7edo: Difference between revisions
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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. | 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. | ||
7edo represents a 7-step closed [[circle of fifths]]. | 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. | |||
=== Prime harmonics === | === Prime harmonics === | ||
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[[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. | [[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. | ||
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 | 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. | ||
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). | 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{{c}}), especially at points of rest. In this manner, the principles of equidistance and harmonic euphony are accommodated within one tonal-harmonic system. | 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. | ||
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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> | 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> | ||
=== Octave stretch === | === Octave stretch === | ||
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! rowspan="2" | [[Cent]]s | ! rowspan="2" | [[Cent]]s | ||
! rowspan="2" | [[Interval region]] | ! rowspan="2" | [[Interval region]] | ||
! colspan=" | ! colspan="4" | Approximated [[JI]] intervals ([[error]] in [[¢]]) | ||
! rowspan="2" | Audio | ! rowspan="2" | Audio | ||
|- | |- | ||
! [[3-limit]] | ! [[3-limit]] | ||
! [[5-limit]] | ! [[5-limit]] | ||
! [[7-limit]] | ! [[7-limit]] | ||
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| Unison (prime) | | Unison (prime) | ||
| [[1/1]] (just) | | [[1/1]] (just) | ||
| | | | ||
| | | | ||
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| 171.429 | | 171.429 | ||
| Submajor second | | Submajor second | ||
| | | | ||
| [[10/9]] (-10.975) | | [[10/9]] (-10.975) | ||
| [[54/49]] (+3.215) | | [[54/49]] (+3.215) | ||
| [[11/10]] (+6.424)<br | | [[11/10]] (+6.424)<br>[[32/29]] (-1.006) | ||
| [[File:0-171,43 second (7-EDO).mp3|frameless]] | | [[File:0-171,43 second (7-EDO).mp3|frameless]] | ||
|- | |- | ||
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| Neutral third | | Neutral third | ||
| | | | ||
| | | | ||
| [[128/105]] (+0.048) | | [[128/105]] (+0.048) | ||
| <br />[[11/9]] (-4.551) | | [[39/32]] (+0.374)<br>[[16/13]] (-16.6)<br>[[11/9]] (-4.551) | ||
| [[File:piano_2_7edo.mp3]] | | [[File:piano_2_7edo.mp3]] | ||
|- | |- | ||
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| Fourth | | Fourth | ||
| [[4/3]] (+16.241) | | [[4/3]] (+16.241) | ||
| [[27/20]] (-5.265) | | [[27/20]] (-5.265) | ||
| | | | ||
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| Fifth | | Fifth | ||
| [[3/2]] (-16.241) | | [[3/2]] (-16.241) | ||
| [[40/27]] (+5.265) | | [[40/27]] (+5.265) | ||
| | | | ||
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| Neutral sixth | | Neutral sixth | ||
| | | | ||
| | |||
| | |||
| [[105/64]] (-0.048) | | [[105/64]] (-0.048) | ||
| [[18/11]] (+4.551)<br /> | | [[18/11]] (+4.551)<br>[[13/8]] (+16.6)<br>[[64/39]] (-0.374) | ||
| [[File:0-857,14 sixth (7-EDO).mp3|frameless]] | | [[File:0-857,14 sixth (7-EDO).mp3|frameless]] | ||
|- | |- | ||
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| Supraminor seventh | | Supraminor seventh | ||
| | | | ||
| [[9/5]] (+10.975) | | [[9/5]] (+10.975) | ||
| [[49/27]] (-3.215) | | [[49/27]] (-3.215) | ||
| [[29/16]] (-1.006)<br | | [[29/16]] (-1.006)<br>[[20/11]] (-6.424) | ||
|[[File:0-1028,57 seventh (7-EDO).mp3|frameless]] | | [[File:0-1028,57 seventh (7-EDO).mp3|frameless]] | ||
|- | |- | ||
| 7 | | 7 | ||
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| Octave | | Octave | ||
| [[2/1]] (just) | | [[2/1]] (just) | ||
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== Approximation to JI == | == Approximation to JI == | ||
[[File:7ed2-001.svg | [[File:7ed2-001.svg]] | ||
== Regular temperament properties == | == Regular temperament properties == | ||
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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. | 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 | 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. | ||
== Instruments == | |||
* [[Lumatone mapping for 7edo]] | |||
== Music == | == Music == | ||
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<references /> | <references /> | ||
[[Category:3-limit record edos|#]] <!-- 1-digit number --> | |||
[[Category:7-tone scales]] | [[Category:7-tone scales]] | ||