;, &amp;quot;pure&amp;quot;, and why you'll hear &amp;quot;pure&amp;quot; used in English as well (and in Czech, Slovak, etc.) Let's call our first tone &amp;quot;Do&amp;quot; and our second tone, a perfect fifth higher, &amp;quot;Sol&amp;quot;.&lt;br /&gt;
+Now, let us play our two example tones together, and we shall see why the German term is &lt;em&gt;Reine&lt;/em&gt;, &amp;quot;pure&amp;quot;, and why you'll hear &amp;quot;pure&amp;quot; used in English and many other languages as well. Let's call our first tone &amp;quot;Do&amp;quot; and our second tone, a perfect fifth higher, &amp;quot;Sol&amp;quot;.&lt;br /&gt;
 &lt;br /&gt;
-Tone.... Frequencies of Partial Number&lt;br /&gt;
+&lt;!-- ws:start:WikiTextCodeRule:0:
-&amp;quot;Do&amp;quot; 100 200 300 400 500 600 700 800 900&lt;br /&gt;
+&amp;lt;pre class=&amp;quot;text&amp;quot;&amp;gt;Tone   Frequencies of partials (Hz)&amp;lt;br/&amp;gt;Do     100  200  300  400  500  600   700   800   900   ...&amp;lt;br/&amp;gt;So     150  300  450  600  750  900  1050  1200  1350   ...&amp;lt;/pre&amp;gt;
-&amp;quot;Sol&amp;quot; 150 300 450 600 750 900&lt;br /&gt;
+ --&gt;
-&lt;br /&gt;
+&lt;style type="text/css"&gt;&lt;!--
-You see that the tones share the frequencies of some of the partials. These partials will &amp;quot;meld&amp;quot; when our Do and Sol are played together. This goes by the wonderful name of &amp;quot;Tonverschmelzung&amp;quot; in German. It is a very distinctinctive &amp;quot;blending&amp;quot; sound. If our Sol was tuned to, for example, 148 Hertz, its second harmonic component would be at 296 Hertz, and the two tones played together would not &amp;quot;meld together&amp;quot; at 300 Hertz, but would &amp;quot;beat&amp;quot;. That is, we would hear a throbbing sound, the &amp;quot;beat rate&amp;quot; of which is found by reckoning the distance in Hertz between the two near-coincident partials. In this case, 300-296=4 Hertz, so we'd hear a beating of four times a second (this is like a rhythm of eighth notes at a metronome marking of 120 beats per minute).&lt;br /&gt;
+/**
+ * GeSHi (C) 2004 - 2007 Nigel McNie, 2007 - 2008 Benny Baumann
+ * (http://qbnz.com/highlighter/ and http://geshi.org/)
+ */
+.text  {font-family:monospace;}
+.text .imp {font-weight: bold; color: red;}
+.text span.xtra { display:block; }
+--&gt;
+&lt;/style&gt;&lt;pre class="text"&gt;Tone   Frequencies of partials (Hz)
+Do     100  200  300  400  500  600   700   800   900   ...
+So     150  300  450  600  750  900  1050  1200  1350   ...&lt;/pre&gt;
+&lt;!-- ws:end:WikiTextCodeRule:0 --&gt;&lt;br /&gt;
+You see that the tones share the frequencies of some of the partials. These partials will &amp;quot;meld&amp;quot; when our Do and Sol are played together. This goes by the wonderful name of &lt;em&gt;Tonverschmelzung&lt;/em&gt; in German. It is a very distinctive &amp;quot;blending&amp;quot; sound. If our Sol was tuned to, for example, 148 Hz, its second harmonic component would be at 296 Hz, and the two tones played together would not &amp;quot;meld together&amp;quot; at 300 Hz, but would &amp;quot;beat&amp;quot;. That is, we would hear a throbbing sound, the &amp;quot;beat rate&amp;quot; of which is found by reckoning the distance in Hertz between the two near-coincident partials. In this case, 300 - 296 = 4 Hz, so we'd hear a beating of four times a second (this is like a rhythm of eighth notes at a metronome marking of 120 beats per minute).&lt;br /&gt;
 &lt;br /&gt;
 One does not need to know of the harmonic series, nor even know how to read, or even count, to sing this.&lt;br /&gt;
@@ Line 151: / Line 160: @@
 There is more to it than this, of course, but the basic principles of Just Intonation are very simple. Hundreds of years ago, when the intonation of a few well-known intervals were the concern, understanding and defining &amp;quot;Just&amp;quot; was not difficult. These days, though, and going on from these basics, it can get a bit more complicated...&lt;br /&gt;
 &lt;br /&gt;
-If you are used to speaking only in note names (e.g. the first 7 letters of the alphabet), you may need to study the relation between frequency and &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Pitch_%28music%29" rel="nofollow"&gt;pitch&lt;/a&gt;. Kyle Gann's &lt;em&gt;&lt;a class="wiki_link_ext" href="http://www.kylegann.com/tuning.html" rel="nofollow"&gt;Just Intonation Explained&lt;/a&gt;&lt;/em&gt; is one good reference. A transparent illustration and one of just intonation's acoustic bases is the &lt;a class="wiki_link" href="/OverToneSeries"&gt;harmonic series&lt;/a&gt;.&lt;br /&gt;
