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]].
== Theory ==
[[File:7edo scale.mp3|thumb|A chromatic 7edo scale on C.]]
Equal-heptatonic scales are used in non-western music in African cultures as well as an integral part of early Thai and early Chinese music. 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).
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.
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). As a temperament, William Lynch gives it the name "Neutron[7]" just as the whole tone scale of 12 ET is known as "Hexe[6]".
Typically, 7-edo exists as the tuning for pentatonic scales in traditional thai music with the other two pitches acting as auxiliary tones. However, it can be used as an interesting diatonic scale choice as well in tunings such as 14 EDO or 21 EDO.
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]].
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.
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.
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.
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.
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.
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.
=Intervals in 7-edo=
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.
|| **Interval** || **Interval**
**size**
**in ¢ (coarse, fine), DMS** || **Closest**
**diatonic**
**interval name** || **The "neighborhood" of just intervals** ||
|| 1 || 171.429, 205.714, 51°25'43" || second || 6.424 c (coarse), 7.709 c (fine), 1°55'38" from Ptolemy (neutral) second 11/10
3.215 c (coarse), 3.8585 c (fine), 57'53" from second 54/49
-1.006 c (coarse), -1.207 c (fine), -18'6" from the 29th subharmonic 32/29
-10.975<span style="line-height: 1.5;"> c (coarse), -13.17 c (fine), -3°17'33" from (Didymus) major second (small whole tone) 10/9</span> ||
|| 2 || 342.857, 411.429,
102°51'26" || third || -4.551 c (coarse), -5.461 c (fine), -1°21'55" from neutral third 11/9 ||
|| 3 || 514.286, 617.143,
154°17'9" || fourth || 16.241 c (coarse), 19.489 c (fine), 4°52'20" from just fourth 4/3 (498.045 c coarse, 597.654 c fine)
-5.265 c (coarse), -6.319 c (fine), -1°34'47" from wide fourth 27/20 ||
|| 4 || 685.714, 822.857,
205°42'51" || fifth || 5.265 c (coarse), 6.319 c (fine), 1°34'47" from narrow fifth 40/27
-16.241 c (coarse), -19.489 c (fine)<span style="line-height: 1.5;">, -4°52'20" from just fifth 3/2 (701.955 c coarse, 842.346 c fine)</span> ||
|| 5 || 857.143, 1028.571,
257°8'34" || sixth || 4.551 c (coarse), 5.461 c (fine), 1°21'55" from neutral sixth 18/11 ||
<span style="line-height: 1.5;"> 308°34'17"1</span> || seventh || 10.975 c (coarse), 13.17 c (fine), 3°17'33" from (Didymus) minor seventh 9/5
-6.424 c (coarse), -7.709 c (fine), -1°55'38" from neutral seventh 20/11
1.006 c (coarse), 1.207 c (fine), 18'6" from the 29th harmonic 29/16
-3.215 c (coarse), -3.8585 c (fine), -57'53" from seventh 49/27 ||
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.
[[file:7ed2-001.svg]]
=== Prime harmonics ===
{{Harmonics in equal|7}}
=== In non-Western traditions ===
[[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.
=Related scales=
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.
* Related in a lateral way to traditional Thai music. Subset of 14 EDO and 21 EDO.
==Notation==
* Rotatable 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]].
=Alphabet=
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).
William Lynch proposes using numbers 1 through 7 as the nominals of 7 ET with # signs being possible to expand to 14 EDO or even 21 EDO.
The alphabet can be written in three ways:
Plain numbers: 1 2 3 4 5 6 7
In Japanese number abbreviations: い に さん し ご ろ な い
In Chinese/Kanji numbers: 一 二 三 四 五 六七
=== ===
=Music=
* [[http://sethares.engr.wisc.edu/mp3s/PagansRevenge.mp3|Pagan's Revenge]] by Bill Sethares (synthetic gamelan)
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.
* [[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/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]]
* [[https://soundcloud.com/santiagocosentino/rock-in-7edo|Rock in 7edo by Santiago Cosentino]]
=Ear Training=
A [[Ugandan]], [[Chopi]] xylophone measured by Haddon (1952) was also tuned something close to this.
