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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | {{Interwiki |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| | | en = 64/63 |
| : This revision was by author [[User:hstraub|hstraub]] and made on <tt>2013-05-24 06:36:22 UTC</tt>.<br> | | | de = 64/63 |
| : The original revision id was <tt>434004772</tt>.<br>
| | }} |
| : The revision comment was: <tt></tt><br>
| | : ''"Septimal comma" redirects here. For non-idiomatic usages, see [[Septimal]] and [[Comma]].'' |
| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
| | {{Infobox Interval |
| <h4>Original Wikitext content:</h4>
| | | Name = septimal comma, Archytas' comma |
| <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">The interval 64/63, called **septimal** or **Archytas comma** (in german **Leipziger Komma**), is a [[xenharmonic/superparticular|superparticular ratio]] which equates [[xenharmonic/9_8|9/8]] and [[xenharmonic/8_7|8/7]] if tempered out and has the eighth square number as a numerator. It also equates [[xenharmonic/7_4|7/4]] with [[xenharmonic/16_9|16/9]], so that the just dominant seventh chord, 1-5/4-3/2-16/9, and the otonal tetrad, 1-5/4-3/2-7/4, are equated to the same chord when 64/63 is tempered out. Equal divisions of the octave tempering out 64/63 include 12, 15, 22, 27, 37, 49 and 59.
| | | Color name = r1, ru unison,<br/>rM, ruma |
| | | Sound = Ji-64-63-csound-foscil-220hz.mp3 |
| | | Comma = yes |
| | }} |
| | {{Wikipedia|Septimal comma}} |
| | '''64/63''', the '''septimal comma''' (also '''Archytas' comma''', or more simply and systematically the '''archytas comma''' or '''archy comma'''), is a [[small comma|small]] [[7-limit]] [[superparticular]] [[comma]] which separates [[9/8]] and [[8/7]] and has the eighth square number as a numerator. It can be considered the [[2.3.7 subgroup|2.3.7-]][[subgroup]] equivalent of the [[syntonic comma]], and seperates complex pythagorean intervals from simpler 7-limit ones. For example, it is the difference between [[32/27]] and [[7/6]], and the difference between [[81/64]] and [[9/7]]. Since its numerator is a power of 2, it is a [[Mersenne comma]]. |
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| The Archytas comma is a 7-limit comma with monzo | 6 -2 0 -1 >.
| | == Temperaments == |
| | [[Tempering out]] this comma leads to [[superpyth]] temperament (sometimes called ''archy'' in the 2.3.7-subgroup), which equates 9/8 and 8/7, and also equates [[7/4]] with [[16/9]]. This means that the just dominant seventh chord, [[36:45:54:64|1–5/4–3/2–16/9]], and the harmonic seventh chord, [[4:5:6:7|1–5/4–3/2–7/4]], are equated to the same chord. Equal temperaments tempering out 64/63 include {{EDOs| 12, 15, 17, 22, 27, 37, 49 and 59 }}. |
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| It is similar to the Didymus or syntonic comma, 81/80, in that when it is tempered out it makes a stack of four fifths equal a major third (octave equivalent). In the case of 81/80, however, the major third is 5/4, while with the Archytas comma, the major third is 9/7. (Note that [[Porcupine family|Porcupine]], which tempers out 64/63, uses a minor tone as a generator and generally is considered to have 5/4 major thirds, so it doesn't depend on this equivalency.)