-&lt;br /&gt;
-*All manner of bells, gongs, percussion instruments, synthesizer sounds, have spectra that follow their own rules, usually very complex. Inharmonic tones can be found in otherwise harmonic spectra, and instruments with harmonic spectra may have inharmonic spectra during the attack portion of the sound. Loudly played brass instruments, for example, have a moment of extremely complex sound not unlike that of striking a piece of metal, followed by a moment in which the partials are &amp;quot;stretched&amp;quot; according to a more complex rule than simply multiplying by, 1, 2, 3, etc., before settling down into a harmonic series accompanied by various amounts of characteristic &amp;quot;noise&amp;quot;. A breathily played flute has a large addition of inharmonic material, a &amp;quot;jinashi&amp;quot; shakuhachi flute is an excellent example of an instrument of varying harmonicity and inharmonicity.&lt;br /&gt;
 &lt;br /&gt;
-&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Just Intonation in use"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Just Intonation in use&lt;/h1&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="x&amp;quot;Pure&amp;quot;? &amp;quot;Rational&amp;quot;? Various shades of &amp;quot;Just&amp;quot; Intonation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;&amp;quot;Pure&amp;quot;? &amp;quot;Rational&amp;quot;? Various shades of &amp;quot;Just&amp;quot; Intonation&lt;/h1&gt;
   &lt;br /&gt;
-Just Intonation is sometimes distinguished from &lt;em&gt;rational intonation,&lt;/em&gt; by requiring that the ratios be lower than some arbitrary complexity (as for example measured by &lt;a class="wiki_link" href="/Tenney%20height"&gt;Tenney height&lt;/a&gt;).&lt;br /&gt;
+To start off your exploration of just intonation scales, the &lt;a class="wiki_link" href="/Gallery%20of%2012-tone%20Just%20Intonation%20Scales"&gt;Gallery of 12-tone Just Intonation Scales&lt;/a&gt; is a good place to start.&lt;br /&gt;
-&lt;br /&gt;
-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 however, it casually suggests a rank two 7-limit &lt;a class="wiki_link" href="/Microtempering"&gt;microtempering&lt;/a&gt; system because of 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 &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_minor_third" rel="nofollow"&gt;septimal minor third&lt;/a&gt;.&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Just Intonation in use"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Just Intonation in use&lt;/h1&gt;
- &lt;br /&gt;
-To start off your exploration of just intonation scales, &lt;a class="wiki_link" href="/Gallery%20of%2012-tone%20Just%20Intonation%20Scales"&gt;this&lt;/a&gt; is a good place to start.&lt;br /&gt;
 &lt;br /&gt;
 The use of just intonation could be divided into these two flavors:&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Just Intonation in use-Free Style Just"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Free Style Just&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Just Intonation in use-Free Style Just"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Free Style Just&lt;/h2&gt;
   &lt;br /&gt;
-&lt;a class="wiki_link" href="/Lou%20Harrison"&gt;Lou Harrison&lt;/a&gt; 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 -&amp;gt; &lt;a class="wiki_link" href="/FreeStyleJI"&gt;FreeStyleJI&lt;/a&gt;&lt;br /&gt;
+&lt;a class="wiki_link" href="/Lou%20Harrison"&gt;Lou Harrison&lt;/a&gt; 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 -&amp;gt; &lt;a class="wiki_link" href="/FreeStyleJI"&gt;FreeStyleJI&lt;/a&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Just Intonation in use-Constrained Just"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Constrained Just&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Just Intonation in use-Constrained Just"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Constrained Just&lt;/h2&gt;
   (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, &amp;quot;Differential Coherence&amp;quot;, &lt;em&gt;1/1&lt;/em&gt; vol. 11, no. 2: p.1):&lt;br /&gt;
 &lt;br /&gt;
