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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>
| | | de = 7-EDO |
| : This revision was by author [[User:Cenobyte|Cenobyte]] and made on <tt>2012-10-07 04:10:49 UTC</tt>.<br>
| | | en = 7edo |
| : The original revision id was <tt>370839812</tt>.<br>
| | | es = 7 EDO |
| : The revision comment was: <tt></tt><br>
| | | ja = 7平均律 |
| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
| | }} |
| <h4>Original Wikitext content:</h4>
| | {{Infobox ET}} |
| <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]]
| | {{ED intro}} |
|
| |
|
| **7 Equal Divisions of the Octave**
| | == Theory == |
| | [[File:7edo scale.mp3|thumb|A chromatic 7edo scale on C.]] |
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| |
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| ="neutral diatonic"=
| | 7edo is the basic example of an [[equiheptatonic]] scale, and in terms of tunings with perfect fifths, is essentially the next size up from [[5edo]]. The 7-form is notable as a common structure for many [[5-limit]] systems, including all seven modes of the [[5L 2s|diatonic]] scale—Ionian, Dorian, Phrygian, Lydian, Mixolydian, Aeolian, and Locrian; in 7edo itself, the two sizes of interval in any heptatonic MOS scale are equated, resulting in a [[interval quality|neutral]] feel. All triads are neutral (except if you use suspended triads, which are particularly harsh in 7edo due to the narrowed major second), so functional harmony is almost entirely based on the positions of the chords in the 7edo scale. |
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| 7-edo divides the 1200-cent [[octave]] into 7 equal parts, making its smallest interval [[cent|171.428¢]], or the seventh root of 2. It is the fourth [[prime numbers|prime]] edo, after [[2edo]], [[3edo]] and [[5edo]].
| | The second (171.429{{c}}) works well as a basic step for melodic progression. The step from seventh to octave is too large as a leading tone - possibly lending itself to a "sevenplus" scale similar to [[elevenplus]]. |
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| Equi-heptatonic scales are used in non-western music in African cultures and it has been speculated in "Indian music:history and structure", that the Indian three-sruti interval of 165 cents is close enough to be mistaken for 171 cents. (or 1.71 semitones).
| | In terms of just intonation, the 3/2 is flat but usable, but we don't find particularly accurate intervals in pure harmonics outside the 3-limit, which suggests a more melodic approach to just intonation; intervals approximated by each of 7edo's steps include 10/9 for 1 step, 11/9 for 2 steps, 4/3 for 3 steps, and their octave complements. Interestingly, this renders an 8:9:10:11:12 pentad equidistant, from which it can be derived that 7edo supports [[meantone]] (equating the major seconds 10/9 and 9/8) and [[porcupine]] (splitting 4/3 into three equal submajor seconds which simultaneously represent 12/11, 11/10, and 10/9), and is the unique system to do so. |
| 7-tet can be thought of as result of stacking seven 11/9s on top of each other, and then tempering to remove the comma 2^(-2) 3^(-14) 11^7.
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| The seventh of 7-edo is almost exactly the 29th harmonic ([[29_16|29/16]]), which can have a very agreeable sound with harmonic timbres. However it also finds itself nested between ratios such as 20/11 and 9/5, which gives it considerably higher [[harmonic entropy]] than [[harmonic seventh|7/4]], a much simpler overtone seventh.
| | Due to 7edo's inaccurately tuned [[5/4]] [[major third]] (which is flat by over 40 cents), it supports several exotemperaments in the 5-limit, such as [[dicot]] (which splits the fifth into two equal [[neutral third]]<nowiki/>s, simultaneously representing 5/4 and the [[minor third]] [[6/5]]) and [[mavila]] (which flattens the fifth so that the diatonic "major third" actually approximates 6/5); 6/5 is a slightly more reasonable interpretation of 7edo's third than 5/4, leading to an overall slightly [[minor]] sound. |
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| Similarly, in equi-heptatonic systems the desire for harmonic sound may dictate constant adjustments of intonation away from the theoretical interval of 171 cents. One of the most impressive areas in Africa in which a pen-equidistant heptatonic scale is combined with a distinctively harmonic style based on singing in intervals of thirds plus fifths, or thirds plus fourths, is the eastern Angolan culture area. This music is heptatonic and non-modal; i.e., there is no concept of major or minor thirds as distinctive intervals. In principle all the thirds are neutral, but in practice the thirds rendered by the singers often approximate natural major thirds (386 cents), especially at points of rest. In this manner, the principles of equidistance and harmonic euphony are accommodated within one tonal-harmonic system. For the notation of such music, a seven-line stave is most appropriate, with each horizontal line representing one pitch level.
| | In higher limits, this third takes on a new role: as a neutral third, it is a decent approximation of the 13th subharmonic, and as such 7edo can be seen as a 2.3.13 temperament. This third is a near perfect approximation of the interval [[39/32]]; the equation of 16/13 and 39/32 is called [[512/507|harmoneutral]] temperament. In general, the inclusion of 13 allows the pentad discussed earlier to be continued to an 8:9:10:11:12:13 hexad, although the specific interval 13/12 is inaccurate due to the errors adding up in the same direction. |
| __//("African music." Encyclopædia Britannica. 2009. Encyclopædia Britannica Online. 05 Jul. 2009 <http://www.britannica.com/EBchecked/topic/719112/African-music>.//__)
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| |
|
| A Thai xylophone measured by Morton (1974) "varied only plus or minus 5 cents," from 7-TET. A Ugandan Chopi xylophone measured by Haddon (1952) was also tuned to this system.
