Ed5/3: Difference between revisions

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== Properties ==
== Properties ==
Division of 5:3 into equal parts can be conceived of as to directly use this interval as an equivalence, or not. The question of [[equivalence]] has not even been posed yet. The utility of 5:3, 11:7 or another sixth as a base though, is apparent by being named directly in the standard definition of such as the octave based [[Sensi|sensi]] temperament or factoring into chord inversions. 5/3 is also the most consonant interval in the range between 3/2 and 2/1, which makes the equivalence easier to hear than for more complex ratios. Many, though not all, of these scales have a pseudo (false) octave, with various degrees of accuracy, but which context(s), if any, it is very perceptually important in is as yet an open question.
Division of 5/3 into equal parts does not necessarily imply directly using this interval as an [[equivalence]]. Many, though not all, ed5/3 scales have a perceptually important [[Pseudo-octave|false octave]], with various degrees of accuracy.


Incidentally, one way to treat 5/3 as an equivalence is the use of the 6:7:8:(10) chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in meantone. Whereas in meantone it takes four 3/2 to get to 5/1, here it takes four 4/3 to get to 8/7 (tempering out the comma 225/224). So, doing this yields 7, 9, and 16 note MOS either way, the 16 note MOS being 7L 9s. While the notes are rather closer together, the scheme is uncannily similar to meantone. "Microdiatonic" might be a good term for it (even better than for edfs as the generator it uses is an excellent fit for heptatonic MOS) if it hasn't been named yet.
5/3 is the most consonant interval in between 1/1 and 2/1, so this suggests it could be useful either as an equivalence, or as just an important structural feature.


If we instead opt to continue using 4:5:6 as the fundamental sonority, then it will take three 3/2 to get to 5/4, resulting in [[Blackcomb]] temperament that tempers out the comma 250/243. This yields MOS scales of 4, 5, 6, 11, 16, and 21 notes. Although, it should be noted that doing this will often create a pseudo-octave unlike the 6:7:8 approach.
[[Joseph Ruhf]] suggested the use of the 6:7:8:(10) chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in meantone as a way to evoke 5/3-equivalence. Though it could also be used just as a useful sonority, even without equivalence. Whereas in meantone it takes four 3/2 to get to 5/1, here it takes four 4/3 to get to 8/7 (tempering out the comma 225/224). So, doing this yields 7-, 9-, and 16-note [[mos]] either way, the 16-note mos being 7L 9s. While the notes are rather closer together, the scheme is uncannily similar to meantone. "Microdiatonic" might be a good term for it (even better than for [[edf]]s as the generator it uses is an excellent fit for heptatonic mos) though it is, technically speaking, micro-[[7L 2s|armotonic]].


== Individual pages for ED5/3's ==
If we instead opt to continue using 4:5:6 as the fundamental sonority, then it will take three 3/2 to get to 5/4, resulting in [[Blackcomb]] temperament that tempers out the comma 250/243. This yields mos scales of 4, 5, 6, 11, 16, and 21 notes. Although, it should be noted that doing this will often create a pseudo-octave unlike the 6:7:8 approach.
* [[2ed5/3]]
* [[3ed5/3]]
* [[7ed5/3]]
* [[8ed5/3]]
* [[9ed5/3]]
* [[16ed5/3]]
* [[23ed5/3]]


[[Category:Ed5/3| ]] <!-- main article -->
ED5/3 tuning systems that accurately represent the intervals 5/4 and 4/3 include: [[7ed5/3]] (7.30 cent error), [[9ed5/3]] (6.73 cent error), and [[16ed5/3]] (0.59 cent error).
[[Category:Equal-step tuning]]
 
[[7ed5/3]], [[9ed5/3]], and [[16ed5/3]] are to the [[Ed5/3|division of 5/3]] what [[5edo]], [[7edo]], and [[12edo]] are to the [[EDO|division of 2/1]].
 