7 EDO ear-training exercises by Alex Ness available [[@https://drive.google.com/folderview?id=0BwsXD8q2VCYUT3VEZUVmeVZUcmc&usp=drive_web#list|here]].
=Commas=
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>
7 EDO tempers out the following commas. (Note: This assumes val < [[tel/7 11 16 20 24 26|7 11 16 20 24 26]] |.)
||~ Comma ||~ Monzo ||~ Cents ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
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.
||= 135/128 || | -7 3 1 > ||> 92.18 ||= Major Chroma ||= Major Limma ||= Pelagic Comma ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc0"><a name="x&quot;Neutral diatonic&quot; or &quot;Neutron[7]&quot;"></a><!-- ws:end:WikiTextHeadingRule:2 -->&quot;Neutral diatonic&quot; or &quot;Neutron[7]&quot;</h1>
! rowspan="2" | [[Interval region]]
<br />
! colspan="4" | Approximated [[JI]] intervals ([[error]] in [[¢]])
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 />
! rowspan="2" | Audio
<br />
|-
Equal-heptatonic scales are used in non-western music in African cultures as well as an integral part of early Thai and early Chinese music. 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 />
! [[3-limit]]
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). As a temperament, William Lynch gives it the name &quot;Neutron[7]&quot; just as the whole tone scale of 12 ET is known as &quot;Hexe[6]&quot;.<br />
! [[5-limit]]
<br />
! [[7-limit]]
Typically, 7-edo exists as the tuning for pentatonic scales in traditional thai music with the other two pitches acting as auxiliary tones. However, it can be used as an interesting diatonic scale choice as well in tunings such as 14 EDO or 21 EDO.<br />
! Other
<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 />
| 0
<br />
| 0
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 />
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:4:&lt;h1&gt; --><h1 id="toc1"><a name="Intervals in 7-edo"></a><!-- ws:end:WikiTextHeadingRule:4 -->Intervals in 7-edo</h1>
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]].
<td>6.424 c (coarse), 7.709 c (fine), 1°55'38&quot; from Ptolemy (neutral) second 11/10<br />
3.215 c (coarse), 3.8585 c (fine), 57'53&quot; from second 54/49<br />
-1.006 c (coarse), -1.207 c (fine), -18'6&quot; from the 29th subharmonic 32/29<br />
-10.975<span style="line-height: 1.5;"> c (coarse), -13.17 c (fine), -3°17'33&quot; from (Didymus) major second (small whole tone) 10/9</span><br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>342.857, 411.429,<br />
102°51'26&quot;<br />
</td>
<td>third<br />
</td>
<td>-4.551 c (coarse), -5.461 c (fine), -1°21'55&quot; from neutral third 11/9<br />
</td>
</tr>
<tr>
<td>3<br />
</td>
<td>514.286, 617.143,<br />
154°17'9&quot;<br />
</td>
<td>fourth<br />
</td>
<td>16.241 c (coarse), 19.489 c (fine), 4°52'20&quot; from just fourth 4/3 (498.045 c coarse, 597.654 c fine)<br />
-5.265 c (coarse), -6.319 c (fine), -1°34'47&quot; from wide fourth 27/20<br />
</td>
</tr>
<tr>
<td>4<br />
</td>
<td>685.714, 822.857,<br />
205°42'51&quot;<br />
</td>
<td>fifth<br />
</td>
<td>5.265 c (coarse), 6.319 c (fine), 1°34'47&quot; from narrow fifth 40/27<br />
-16.241 c (coarse), -19.489 c (fine)<span style="line-height: 1.5;">, -4°52'20&quot; from just fifth 3/2 (701.955 c coarse, 842.346 c fine)</span><br />
</td>
</tr>
<tr>
<td>5<br />
</td>
<td>857.143, 1028.571,<br />
257°8'34&quot;<br />
</td>
<td>sixth<br />
</td>
<td>4.551 c (coarse), 5.461 c (fine), 1°21'55&quot; from neutral sixth 18/11<br />
<td>10.975 c (coarse), 13.17 c (fine), 3°17'33&quot; from (Didymus) minor seventh 9/5<br />
-6.424 c (coarse), -7.709 c (fine), -1°55'38&quot; from neutral seventh 20/11<br />
1.006 c (coarse), 1.207 c (fine), 18'6&quot; from the 29th harmonic 29/16<br />
-3.215 c (coarse), -3.8585 c (fine), -57'53&quot; from seventh 49/27<br />
</td>
</tr>
<tr>
<td>7 (0)<br />
</td>
<td>1200.0, 360º<br />
</td>
<td>eighth<br />
</td>
<td>exactly 2/1<br />
</td>
</tr>
</table>
<br />
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.