| | The seventh of 7edo is almost exactly the 29th harmonic (29/16), which can have a very agreeable sound with harmonic timbres. However it also finds itself nested between ratios such as 20/11 and 9/5, which gives it considerably higher harmonic entropy than 7/4, a much simpler overtone seventh. |
|
| |
|
| =Intervals in 7-edo=
| | 7edo represents a 7-step closed [[circle of fifths]], tempering out the Pythagorean chromatic semitone. However, it can also be seen as a circle of neutral thirds, which can be interpreted as 11/9; this is called [[neutron]] temperament. |
| || **Interval** || **Interval**
| |
| **size**
| |
| **in ¢** || **Closest**
| |
| **diatonic**
| |
| **interval name** || **The "neighborhood" of just intervals** ||
| |
| || 0 || 0.0 || unison / prime || exactly 1/1 ||
| |
| || 1 || 171,429 || second || 6,424 c from Ptolemy (neutral) second 11/10
| |
| 3,215 c from second 54/49
| |
| -10,976 c from (Didymus) major second (small whole tone) 10/9 ||
| |
| || 2 || 342,857 || third || -4,551 c from neutral third 11/9 ||
| |
| || 3 || 514,286 || fourth || 16,241 c from just fourth 4/3 (498,045 c)
| |
| -5,265 c from wide fourth 27/20 ||
| |
| || 4 || 685,714 || fifth || 5,265 c from narrow fifth 40/27
| |
| -16,241 c from just fifth 3/2 (701,955 c) ||
| |
| || 5 || 857,143 || sixth || 4,551 c from neutral sixth 18/11 ||
| |
| || 6 || 1028,571 || seventh || 10,975 c from (Didymus) minor seventh 9/5
| |
| -6,425 c from neutral seventh 20/11
| |
| 1,006 c from the 29th harmonic 29/16
| |
| -3,216 c from seventh 49/27 ||
| |
| || 7 (0) || 1200.0 || eighth || exactly 2/1 ||
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|
| =Related scales= | | === Prime harmonics === |
| * Related in a lateral way to traditional Thai music.
| | {{Harmonics in equal|7}} |
| ==Notation== | |
| * Notatable on a five-line staff without accidentals.
| |
| ==Harmony==
| |
| There is no distinction between Major or Minor; each pitch class is unique.
| |
| ==Melody==
| |
| There is a neutral feel between whole tone scale and major/minor diatonic scale. The second 171,429 c works well as a basic step for melodic progression.
| |
| The step from seventh to octave is too large for the leading tone.
| |
| ==Relative tuning accuracy==
| |
| 7-edo is the third [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]]. | |
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| |
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| === === | | === In non-Western traditions === |
| =Music=
| | [[Equiheptatonic]] scales close to 7edo are used in non-western music in some [[African]] cultures<ref>[https://www.britannica.com/art/African-music ''African music'', Encyclopedia Britannica.]</ref> as well as an integral part of early [[Chinese]] music<ref>Robotham, Donald Keith and Gerhard Kubik.</ref>. Also [[Georgian]] music seems to be based on near-equal 7-step scales. |
|
| |
|
| [[http://sethares.engr.wisc.edu/mp3s/PagansRevenge.mp3|Pagan's Revenge]] by Bill Sethares (synthetic gamelan)
| | It has been speculated in ''Indian music: history and structure''<ref>Nambiyathiri, Tarjani. ''[https://archive.org/details/indianmusichistoryandstructureemmietenijenhuisbrill Indian Music History And Structure Emmie Te Nijenhuis Brill]''</ref> that the [[Indian]] three-sruti interval of 165 cents is very similar to one 171-cent step of 7edo. |
| [[http://www.akjmusic.com/audio/i_dream_of_tibet.mp3|I dream of Tibet]] by [[http://www.akjmusic.com/works.html|Aaron K. Johnson]] (electronic swirlies) | |
| [[http://micro.soonlabel.com/gene_ward_smith/Others/Walker/Seven%20Equal%20Trio.mp3|Seven Equal Trio]] by [[http://robertinventor.tripod.com/tunes/tunes.htm#7_equal_trio|Robert Walker]] ((synth) violin, viola, glockenspiel) | |
| [[http://micro.soonlabel.com/gene_ward_smith/Others/Hunt/7ET.mp3|Two-part Invention in 7TET]] by [[http://www.h-pi.com/musicFiles.html|Aaron Hunt]]
| |
| [[http://www.uvnitr.cz/flaoyg/flao_yg/pavouci.html|Pavouci]], [[http://www.uvnitr.cz/flaoyg/flao_yg/kelt.html|Kelt]] by Milan Guštar
| |
| [[http://www.seraph.it/dep/det/7edo%20dance.mp3|7edo Dance]] by Carlo Serafini
| |
| [[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/06%20-%206.%207%20octave.mp3|Comets Over Flatland 6]] by [[Randy Winchester]]
| |
| [[http://media.soundcloud.com/stream/IUcgFYhtu3Rk?stream_token=4QIvd|Sävelmä]] by [[http://soundcloud.com/juhani-nuorvala/s-velm-long-version|Juhani Nuorvala]]
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|
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| =Commas=
| | In [[equiheptatonic]] systems the desire for harmonic sound may dictate constant adjustments of intonation away from the theoretical interval of 171 cents. (Similar to [[adaptive just intonation]] but with equal tuning instead). |
| 7 EDO tempers out the following commas. (Note: This assumes val < 7 11 16 20 24 26 |.)
| |
|
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|
| ||~ Comma ||~ Monzo ||~ Cents ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
| | One region of Africa in which a pen-equidistant heptatonic scale is combined with a distinctively harmonic style based on singing in intervals of thirds plus fifths, or thirds plus fourths, is the eastern [[Angolan]] area. This music is [[heptatonic]] and non-modal; i.e., there is no concept of major or minor thirds as distinctive intervals. In principle all the thirds are neutral, but in practice the thirds rendered by the singers often approximate natural major thirds ([[5/4]], 386{{c}}), especially at points of rest. In this manner, the principles of equidistance and harmonic euphony are accommodated within one tonal-harmonic system. |