== Individual pages for ed5/3's ==
{| class="wikitable center-all"
|+ style=white-space:nowrap | 0…49
| [[0ed5/3|0]]
| [[1ed5/3|1]]
| [[2ed5/3|2]]
| [[3ed5/3|3]]
| [[4ed5/3|4]]
| [[5ed5/3|5]]
| [[6ed5/3|6]]
| [[7ed5/3|7]]
| [[8ed5/3|8]]
| [[9ed5/3|9]]
|-
| [[10ed5/3|10]]
| [[11ed5/3|11]]
| [[12ed5/3|12]]
| [[13ed5/3|13]]
| [[14ed5/3|14]]
| [[15ed5/3|15]]
| [[16ed5/3|16]]
| [[17ed5/3|17]]
| [[18ed5/3|18]]
| [[19ed5/3|19]]
|-
| [[20ed5/3|20]]
| [[21ed5/3|21]]
| [[22ed5/3|22]]
| [[23ed5/3|23]]
| [[24ed5/3|24]]
| [[25ed5/3|25]]
| [[26ed5/3|26]]
| [[27ed5/3|27]]
| [[28ed5/3|28]]
| [[29ed5/3|29]]
|-
| [[30ed5/3|30]]
| [[31ed5/3|31]]
| [[32ed5/3|32]]
| [[33ed5/3|33]]
| [[34ed5/3|34]]
| [[35ed5/3|35]]
| [[36ed5/3|36]]
| [[37ed5/3|37]]
| [[38ed5/3|38]]
| [[39ed5/3|39]]
|-
| [[40ed5/3|40]]
| [[41ed5/3|41]]
| [[42ed5/3|42]]
| [[43ed5/3|43]]
| [[44ed5/3|44]]
| [[45ed5/3|45]]
| [[46ed5/3|46]]
| [[47ed5/3|47]]
| [[48ed5/3|48]]
| [[49ed5/3|49]]
|}
 
[[Category:Ed5/3's| ]]
<!-- main article -->
[[Category:Edonoi]]
[[Category:Lists of scales]]
 
 
{{todo|inline=1|cleanup|explain edonoi|text=Most people do not think 5/3 sounds like an equivalence, so there must be some other reason why people are dividing it — some property ''other than'' equivalence that makes people want to divide it. Please add to this page an explanation of what that reason is... The page also needs a general overall cleanup.}}

Latest revision as of 18:40, 1 August 2025

The equal division of 5/3 (ed5/3) is a tuning obtained by dividing the just major sixth (5/3) into a number of equal steps.

Properties

Division of 5/3 into equal parts does not necessarily imply directly using this interval as an equivalence. Many, though not all, ed5/3 scales have a perceptually important false octave, with various degrees of accuracy.

5/3 is the most consonant interval in between 1/1 and 2/1, so this suggests it could be useful either as an equivalence, or as just an important structural feature.

Joseph Ruhf suggested the use of the 6:7:8:(10) chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in meantone as a way to evoke 5/3-equivalence. Though it could also be used just as a useful sonority, even without equivalence. Whereas in meantone it takes four 3/2 to get to 5/1, here it takes four 4/3 to get to 8/7 (tempering out the comma 225/224). So, doing this yields 7-, 9-, and 16-note mos either way, the 16-note mos being 7L 9s. While the notes are rather closer together, the scheme is uncannily similar to meantone. "Microdiatonic" might be a good term for it (even better than for edfs as the generator it uses is an excellent fit for heptatonic mos) though it is, technically speaking, micro-armotonic.

If we instead opt to continue using 4:5:6 as the fundamental sonority, then it will take three 3/2 to get to 5/4, resulting in Blackcomb temperament that tempers out the comma 250/243. This yields mos scales of 4, 5, 6, 11, 16, and 21 notes. Although, it should be noted that doing this will often create a pseudo-octave unlike the 6:7:8 approach.

ED5/3 tuning systems that accurately represent the intervals 5/4 and 4/3 include: 7ed5/3 (7.30 cent error), 9ed5/3 (6.73 cent error), and 16ed5/3 (0.59 cent error).

7ed5/3, 9ed5/3, and 16ed5/3 are to the division of 5/3 what 5edo, 7edo, and 12edo are to the division of 2/1.

Individual pages for ed5/3's

0…49
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49


Todo: cleanup , explain edonoi

Most people do not think 5/3 sounds like an equivalence, so there must be some other reason why people are dividing it — some property other than equivalence that makes people want to divide it. Please add to this page an explanation of what that reason is... The page also needs a general overall cleanup.

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