| | Archytas' comma is similar to Didymus' or the syntonic comma, 81/80, in that when it is tempered out it makes a stack of four fifths [[octave reduction|octave reduced]] equal a relatively consonant major third. In the case of 81/80, the major third is [[5/4]], while with Archytas' comma, the major third is [[9/7]]. |
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| If you are using 9/7 major thirds, this also implies that the major third is split into two equal steps that represent both 9/8 and 8/7: If a stack of four fifths gets you to (octave-equivalent) 9/7, and a stack of two fifths gets you to 9/8, then the difference must be (9/7)/(9/8) = 8/7. The 8/7 and 9/8 intervals are equal, however, as a result of the generation process. | | If one is using 9/7 major thirds, this also implies that the major third is split into two equal steps that represent both [[9/8]] and [[8/7]]: if a stack of four fifths (octave-reduced) reaches the interval 9/7, and a stack of two fifths reaches 9/8, then the difference must be (9/7)/(9/8) = 8/7. The 8/7 and 9/8 intervals are equated, however, as a result of the generation process. |
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| | See [[Archytas family]] for the family of rank-3 temperaments where it is tempered out. See [[Archytas clan]] for the clan of rank-2 temperaments where it is tempered out. |
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| [[http://en.wikipedia.org/wiki/Septimal_comma]]</pre></div>
| | == Comma pumps == |
| <h4>Original HTML content:</h4>
| | The septimal version of the common vi–ii–V–I progression, which uses the 6:7:9 subminor and 14:18:21 supermajor triads, requires that 64/63 be tempered out in order to avoid shifting the root. If 64/63 is not tempered out and intervals are kept pure, the root in the final I chord will be 64/63 higher than the root in the vi chord. |
| <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>64_63</title></head><body>The interval 64/63, called <strong>septimal</strong> or <strong>Archytas comma</strong> (in german <strong>Leipziger Komma</strong>), is a <a class="wiki_link" href="http://xenharmonic.wikispaces.com/superparticular">superparticular ratio</a> which equates <a class="wiki_link" href="http://xenharmonic.wikispaces.com/9_8">9/8</a> and <a class="wiki_link" href="http://xenharmonic.wikispaces.com/8_7">8/7</a> if tempered out and has the eighth square number as a numerator. It also equates <a class="wiki_link" href="http://xenharmonic.wikispaces.com/7_4">7/4</a> with <a class="wiki_link" href="http://xenharmonic.wikispaces.com/16_9">16/9</a>, so that the just dominant seventh chord, 1-5/4-3/2-16/9, and the otonal tetrad, 1-5/4-3/2-7/4, are equated to the same chord when 64/63 is tempered out. Equal divisions of the octave tempering out 64/63 include 12, 15, 22, 27, 37, 49 and 59.<br />
| | {{todo|add sound example}} |
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| The Archytas comma is a 7-limit comma with monzo | 6 -2 0 -1 &gt;.<br />
| | == Notation == |
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| | This interval is significant in the [[Functional Just System]] and [[Helmholtz–Ellis notation]] as the septimal formal comma which translates a Pythagorean interval to a nearby septimal interval. |
| It is similar to the Didymus or syntonic comma, 81/80, in that when it is tempered out it makes a stack of four fifths equal a major third (octave equivalent). In the case of 81/80, however, the major third is 5/4, while with the Archytas comma, the major third is 9/7. (Note that <a class="wiki_link" href="/Porcupine%20family">Porcupine</a>, which tempers out 64/63, uses a minor tone as a generator and generally is considered to have 5/4 major thirds, so it doesn't depend on this equivalency.)<br />
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| | === Sagittal notation === |
| If you are using 9/7 major thirds, this also implies that the major third is split into two equal steps that represent both 9/8 and 8/7: If a stack of four fifths gets you to (octave-equivalent) 9/7, and a stack of two fifths gets you to 9/8, then the difference must be (9/7)/(9/8) = 8/7. The 8/7 and 9/8 intervals are equal, however, as a result of the generation process.<br />
| | In the [[Sagittal]] system, the downward version of this comma (possibly tempered) is represented by the sagittal {{sagittal | !) }} and is called the '''7 comma''', or '''7C''' for short, because the simplest interval it notates is 7/1 (equiv. 7/4), as for example in G–F{{nbhsp}}{{sagittal | !) }}. The upward version is called '''1/7C''' or '''7C up''' and is represented by {{sagittal| |) }}. |
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| | == Approximation == |
| <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_comma" rel="nofollow">http://en.wikipedia.org/wiki/Septimal_comma</a></body></html></pre></div>
| | If one wants to treat Archytas' comma as a musical interval in its own right as opposed to tempering it out, you will find that it acts as a sort of chroma – specifically, it functions as a septimal equivalent of [[55/54]], from which it differs by a [[385/384|keenanisma]], or of [[56/55]], from which it differs by a [[441/440|werckisma]]. In addition, its incredible proximity to 1/44th of the octave – to the point where the [[septimal ruthenia|44-64/63 comma]] is tempered out in edos as large as tens of thousands – enables the tuning of [[ruthenium]] temperament. As a result, the major second of [[22edo]] is a good approximation to [[17/15]], due to it being the [[mediant]] of [[9/8]] and [[8/7]], so that the ~7:8:9 chord is much more accurately a 17/15–17/15 chord, with the outer interval as 9/7, by tempering out [[2025/2023]]. |
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| | == See also == |
| | * [[Gallery of just intervals]] |
| | * [[List of superparticular intervals]] |
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| | [[Category:Commas named for their regular temperament properties]] |
| | [[Category:Commas named after polymaths]] |