| ||= 2187/2048 || | -11 7 > ||> 113.69 ||= Apotome ||= ||= ||
| |
| ||= 135/128 || | -7 3 1 > ||> 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
| |
| ||= 250/243 || | 1 -5 3 > ||> 49.17 ||= Maximal Diesis ||= Porcupine Comma ||= ||
| |
| ||= 20000/19683 || | 5 -9 4 > ||> 27.66 ||= Minimal Diesis ||= Tetracot Comma ||= ||
| |
| ||= 81/80 || | -4 4 -1 > ||> 21.51 ||= Syntonic Comma ||= Didymos Comma ||= Meantone Comma ||
| |
| ||= 1600000/1594323 || | 9 -13 5 > ||> 6.15 ||= Amity Comma ||= ||= ||
| |
| ||= 36/35 || | 2 2 -1 -1 > ||> 48.77 ||= Septimal Quarter Tone ||= ||= ||
| |
| ||= 525/512 || | -9 1 2 1 > ||> 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||= ||
| |
| ||= 64/63 || | 6 -2 0 -1 > ||> 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma ||
| |
| ||= 875/864 || | -5 -3 3 1 > ||> 21.90 ||= Keema ||= ||= ||
| |
| ||= 5120/5103 || | 10 -6 1 -1 > ||> 5.76 ||= Hemifamity ||= ||= ||
| |
| ||= 6144/6125 || | 11 1 -3 -2 > ||> 5.36 ||= Porwell ||= ||= ||
| |
| ||= 4375/4374 || | -1 -7 4 1 > ||> 0.40 ||= Ragisma ||= ||= ||
| |
| ||= 394839/394762 || | 47 -7 -7 -7 > ||> 0.34 ||= Akjaysma ||= 5\7 Octave Comma ||= ||
| |
| ||= 100/99 || | 2 -2 2 0 -1 > ||> 17.40 ||= Ptolemisma ||= ||= ||
| |
| ||= 121/120 || | -3 -1 -1 0 2 > ||> 14.37 ||= Biyatisma ||= ||= ||
| |
| ||= 176/175 || | 4 0 -2 -1 1 > ||> 9.86 ||= Valinorsma ||= ||= ||
| |
| ||= 65536/65219 || | 16 0 0 -2 -3 > ||> 8.39 ||= Orgonisma ||= ||= ||
| |
| ||= 243/242 || | -1 5 0 0 -2 > ||> 7.14 ||= Rastma ||= ||= ||
| |
| ||= 385/384 || | -7 -1 1 1 1 > ||> 4.50 ||= Keenanisma ||= ||= ||
| |
| ||= 4000/3993 || | 5 -1 3 0 -3 > ||> 3.03 ||= Wizardharry ||= ||= ||</pre></div>
| |
| <h4>Original HTML content:</h4>
| |
| <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>7edo</title></head><body><!-- ws:start:WikiTextTocRule:20:&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:20 --><!-- ws:start:WikiTextTocRule:21: --><a href="#x&quot;neutral diatonic&quot;">&quot;neutral diatonic&quot;</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --> | <a href="#Intervals in 7-edo">Intervals in 7-edo</a><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --> | <a href="#Related scales">Related scales</a><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --><!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --><!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --> | <a href="#Music">Music</a><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --> | <a href="#Commas">Commas</a><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: -->
| |
| <!-- ws:end:WikiTextTocRule:31 --><br />
| |
| <strong>7 Equal Divisions of the Octave</strong><br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x&quot;neutral diatonic&quot;"></a><!-- ws:end:WikiTextHeadingRule:0 -->&quot;neutral diatonic&quot;</h1>
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| <br />
| |
| 7-edo divides the 1200-cent <a class="wiki_link" href="/octave">octave</a> into 7 equal parts, making its smallest interval <a class="wiki_link" href="/cent">171.428¢</a>, or the seventh root of 2. It is the fourth <a class="wiki_link" href="/prime%20numbers">prime</a> edo, after <a class="wiki_link" href="/2edo">2edo</a>, <a class="wiki_link" href="/3edo">3edo</a> and <a class="wiki_link" href="/5edo">5edo</a>.<br />
| |
| <br />
| |
| Equi-heptatonic scales are used in non-western music in African cultures and it has been speculated in &quot;Indian music:history and structure&quot;, that the Indian three-sruti interval of 165 cents is close enough to be mistaken for 171 cents. (or 1.71 semitones).<br />
| |
| 7-tet can be thought of as result of stacking seven 11/9s on top of each other, and then tempering to remove the comma 2^(-2) 3^(-14) 11^7.<br />
| |
| <br />
| |
| The seventh of 7-edo is almost exactly the 29th harmonic (<a class="wiki_link" href="/29_16">29/16</a>), which can have a very agreeable sound with harmonic timbres. However it also finds itself nested between ratios such as 20/11 and 9/5, which gives it considerably higher <a class="wiki_link" href="/harmonic%20entropy">harmonic entropy</a> than <a class="wiki_link" href="/harmonic%20seventh">7/4</a>, a much simpler overtone seventh.<br />
| |
| <br />
| |
| Similarly, in equi-heptatonic systems the desire for harmonic sound may dictate constant adjustments of intonation away from the theoretical interval of 171 cents. One of the most impressive areas in Africa in which a pen-equidistant heptatonic scale is combined with a distinctively harmonic style based on singing in intervals of thirds plus fifths, or thirds plus fourths, is the eastern Angolan culture area. This music is heptatonic and non-modal; i.e., there is no concept of major or minor thirds as distinctive intervals. In principle all the thirds are neutral, but in practice the thirds rendered by the singers often approximate natural major thirds (386 cents), especially at points of rest. In this manner, the principles of equidistance and harmonic euphony are accommodated within one tonal-harmonic system. For the notation of such music, a seven-line stave is most appropriate, with each horizontal line representing one pitch level.<br />
| |
| <u><em>(&quot;African music.&quot; Encyclopædia Britannica. 2009. Encyclopædia Britannica Online. 05 Jul. 2009 &lt;<!-- ws:start:WikiTextUrlRule:679:http://www.britannica.com/EBchecked/topic/719112/African-music --><a class="wiki_link_ext" href="http://www.britannica.com/EBchecked/topic/719112/African-music" rel="nofollow">http://www.britannica.com/EBchecked/topic/719112/African-music</a><!-- ws:end:WikiTextUrlRule:679 -->&gt;.</em></u>)<br />
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| <br />
| |