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<ul><li>Related in a lateral way to traditional Thai music. Subset of 14 EDO and 21 EDO.</li></ul><!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc3"><a name="Related scales-Notation"></a><!-- ws:end:WikiTextHeadingRule:8 -->Notation</h2>
<ul><li>Rotatable on a five-line staff without accidentals.</li></ul><!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc4"><a name="Related scales-Harmony"></a><!-- ws:end:WikiTextHeadingRule:10 -->Harmony</h2>
There is no distinction between Major or Minor; each pitch class is unique.<br />
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 />
7-edo is the third <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists">zeta integral edo</a>.<br />
7 EDO ear-training exercises by Alex Ness available <a class="wiki_link_ext" href="https://drive.google.com/folderview?id=0BwsXD8q2VCYUT3VEZUVmeVZUcmc&amp;usp=drive_web#list" rel="nofollow" target="_blank">here</a>.<br />
7 EDO tempers out the following commas. (Note: This assumes val &lt; <a class="wiki_link" href="http://tel.wikispaces.com/7%2011%2016%2020%2024%2026">7 11 16 20 24 26</a> |.)<br />
<br />
{| 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'''
|}
<table class="wiki_table">
In 7edo:
<tr>
* [[Ups and downs notation]] is identical to circle-of-fifths notation;
<th>Comma<br />
* Mixed and pure [[sagittal notation]] are identical to circle-of-fifths 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]].
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.
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)}}
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)}}
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)}}
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)}}
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)}}
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 ==
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== References ==
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[[Category:3-limit record edos|#]] <!-- 1-digit number -->
7 equal divisions of the octave (abbreviated 7edo or 7ed2), also called 7-tone equal temperament (7tet) or 7 equal temperament (7et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 7 equal parts of about 171 ¢ each. Each step represents a frequency ratio of 21/7, or the 7th root of 2.
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 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 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.
The second (171.429 ¢) 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.
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.
Due to 7edo's inaccurately tuned 5/4major 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 thirds, simultaneously representing 5/4 and the minor third6/5) 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.
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 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.
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.
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.
Equiheptatonic scales close to 7edo are used in non-western music in some African cultures[1] as well as an integral part of early Chinese music[2]. Also Georgian music seems to be based on near-equal 7-step scales.
It has been speculated in Indian music: history and structure[3] that the Indian three-sruti interval of 165 cents is very similar to one 171-cent step of 7edo.
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).
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 ¢), especially at points of rest. In this manner, the principles of equidistance and harmonic euphony are accommodated within one tonal-harmonic system.
A Ugandan, Chopi xylophone measured by Haddon (1952) was also tuned something close to this.
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.[4]
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.
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.
7edo is the first edo in which regular temperament theory starts to make sense as a way of subdividing the steps into mos scales, 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-1Lns 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 ¢, octave size: 1200.0 ¢
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.
Stretching the octave of 7edo by around 4 ¢ 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.
Stretching the octave of 7edo by around 7 ¢ 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.
Stretching the octave of 7edo by around 7.5 ¢ results in much improved primes 3, 5 and 11, but much worse primes 2, 7 and 13. The tuning 15zpi does this.
Stretching the octave of 7edo by around NNN ¢ results in much improved primes 3, 5 and 11, but much worse primes 2, 7 and 13. The tuning 11edt does this.