 &lt;em&gt;1. The principle of &amp;quot;&lt;a class="wiki_link" href="/Harmonic%20Limit"&gt;harmonic limits&lt;/a&gt;,&amp;quot; which sets a threshold in order to place a limit on the largest prime number in any ratio (cf: Tanner's &amp;quot;psycharithmes&amp;quot; and his ordering by complexity; Gioseffe Zarlino's five-limit &amp;quot;senario,&amp;quot; and the like; Helmholtz's theory of consonance with its &amp;quot;blending of partials,&amp;quot; which, like the others, results in giving priority to the lowest prime numbers). See &lt;a class="wiki_link" href="/3-limit"&gt;3-limit&lt;/a&gt;, &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt;, &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt;, &lt;a class="wiki_link" href="/11-limit"&gt;11-limit&lt;/a&gt;, &lt;a class="wiki_link" href="/13-limit"&gt;13-limit&lt;/a&gt;.&lt;/em&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;em&gt;2. Restrictions on the combinations of numbers that make up the numerator and denominator of the ratios under consideration, such as the &amp;quot;monophonic&amp;quot; system of &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Harry_Partch" rel="nofollow"&gt;Harry Partch&lt;/a&gt;'s &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Pitch_%28music%29" rel="nofollow"&gt;tonality diamond&lt;/a&gt;. 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.&lt;/em&gt;&lt;br /&gt;
+&lt;em&gt;2. Restrictions on the combinations of numbers that make up the numerator and denominator of the ratios under consideration, such as the &amp;quot;monophonic&amp;quot; system of &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Harry_Partch" rel="nofollow"&gt;Harry Partch&lt;/a&gt;'s &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Pitch_%28music%29" rel="nofollow"&gt;tonality diamond&lt;/a&gt;. This, incidentally, 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.&lt;/em&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;em&gt;3. Other theorists who, in contrast to the above, advocate the use of &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Hexany" rel="nofollow"&gt;products sets&lt;/a&gt; of given arrays of prime numbers, such as &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Erv_Wilson" rel="nofollow"&gt;Ervin Wilson&lt;/a&gt;,&lt;/em&gt;&lt;em&gt;Robert Dussaut,&lt;/em&gt; &lt;em&gt;and others.&lt;/em&gt;&lt;br /&gt;
+&lt;em&gt;3. Other theorists who, in contrast to the above, advocate the use of &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Hexany" rel="nofollow"&gt;products sets&lt;/a&gt; of given arrays of prime numbers, such as &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Erv_Wilson" rel="nofollow"&gt;Ervin Wilson&lt;/a&gt;, Robert Dussaut, and others.&lt;/em&gt;&lt;br /&gt;
 &lt;br /&gt;
 &lt;em&gt;4. &lt;a class="wiki_link" href="/Just%20intonation%20subgroups"&gt;Restrictions on the variety of prime numbers&lt;/a&gt; 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.&lt;/em&gt;&lt;br /&gt;
@@ Line 189: / Line 187: @@
 &lt;br /&gt;
 to this can be added&lt;br /&gt;
-&lt;em&gt;7. The use of&lt;/em&gt; &lt;em&gt;harmonic and arithmetic&lt;/em&gt; &lt;em&gt;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.&lt;/em&gt;&lt;br /&gt;
+&lt;em&gt;7. 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.&lt;/em&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;em&gt;8. While related to the above, the use of recurrent sequences is by some included under JI as it involves whole numbers. Wilson's &lt;a class="wiki_link_ext" href="http://anaphoria.com/wilsonintroMERU.html" rel="nofollow"&gt;Meru scales&lt;/a&gt; are a good example as well as Jacques Dudon&lt;/em&gt;&lt;br /&gt;
+&lt;em&gt;8. While related to the above, the use of recurrent sequences is by some included under JI as it involves whole numbers. Wilson's &lt;a class="wiki_link_ext" href="http://anaphoria.com/wilsonintroMERU.html" rel="nofollow"&gt;Meru scales&lt;/a&gt; are a good example.&lt;/em&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Variations on 'Just'"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Variations on 'Just'&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Variations on 'Just'"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Variations on 'Just'&lt;/h1&gt;