| A Thai xylophone measured by Morton (1974) &quot;varied only plus or minus 5 cents,&quot; from 7-TET. A Ugandan Chopi xylophone measured by Haddon (1952) was also tuned to this system.<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Intervals in 7-edo"></a><!-- ws:end:WikiTextHeadingRule:2 -->Intervals in 7-edo</h1>
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|
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|
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|
| <table class="wiki_table">
| | A [[Ugandan]], [[Chopi]] xylophone measured by Haddon (1952) was also tuned something close to this. |
| <tr>
| |
| <td><strong>Interval</strong><br />
| |
| </td>
| |
| <td><strong>Interval</strong><br />
| |
| <strong>size</strong><br />
| |
| <strong>in ¢</strong><br />
| |
| </td>
| |
| <td><strong>Closest</strong><br />
| |
| <strong>diatonic</strong><br />
| |
| <strong>interval name</strong><br />
| |
| </td>
| |
| <td><strong>The &quot;neighborhood&quot; of just intervals</strong><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>0<br />
| |
| </td>
| |
| <td>0.0<br />
| |
| </td>
| |
| <td>unison / prime<br />
| |
| </td>
| |
| <td>exactly 1/1<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
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| <td>1<br />
| |
| </td>
| |
| <td>171,429<br />
| |
| </td>
| |
| <td>second<br />
| |
| </td>
| |
| <td>6,424 c from Ptolemy (neutral) second 11/10<br />
| |
| 3,215 c from second 54/49<br />
| |
| -10,976 c from (Didymus) major second (small whole tone) 10/9<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>2<br />
| |
| </td>
| |
| <td>342,857<br />
| |
| </td>
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| <td>third<br />
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| </td>
| |
| <td>-4,551 c from neutral third 11/9<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
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| <td>3<br />
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| </td>
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| <td>514,286<br />
| |
| </td>
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| <td>fourth<br />
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| </td>
| |
| <td>16,241 c from just fourth 4/3 (498,045 c)<br />
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| -5,265 c from wide fourth 27/20<br />
| |
| </td>
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| </tr>
| |
| <tr>
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| <td>4<br />
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| </td>
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| <td>685,714<br />
| |
| </td>
| |
| <td>fifth<br />
| |
| </td>
| |
| <td>5,265 c from narrow fifth 40/27<br />
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| -16,241 c from just fifth 3/2 (701,955 c)<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>5<br />
| |
| </td>
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| <td>857,143<br />
| |
| </td>
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| <td>sixth<br />
| |
| </td>
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| <td>4,551 c from neutral sixth 18/11<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
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| <td>6<br />
| |
| </td>
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| <td>1028,571<br />
| |
| </td>
| |
| <td>seventh<br />
| |
| </td>
| |
| <td>10,975 c from (Didymus) minor seventh 9/5<br />
| |
| -6,425 c from neutral seventh 20/11<br />
| |
| 1,006 c from the 29th harmonic 29/16<br />
| |
| -3,216 c from seventh 49/27<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>7 (0)<br />
| |
| </td>
| |
| <td>1200.0<br />
| |
| </td>
| |
| <td>eighth<br />
| |
| </td>
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| <td>exactly 2/1<br />
| |
| </td>
| |
| </tr>
| |
| </table>
| |
|
| |
|
| <br />
| | It has often been stated that 7edo approximates tunings used in [[Thai]] classical music, though this is a myth unsupported by [[empirical]] studies of the instruments.<ref>Garzoli, John. [http://iftawm.org/journal/oldsite/articles/2015b/Garzoli_AAWM_Vol_4_2.pdf ''The Myth of Equidistance in Thai Tuning.'']</ref> |
| <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Related scales"></a><!-- ws:end:WikiTextHeadingRule:4 -->Related scales</h1>
| |
| <ul><li>Related in a lateral way to traditional Thai music.</li></ul><!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Related scales-Notation"></a><!-- ws:end:WikiTextHeadingRule:6 -->Notation</h2>
| |
| <ul><li>Notatable on a five-line staff without accidentals.</li></ul><!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Related scales-Harmony"></a><!-- ws:end:WikiTextHeadingRule:8 -->Harmony</h2>
| |
| There is no distinction between Major or Minor; each pitch class is unique.<br />
| |
| <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Related scales-Melody"></a><!-- ws:end:WikiTextHeadingRule:10 -->Melody</h2>
| |
| There is a neutral feel between whole tone scale and major/minor diatonic scale. The second 171,429 c works well as a basic step for melodic progression.<br />
| |
| The step from seventh to octave is too large for the leading tone.<br />
| |
| <!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Related scales-Relative tuning accuracy"></a><!-- ws:end:WikiTextHeadingRule:12 -->Relative tuning accuracy</h2>
| |
| 7-edo is the third <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists">zeta integral edo</a>.<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><!-- ws:end:WikiTextHeadingRule:14 --> </h3>
| |
| <!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc8"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:16 -->Music</h1>
