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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! 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]] | ||