   &lt;a class="wiki_link" href="/Regular%20Temperaments"&gt;Regular Temperaments&lt;/a&gt; are just intonation systems of various &lt;a class="wiki_link" href="/harmonic%20limits"&gt;harmonic limits&lt;/a&gt; with certain commas 'tempered out'&lt;br /&gt;
 &lt;a class="wiki_link" href="/AdaptiveJI"&gt;Adaptive JI&lt;/a&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Links"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Links&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Links"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;Links&lt;/h1&gt;
-  &lt;a class="wiki_link" href="/hypergenesis58.scl"&gt;58 note 11 limit JI&lt;/a&gt; - hyper-Partchian!&lt;br /&gt;
+  &lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&gt;Gallery of Just Intervals&lt;/a&gt;&lt;br /&gt;
+&lt;a class="wiki_link" href="/Gallery%20of%2012-tone%20Just%20Intonation%20Scales"&gt;Gallery of 12-tone Just Intonation Scales&lt;/a&gt;&lt;br /&gt;
+&lt;a class="wiki_link" href="/Gallery%20of%20pentatonics"&gt;Gallery of pentatonics&lt;/a&gt;&lt;br /&gt;
+&lt;a class="wiki_link" href="/List%20of%20root-3rd-P5%20triads%20in%20JI"&gt;List of root-3rd-P5 triads in JI&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link" href="/Hahn%20distance"&gt;Hahn distance&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&gt;Gallery of Just Intervals&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link" href="/Gallery%20of%2012-tone%20Just%20Intonation%20Scales"&gt;Gallery of 12-tone Just Intonation Scales&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link" href="/boogiewoogiescale"&gt;Boogie woogie scale&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link" href="/Arnold%20Dreyblatt"&gt;Arnold Dreyblatt&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link" href="/Gallery%20of%20pentatonics"&gt;Gallery of pentatonics&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link" href="/FiniteSubsetJI"&gt;FiniteSubsetJI&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link" href="/List%20of%20root-3rd-P5%20triads%20in%20JI"&gt;List of root-3rd-P5 triads in JI&lt;/a&gt;&lt;br /&gt;
+&lt;a class="wiki_link" href="/hypergenesis58.scl"&gt;58 note 11 limit JI scale - hyper-Partchian!&lt;/a&gt;&lt;br /&gt;
+&lt;a class="wiki_link" href="/boogiewoogiescale"&gt;Boogie woogie scale&lt;/a&gt;&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Articles"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;Articles&lt;/h1&gt;
+ &lt;ul&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Just_intonation" rel="nofollow"&gt;Wikipedia article on just intonation&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://nowitzky.hostwebs.com/justint/" rel="nofollow"&gt;Just Intonation&lt;/a&gt; by Mark Nowitzky &lt;a class="wiki_link_ext" href="http://www.webcitation.org/5xeAm2lPL" rel="nofollow"&gt;Permalink&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://www.kylegann.com/tuning.html" rel="nofollow"&gt;Just Intonation Explained&lt;/a&gt; by Kyle Gann &lt;a class="wiki_link_ext" href="http://www.webcitation.org/5xe2iC7Nq" rel="nofollow"&gt;Permalink&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://www.kylegann.com/Octave.html" rel="nofollow"&gt;Anatomy of an Octave&lt;/a&gt; by Kyle Gann &lt;a class="wiki_link_ext" href="http://www.webcitation.org/5xe30LCev" rel="nofollow"&gt;Permalink&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://www.dbdoty.com/Words/What-is-Just-Intonation.html" rel="nofollow"&gt;What is Just Intonation?&lt;/a&gt; by David B. Doty &lt;a class="wiki_link_ext" href="http://www.webcitation.org/5xe3MeWVq" rel="nofollow"&gt;Permalink&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://lumma.org/tuning/faq/#whatisJI" rel="nofollow"&gt;What is &amp;quot;just intonation&amp;quot;?&lt;/a&gt; by Carl Lumma &lt;a class="wiki_link_ext" href="http://www.webcitation.org/65NwFAKLh" rel="nofollow"&gt;Permalink&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://www.dbdoty.com/Words/werntz.html" rel="nofollow"&gt;A Response to Julia Werntz&lt;/a&gt; by David B. Doty &lt;a class="wiki_link_ext" href="http://www.webcitation.org/5xe38KWx4" rel="nofollow"&gt;Permalink&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://lumma.org/tuning/gws/commaseq.htm" rel="nofollow"&gt;Comma Sequences&lt;/a&gt; by Gene Ward Smith &lt;a class="wiki_link_ext" href="http://www.webcitation.org/5xe4rPLZ0" rel="nofollow"&gt;Permalink&lt;/a&gt;&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