| |
| <br />
| |
| <a class="wiki_link_ext" href="http://sethares.engr.wisc.edu/mp3s/PagansRevenge.mp3" rel="nofollow">Pagan's Revenge</a> by Bill Sethares (synthetic gamelan)<br />
| |
| <a class="wiki_link_ext" href="http://www.akjmusic.com/audio/i_dream_of_tibet.mp3" rel="nofollow">I dream of Tibet</a> by <a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow">Aaron K. Johnson</a> (electronic swirlies)<br />
| |
| <a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Walker/Seven%20Equal%20Trio.mp3" rel="nofollow">Seven Equal Trio</a> by <a class="wiki_link_ext" href="http://robertinventor.tripod.com/tunes/tunes.htm#7_equal_trio" rel="nofollow">Robert Walker</a> ((synth) violin, viola, glockenspiel)<br />
| |
| <a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Hunt/7ET.mp3" rel="nofollow">Two-part Invention in 7TET</a> by <a class="wiki_link_ext" href="http://www.h-pi.com/musicFiles.html" rel="nofollow">Aaron Hunt</a><br />
| |
| <a class="wiki_link_ext" href="http://www.uvnitr.cz/flaoyg/flao_yg/pavouci.html" rel="nofollow">Pavouci</a>, <a class="wiki_link_ext" href="http://www.uvnitr.cz/flaoyg/flao_yg/kelt.html" rel="nofollow">Kelt</a> by Milan Guštar<br />
| |
| <a class="wiki_link_ext" href="http://www.seraph.it/dep/det/7edo%20dance.mp3" rel="nofollow">7edo Dance</a> by Carlo Serafini<br />
| |
| <a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/06%20-%206.%207%20octave.mp3" rel="nofollow">Comets Over Flatland 6</a> by <a class="wiki_link" href="/Randy%20Winchester">Randy Winchester</a><br />
| |
| <a class="wiki_link_ext" href="http://media.soundcloud.com/stream/IUcgFYhtu3Rk?stream_token=4QIvd" rel="nofollow">Sävelmä</a> by <a class="wiki_link_ext" href="http://soundcloud.com/juhani-nuorvala/s-velm-long-version" rel="nofollow">Juhani Nuorvala</a><br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:18:&lt;h1&gt; --><h1 id="toc9"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:18 -->Commas</h1>
| |
| 7 EDO tempers out the following commas. (Note: This assumes val &lt; 7 11 16 20 24 26 |.)<br />
| |
| <br />
| |
|
| |
|
| | === Subsets and supersets === |
| | 7edo is the 4th [[prime edo]], after [[5edo]] and before [[11edo]]. It does not contain any nontrivial subset edos, though it contains [[7ed4]]. Multiples such as [[14edo]], [[21edo]], … up to [[35edo]], share the same tuning of the perfect fifth as 7edo, while improving on other intervals. |
|
| |
|
| <table class="wiki_table">
| | == Intervals == |
| <tr>
| | {| class="wikitable center-all" |
| <th>Comma<br />
| | |+ style="font-size: 105%;" | Intervals of 7edo |
| </th>
| | |- |
| <th>Monzo<br />
| | ! rowspan="2" | [[Degree]] |
| </th>
| | ! rowspan="2" | [[Cent]]s |
| <th>Cents<br />
| | ! rowspan="2" | [[Interval region]] |
| </th>
| | ! colspan="4" | Approximated [[JI]] intervals ([[error]] in [[¢]]) |
| <th>Name 1<br />
| | ! rowspan="2" | Audio |
| </th>
| | |- |
| <th>Name 2<br />
| | ! [[3-limit]] |
| </th>
| | ! [[5-limit]] |
| <th>Name 3<br />
| | ! [[7-limit]] |
| </th>
| | ! Other |
| </tr>
| | |- |
| <tr>
| | | 0 |
| <td style="text-align: center;">2187/2048<br />
| | | 0 |
| </td>
| | | Unison (prime) |
| <td>| -11 7 &gt;<br />
| | | [[1/1]] (just) |
| </td>
| | | |
| <td style="text-align: right;">113.69<br />
| | | |
| </td>
| | | |
| <td style="text-align: center;">Apotome<br />
| | | [[File:0-0 unison.mp3|frameless]] |
| </td>
| | |- |
| <td style="text-align: center;"><br />
| | | 1 |
| </td>
| | | 171.429 |
| <td style="text-align: center;"><br />
| | | Submajor second |
| </td>
| | | |
| </tr>
| | | [[10/9]] (-10.975) |
| <tr>
| | | [[54/49]] (+3.215) |
| <td style="text-align: center;">135/128<br />
| | | [[11/10]] (+6.424)<br>[[32/29]] (-1.006) |
| </td>
| | | [[File:0-171,43 second (7-EDO).mp3|frameless]] |
| <td>| -7 3 1 &gt;<br />
| | |- |
| </td>
| | | 2 |
| <td style="text-align: right;">92.18<br />
| | | 342.857 |
| </td>
| | | Neutral third |
| <td style="text-align: center;">Major Chroma<br />
| | | |
| </td>
| | | |
| <td style="text-align: center;">Major Limma<br />
| | | [[128/105]] (+0.048) |
| </td>
| | | [[39/32]] (+0.374)<br>[[16/13]] (-16.6)<br>[[11/9]] (-4.551) |
| <td style="text-align: center;">Pelogic Comma<br />
| | | [[File:piano_2_7edo.mp3]] |
| </td>
| | |- |
| </tr>
| | | 3 |
| <tr>
| | | 514.286 |
| <td style="text-align: center;">250/243<br />
| | | Fourth |
| </td>
| | | [[4/3]] (+16.241) |
| <td>| 1 -5 3 &gt;<br />
| | | [[27/20]] (-5.265) |
| </td>
| | | |
| <td style="text-align: right;">49.17<br />
| | | [[35/26]] (-0.326) |
| </td>
| | | [[File:0-514,29 fourth (7-EDO).mp3|frameless]] |
| <td style="text-align: center;">Maximal Diesis<br />
| | |- |
| </td>
| | | 4 |
| <td style="text-align: center;">Porcupine Comma<br />
| | | 685.714 |
| </td>
| | | Fifth |
| <td style="text-align: center;"><br />
| | | [[3/2]] (-16.241) |
| </td>
| | | [[40/27]] (+5.265) |
| </tr>
| | | |
| <tr>
| | | [[52/35]] (+0.326) |
| <td style="text-align: center;">20000/19683<br />
| | | [[File:0-685,71 fifth (7-EDO).mp3|frameless]] |
| </td>
| | |- |
| <td>| 5 -9 4 &gt;<br />
| | | 5 |
| </td>
| | | 857.143 |
| <td style="text-align: right;">27.66<br />
| | | Neutral sixth |
| </td>
| | | |
| <td style="text-align: center;">Minimal Diesis<br />
| | | |
| </td>
| | | [[105/64]] (-0.048) |
| <td style="text-align: center;">Tetracot Comma<br />
| | | [[18/11]] (+4.551)<br>[[13/8]] (+16.6)<br>[[64/39]] (-0.374) |
| </td>
| | | [[File:0-857,14 sixth (7-EDO).mp3|frameless]] |
| <td style="text-align: center;"><br />
| | |- |
| </td>
| | | 6 |
| </tr>
| | | 1028.571 |
| <tr>
| | | Supraminor seventh |
| <td style="text-align: center;">81/80<br />
| | | |
| </td>
| | | [[9/5]] (+10.975) |
| <td>| -4 4 -1 &gt;<br />
| | | [[49/27]] (-3.215) |
| </td>
| | | [[29/16]] (-1.006)<br>[[20/11]] (-6.424) |
| <td style="text-align: right;">21.51<br />
| | | [[File:0-1028,57 seventh (7-EDO).mp3|frameless]] |
| </td>
| | |- |
| <td style="text-align: center;">Syntonic Comma<br />
| | | 7 |
| </td>
| | | 1200 |
| <td style="text-align: center;">Didymos Comma<br />
| | | Octave |
| </td>
| | | [[2/1]] (just) |
| <td style="text-align: center;">Meantone Comma<br />
| | | |
| </td>
| | | |
| </tr>
| | | |
| <tr>
| | | [[File:0-1200 octave.mp3|frameless]] |
| <td style="text-align: center;">1600000/1594323<br />
| | |} |
| </td>
| |
| <td>| 9 -13 5 &gt;<br />
| |
| </td>
| |
| <td style="text-align: right;">6.15<br />
| |
| </td>
| |
| <td style="text-align: center;">Amity Comma<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">36/35<br />
| |
| </td>
| |
| <td>| 2 2 -1 -1 &gt;<br />
| |
| </td>
| |
| <td style="text-align: right;">48.77<br />
| |
| </td>
| |
| <td style="text-align: center;">Septimal Quarter Tone<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">525/512<br />
| |
| </td>
| |
| <td>| -9 1 2 1 &gt;<br />
| |
| </td>
| |
| <td style="text-align: right;">43.41<br />
| |
| </td>
| |
| <td style="text-align: center;">Avicennma<br />
| |
| </td>
| |
| <td style="text-align: center;">Avicenna's Enharmonic Diesis<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">64/63<br />
| |
| </td>
| |
| <td>| 6 -2 0 -1 &gt;<br />
| |
| </td>
| |
| <td style="text-align: right;">27.26<br />
| |
| </td>
| |
| <td style="text-align: center;">Septimal Comma<br />
| |
| </td>
| |
| <td style="text-align: center;">Archytas' Comma<br />
| |
| </td>
| |
| <td style="text-align: center;">Leipziger Komma<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">875/864<br />
| |
| </td>
| |
| <td>| -5 -3 3 1 &gt;<br />
| |
| </td>
| |
| <td style="text-align: right;">21.90<br />
| |
| </td>
| |
| <td style="text-align: center;">Keema<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">5120/5103<br />
| |
| </td>
| |
| <td>| 10 -6 1 -1 &gt;<br />
| |
| </td>
| |
| <td style="text-align: right;">5.76<br />
| |
| </td>
| |
| <td style="text-align: center;">Hemifamity<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">6144/6125<br />
| |
| </td>
| |
| <td>| 11 1 -3 -2 &gt;<br />
| |
| </td>
| |
| <td style="text-align: right;">5.36<br />
| |
| </td>
| |
| <td style="text-align: center;">Porwell<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">4375/4374<br />
| |
| </td>
| |
| <td>| -1 -7 4 1 &gt;<br />
| |
| </td>
| |
| <td style="text-align: right;">0.40<br />
| |
| </td>
| |
| <td style="text-align: center;">Ragisma<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">394839/394762<br />
| |
| </td>
| |
| <td>| 47 -7 -7 -7 &gt;<br />
| |
| </td>
| |
| <td style="text-align: right;">0.34<br />
| |
| </td>
| |
| <td style="text-align: center;">Akjaysma<br />
| |
| </td>
| |
| <td style="text-align: center;">5\7 Octave Comma<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">100/99<br />
| |
| </td>
| |
| <td>| 2 -2 2 0 -1 &gt;<br />
| |
| </td>
| |
| <td style="text-align: right;">17.40<br />
| |
| </td>
| |
| <td style="text-align: center;">Ptolemisma<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">121/120<br />
| |
| </td>
| |
| <td>| -3 -1 -1 0 2 &gt;<br />
| |
| </td>
| |
| <td style="text-align: right;">14.37<br />
| |
| </td>
| |
| <td style="text-align: center;">Biyatisma<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">176/175<br />
| |
| </td>
| |
| <td>| 4 0 -2 -1 1 &gt;<br />
| |
| </td>
| |
| <td style="text-align: right;">9.86<br />
| |
| </td>
| |
| <td style="text-align: center;">Valinorsma<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">65536/65219<br />
| |
| </td>
| |
| <td>| 16 0 0 -2 -3 &gt;<br />
| |
| </td>
| |
| <td style="text-align: right;">8.39<br />
| |
| </td>
| |
| <td style="text-align: center;">Orgonisma<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">243/242<br />
| |
| </td>
| |
| <td>| -1 5 0 0 -2 &gt;<br />
| |
| </td>
| |
| <td style="text-align: right;">7.14<br />
| |
| </td>
| |
| <td style="text-align: center;">Rastma<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">385/384<br />
| |
| </td>
| |
| <td>| -7 -1 1 1 1 &gt;<br />
| |
| </td>
| |
| <td style="text-align: right;">4.50<br />
| |
| </td>
| |
| <td style="text-align: center;">Keenanisma<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">4000/3993<br />
| |
| </td>
| |
| <td>| 5 -1 3 0 -3 &gt;<br />
| |
| </td>
| |
| <td style="text-align: right;">3.03<br />
| |
| </td>
| |
| <td style="text-align: center;">Wizardharry<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| </table>
| |
|
| |
|
| </body></html></pre></div>
| | == Notation == |
| | The usual [[Musical notation|notation system]] for 7edo is the [[chain-of-fifths notation]], which is directly derived from the standard notation used in [[12edo]]. |
| | |
| | Because the Pythagorean apotome ([[2187/2048]]) is [[tempered out]], sharps (♯) and flats (♭) are redundant in 7edo. Therefore, 7edo can be notated on a five-line staff without accidentals. Alternatively, a seven-line stave can be used, with each horizontal line representing one pitch level. There is no distinction between major or minor, so every interval has the [[interval quality]] "perfect" instead. |
| | |
| | {| class="wikitable center-all" |
| | |+ style="font-size: 105%;" | Notation of 7edo |
| | |- |
| | ! rowspan="2" | [[Degree]] |
| | ! rowspan="2" | [[Cent]]s |
| | ! colspan="2" | [[Chain-of-fifths notation]] |
| | |- |
| | ! [[5L 2s|Diatonic]] interval names |
| | ! Note names (on D) |