+.&lt;!-- ws:start:WikiTextReferencesRule:210: --&gt;&lt;hr class="references" /&gt;&lt;ol class="references"&gt;
+&lt;li id="cite_note-1"&gt;&lt;a href="#cite_ref-1"&gt;^&lt;/a&gt; Just Intonation is sometimes distinguished from &lt;em&gt;rational intonation,&lt;/em&gt; by requiring that the ratios be lower than some arbitrary complexity (as for example measured by &lt;a class="wiki_link" href="/Tenney%20height"&gt;Tenney height&lt;/a&gt;, &lt;a class="wiki_link" href="/Benedetti%20height"&gt;Benedetti height&lt;/a&gt;, etc.) but there is no clear dividing line. The matter is partially a question of intent.&lt;br /&gt;
+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 however, it casually suggests a rank two 7-limit &lt;a class="wiki_link" href="/Microtempering"&gt;microtempering&lt;/a&gt; system because of 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 &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_minor_third" rel="nofollow"&gt;septimal minor third&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Articles"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Articles&lt;/h1&gt;
+&lt;/li&gt;
- &lt;ul&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Just_intonation" rel="nofollow"&gt;Wikipedia article on just intonation&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://nowitzky.hostwebs.com/justint/" rel="nofollow"&gt;Just Intonation&lt;/a&gt; by Mark Nowitzky &lt;a class="wiki_link_ext" href="http://www.webcitation.org/5xeAm2lPL" rel="nofollow"&gt;Permalink&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://www.kylegann.com/tuning.html" rel="nofollow"&gt;Just Intonation Explained&lt;/a&gt; by Kyle Gann &lt;a class="wiki_link_ext" href="http://www.webcitation.org/5xe2iC7Nq" rel="nofollow"&gt;Permalink&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://www.kylegann.com/Octave.html" rel="nofollow"&gt;Anatomy of an Octave&lt;/a&gt; by Kyle Gann &lt;a class="wiki_link_ext" href="http://www.webcitation.org/5xe30LCev" rel="nofollow"&gt;Permalink&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://www.dbdoty.com/Words/What-is-Just-Intonation.html" rel="nofollow"&gt;What is Just Intonation?&lt;/a&gt; by David B. Doty &lt;a class="wiki_link_ext" href="http://www.webcitation.org/5xe3MeWVq" rel="nofollow"&gt;Permalink&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://lumma.org/tuning/faq/#whatisJI" rel="nofollow"&gt;What is &amp;quot;just intonation&amp;quot;?&lt;/a&gt; by Carl Lumma &lt;a class="wiki_link_ext" href="http://www.webcitation.org/65NwFAKLh" rel="nofollow"&gt;Permalink&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://www.dbdoty.com/Words/werntz.html" rel="nofollow"&gt;A Response to Julia Werntz&lt;/a&gt; by David B. Doty &lt;a class="wiki_link_ext" href="http://www.webcitation.org/5xe38KWx4" rel="nofollow"&gt;Permalink&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://lumma.org/tuning/gws/commaseq.htm" rel="nofollow"&gt;Comma Sequences&lt;/a&gt; by Gene Ward Smith &lt;a class="wiki_link_ext" href="http://www.webcitation.org/5xe4rPLZ0" rel="nofollow"&gt;Permalink&lt;/a&gt;&lt;/li&gt;&lt;/ul&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
+&lt;li id="cite_note-2"&gt;&lt;a href="#cite_ref-2"&gt;^&lt;/a&gt; [2] All manner of bells, gongs, percussion instruments, synthesizer sounds, have spectra that follow their own rules, usually very complex. Inharmonic tones can be found in otherwise harmonic spectra, and instruments with harmonic spectra may have inharmonic spectra during the attack portion of the sound. Loudly played brass instruments, for example, have a moment of extremely complex sound not unlike that of striking a piece of metal, followed by a moment in which the partials are &amp;quot;stretched&amp;quot; according to a more complex rule than simply multiplying by, 1, 2, 3, etc., before settling down into a harmonic series accompanied by various amounts of characteristic &amp;quot;noise&amp;quot;. A breathily played flute has a large addition of inharmonic material, a &amp;quot;jinashi&amp;quot; shakuhachi flute is an excellent example of an instrument of varying harmonicity and inharmonicity.&lt;/li&gt;
+&lt;/ol&gt;&lt;!-- ws:end:WikiTextReferencesRule:210 --&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

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