| | |- |
| | | 0 |
| | | 0 |
| | | '''Perfect unison (P1)''' |
| | | '''D''' |
| | |- |
| | | 1 |
| | | 171.429 |
| | | '''Perfect second (P2)''' |
| | | '''E''' |
| | |- |
| | | 2 |
| | | 342.857 |
| | | '''Perfect third (P3)''' |
| | | '''F''' |
| | |- |
| | | 3 |
| | | 514.286 |
| | | '''Perfect fourth (P4)''' |
| | | '''G''' |
| | |- |
| | | 4 |
| | | 685.714 |
| | | '''Perfect fifth (P5)''' |
| | | '''A''' |
| | |- |
| | | 5 |
| | | 857.143 |
| | | '''Perfect sixth (P6)''' |
| | | '''B''' |
| | |- |
| | | 6 |
| | | 1028.571 |
| | | '''Perfect seventh (P7)''' |
| | | '''C''' |
| | |- |
| | | 7 |
| | | 1200 |
| | | '''Perfect octave (P8)''' |
| | | '''D''' |
| | |} |
| | |
| | In 7edo: |
| | * [[Ups and downs notation]] is identical to circle-of-fifths notation; |
| | * Mixed and pure [[sagittal notation]] are identical to circle-of-fifths notation. |
| | |
| | ===Sagittal notation=== |
| | This notation is a subset of the notations for EDOs [[14edo#Sagittal notation|14]], [[21edo#Sagittal notation|21]], [[28edo#Sagittal notation|28]], [[35edo#Sagittal notation|35]], and [[42edo#Second-best fifth notation|42b]]. |
| | |
| | <imagemap> |
| | File:7-EDO_Sagittal.svg |
| | desc none |
| | rect 80 0 246 50 [[Sagittal_notation]] |
| | rect 246 0 406 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation] |
| | rect 20 80 246 106 [[Fractional_3-limit_notation#Bad-fifths_limma-fraction_notation | limma-fraction notation]] |
| | default [[File:7-EDO_Sagittal.svg]] |
| | </imagemap> |
| | |
| | Because it includes no Sagittal symbols, this Sagittal notation is also a conventional notation. |
| | |
| | === Alternative notations === |
| | [[William Lynch]] proposes using numbers 1 through 7 as the nominals of 7edo with sharp signs being possible to expand to 14edo or even 21edo. |
| | |
| | == Solfege == |
| | {| class="wikitable center-all" |
| | |+ <span style="font-size: 105%;">Solfege of 7edo</span> |
| | |- |
| | ! [[Degree]] |
| | ! [[Cents]] |
| | ! Standard [[solfege]]<br />(movable do) |
| | ! [[Uniform solfege]]<br />(1 vowel) |
| | |- |
| | | 0 |
| | | 0 |
| | | Do |
| | | Da |
| | |- |
| | | 1 |
| | | 171.429 |
| | | Re |
| | | Ra |
| | |- |
| | | 2 |
| | | 342.857 |
| | | Mi |
| | | Ma |
| | |- |
| | | 3 |
| | | 514.286 |
| | | Fa |
| | | Fa |
| | |- |
| | | 4 |
| | | 685.714 |
| | | So |
| | | Sa |
| | |- |
| | | 5 |
| | | 857.143 |
| | | La |
| | | La |
| | |- |
| | | 6 |
| | | 1028.571 |
| | | Ti |
| | | Ta |
| | |- |
| | | 7 |
| | | 1200 |
| | | Do |
| | | Da |
| | |} |
| | |
| | == Approximation to JI == |
| | [[File:7ed2-001.svg]] |
| | |
| | == Regular temperament properties == |
| | === Uniform maps === |
| | {{Uniform map|edo=7}} |
| | |
| | === Commas === |
| | 7et [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 7 11 16 20 24 26 }}. |
| | |
| | {| class="commatable wikitable center-1 center-2 right-4 center-5" |
| | |- |
| | ! [[Harmonic limit|Prime<br>limit]] |
| | ! [[Ratio]]<ref group="note">Ratios longer than 10 digits are presented by placeholders with informative hints</ref> |
| | ! [[Monzo]] |
| | ! [[Cent]]s |
| | ! [[Color notation/Temperament names|Color name]] |
| | ! Name(s) |
| | |- |
| | | 3 |
| | | [[2187/2048]] |
| | | {{monzo| -11 7 }} |
| | | 113.69 |
| | | Lawa |
| | | Whitewood comma, apotome, Pythagorean chroma |
| | |- |
| | | 5 |
| | | [[135/128]] |
| | | {{monzo| -7 3 1 }} |
| | | 92.18 |
| | | Layobi |
| | | Mavila comma, major chroma |
| | |- |
| | | 5 |
| | | [[25/24]] |
| | | {{monzo| -3 -1 2 }} |
| | | 70.67 |
| | | Yoyo |
| | | Dicot comma, classic chroma |
| | |- |
| | | 5 |
| | | [[250/243]] |
| | | {{monzo| 1 -5 3 }} |
| | | 49.17 |
| | | Triyo |
| | | Porcupine comma, maximal diesis |
| | |- |
| | | 5 |
| | | [[20000/19683]] |
| | | {{monzo| 5 -9 4 }} |
| | | 27.66 |
| | | Saquadyo |
| | | Tetracot comma, minimal diesis |
| | |- |
| | | 5 |
| | | [[81/80]] |
| | | {{monzo| -4 4 -1 }} |
| | | 21.51 |
| | | Gu |
| | | Syntonic comma, Didymus' comma, meantone comma |
| | |- |
| | | 5 |
| | | [[1600000/1594323|(14 digits)]] |
| | | {{monzo| 9 -13 5 }} |
| | | 6.15 |
| | | Saquinyo |
| | | [[Amity comma]] |
| | |- |
| | | 7 |
| | | <abbr title="1119744/1071875">(14 digits)</abbr> |
| | | {{monzo| 9 7 -5 -3 }} |
| | | 75.64 |
| | | Triru-aquingu |
| | | [[Superpine comma]] |
| | |- |
| | | 7 |
| | | [[36/35]] |
| | | {{monzo| 2 2 -1 -1 }} |
| | | 48.77 |
| | | Rugu |
| | | Mint comma, septimal quartertone |
| | |- |
| | | 7 |
| | | [[525/512]] |
| | | {{monzo| -9 1 2 1 }} |
| | | 43.41 |
| | | Lazoyoyo |
| | | Avicennma, Avicenna's enharmonic diesis |
| | |- |
| | | 7 |
| | | [[64/63]] |
| | | {{monzo| 6 -2 0 -1 }} |
| | | 27.26 |
| | | Ru |
| | | Septimal comma, Archytas' comma, Leipziger Komma |
| | |- |
| | | 7 |
| | | [[875/864]] |
| | | {{monzo| -5 -3 3 1 }} |
| | | 21.90 |
| | | Zotriyo |
| | | Keema |
| | |- |
| | | 7 |
| | | [[5120/5103]] |
| | | {{monzo| 10 -6 1 -1 }} |
| | | 5.76 |
| | | Saruyo |
| | | Hemifamity comma |
| | |- |
| | | 7 |
| | | [[6144/6125]] |
| | | {{monzo| 11 1 -3 -2 }} |
| | | 5.36 |
| | | Sarurutriyo |
| | | Porwell comma |
| | |- |
| | | 7 |
| | | [[4375/4374]] |
| | | {{monzo| -1 -7 4 1 }} |
| | | 0.40 |
| | | Zoquadyo |
| | | Ragisma |
| | |- |
| | | 7 |
| | | <abbr title="140737488355328/140710042265625">(30 digits)</abbr> |
| | | {{monzo| 47 -7 -7 -7 }} |
| | | 0.34 |
| | | Trisa-rugu |
| | | [[Akjaysma]] |
| | |- |
| | | 11 |
| | | [[33/32]] |
| | | {{monzo| -5 1 0 0 1}} |
| | | 53.27 |
| | | Ilo |
| | | Io comma, undecimal quartertone |
| | |- |
| | | 11 |
| | | [[100/99]] |
| | | {{monzo| 2 -2 2 0 -1 }} |
| | | 17.40 |
| | | Luyoyo |
| | | Ptolemisma |
| | |- |
| | | 11 |
| | | [[121/120]] |
| | | {{monzo| -3 -1 -1 0 2 }} |
| | | 14.37 |
| | | Lologu |
| | | Biyatisma |
| | |- |
| | | 11 |
| | | [[176/175]] |
| | | {{monzo| 4 0 -2 -1 1 }} |
| | | 9.86 |
| | | Lurugugu |
| | | Valinorsma |
| | |- |
| | | 11 |
| | | [[65536/65219]] |
| | | {{monzo| 16 0 0 -2 -3 }} |
| | | 8.39 |
| | | Satrilu-aruru |
| | | Orgonisma |
| | |- |
| | | 11 |
| | | [[243/242]] |
| | | {{monzo| -1 5 0 0 -2 }} |
| | | 7.14 |
| | | Lulu |
| | | Rastma |
| | |- |
| | | 11 |
| | | [[385/384]] |
| | | {{monzo| -7 -1 1 1 1 }} |
| | | 4.50 |
| | | Lozoyo |
| | | Keenanisma |
| | |- |
| | | 11 |
| | | [[4000/3993]] |
| | | {{monzo| 5 -1 3 0 -3 }} |
| | | 3.03 |
| | | Triluyo |
| | | Wizardharry comma |
| | |- |
| | | 13 |
| | | [[14641/13312]] |
| | | {{monzo| -10 0 0 0 4 -1 }} |
| | | 164.74 |
| | | |
| | | |
| | |- |
| | | 13 |
| | | [[52/49]] |
| | | {{monzo| 2 0 0 -2 0 1 }} |
| | | 102.87 |
| | | thoruru unison |
| | | Hammerisma |
| | |- |
| | | 13 |
| | | [[27/26]] |
| | | {{monzo| -1 3 0 0 0 -1 }} |
| | | 65.33 |
| | | Thu |
| | | Small tridecimal third tone |
| | |- |
| | | 13 |
| | | [[65/64]] |
| | | {{monzo| -6 0 1 0 0 1 }} |
| | | 26.84 |
| | | |
| | | Wilsorma |
| | |} |
| | |
| | == Temperaments == |
| | 7edo is the first edo in which [[regular temperament theory]] starts to make sense as a way of subdividing the steps into [[mos scale]]s, with three different ways of dividing it, although there is still quite a lot of ambiguity as each step can be considered as the sharp extreme of one temperament or the flat end of another. |
| | |
| | 1/7 can be considered the intersection of sharp [[porcupine]] and flat [[tetracot]] temperaments, as three steps makes a 4th and four a 5th. 2/7 can be interpreted as critically flat [[Mohajira]] or critically sharp [[amity]], and creates mosses of 322 and 2221. |
| | |
| | 3/7 is on the intersection of [[meantone]] and [[mavila]], and has MOS's of 331 and 21211, making 7edo the first edo with a non-equalized, non-1L''n''s [[pentatonic]] mos. This is in part because 7edo is close to low-complexity JI for its size, and is the second edo with a good fifth for its size (after [[5edo]]), the fifth serving as a generator for the edo's meantone and mavila interpertations. |
| | |
| | == Octave stretch == |
| | What follows is a comparison of stretched-octave 7edo tunings. |
| | |
| | ; 7edo |
| | * Step size: 171.429{{c}}, octave size: 1200.0{{c}} |
| | Pure-octaves 7edo approximates the 2nd, 3rd, 11th and 13th harmonics well for its size, but it's arguable whether it approximates 5 - if it does it does so poorly. It doesn't approximate 7. |
| | {{Harmonics in equal|7|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 7edo}} |
| | {{Harmonics in equal|7|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 7edo (continued)}} |
| | |
| | ; [[WE|7et, 2.3.11.13 WE]] |
| | * Step size: 171.993{{c}}, octave size: 1204.0{{c}} |
| | Stretching the octave of 7edo by around 4{{c}} results in much improved primes 3, 5 and 11, but much worse primes 7 and 13. The 2.3.11.13 WE tuning and 2.3.11.13 [[TE]] tuning both do this. |
| | {{Harmonics in cet|171.993|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 7et, 2.3.11.13 WE}} |
| | {{Harmonics in cet|171.993|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 7et, 2.3.11.13 WE (continued)}} |
| | |
| | ; [[18ed6]] |
| | * Step size: 172.331{{c}}, octave size: 1206.3{{c}} |
| | Stretching the octave of 7edo by around 6{{c}} results in much improved primes 3, 5 and 7, but much worse primes 11 and 13. The tuning 18ed6 does this. |
| | {{Harmonics in equal|18|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 18ed6}} |
| | {{Harmonics in equal|18|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 18ed6 (continued)}} |
| | |
| | ; [[WE|7et, 2.3.5.11.13 WE]] |
| | * Step size: 172.390{{c}}, octave size: 1206.7{{c}} |
| | Stretching the octave of 7edo by around 7{{c}} results in much improved primes 3, 5 and 11, but much worse primes 7 and 13. Its 2.3.5.11.13 WE tuning and 2.3.5.11.13 [[TE]] tuning both do this. |
| | {{Harmonics in cet|172.390|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 7et, 2.3.5.11.13 WE}} |
| | {{Harmonics in cet|172.390|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 7et, 2.3.5.11.13 WE (continued)}} |
| | |
| | ; [[zpi|15zpi]] |
| | * Step size: 172.495{{c}}, octave size: 1207.5{{c}} |
| | Stretching the octave of 7edo by around 7.5{{c}} results in much improved primes 3, 5 and 11, but much worse primes 2, 7 and 13. The tuning 15zpi does this. |
| | {{Harmonics in cet|172.495|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 15zpi}} |
| | {{Harmonics in cet|172.495|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 15zpi (continued)}} |
| | |
| | ; [[11edt]] |
| | * Step size: 172.905{{c}}, octave size: 1210.3{{c}} |
| | Stretching the octave of 7edo by around NNN{{c}} results in much improved primes 3, 5 and 11, but much worse primes 2, 7 and 13. The tuning 11edt does this. |
| | {{Harmonics in equal|11|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 11edt}} |
| | {{Harmonics in equal|11|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 11edt (continued)}} |
| | |
| | == Instruments == |
| | * [[Lumatone mapping for 7edo]] |
| | |
| | == Music == |
| | {{Main| 7edo/Music }} |
| | {{Catrel|7edo tracks}} |
| | |
| | == Ear training == |
| | 7edo ear-training exercises by Alex Ness available [https://drive.google.com/folderview?id=0BwsXD8q2VCYUT3VEZUVmeVZUcmc&usp=drive_web#list here]. |
| | |
| | == Notes == |
| | <references group="note" /> |
| | |
| | == References == |
| | <references /> |
| | |
| | [[Category:3-limit record edos|#]] <!-- 1-digit number --> |
| | [[Category:7-tone scales]] |