Ed4/3: Difference between revisions
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An '''equal division of the fourth''' ('''ed4/3''') is an [[equal-step tuning]] in which the perfect fourth ([[4/3]]) is [[Just intonation|justly tuned]] and is divided in a given number of equal steps | An '''equal division of the fourth''' ('''ed4/3''') is an [[equal-step tuning]] in which the perfect fourth ([[4/3]]) is [[Just intonation|justly tuned]] and is divided in a given number of equal steps. | ||
The expression ''equal division of the fourth'' could be interpreted as applying to other [[interval]]s in the region of the fourth (see [[:Category: Fourth]]), such as [[15/11]]. However, these should be named more specifically and be treated on other pages to avoid any confusion. | The expression ''equal division of the fourth'' could be interpreted as applying to other [[interval]]s in the region of the fourth (see [[:Category: Fourth]]), such as [[15/11]]. However, these should be named more specifically and be treated on other pages to avoid any confusion. | ||
The utility of the fourth as | The utility of the fourth as structural scaffolding is apparent by being used at the base of so much Neo-Medieval harmony (see [[tetrachord]]). Division of 4/3 into equal parts does not necessarily imply directly using this interval as an [[equivalence]]. Many, though not all, ed4/3 scales have a perceptually important [[Pseudo-octave|false octave]], with various degrees of accuracy. | ||
One approach to some ed4/3 tunings is the use of the 12:13:14:(16) 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 (an octave-reduced stack of) four [[3/2]] to get to [[5/4]], here it takes (a fourth-reduced stack of) eight [[7/6]] to get to [[13/12]] (tempering out the comma [[5764801/5750784]]). So, doing this yields 13-, 15-, and 28-note [[mos scale]]s for ed4/3's. While the notes are rather closer together, the scheme is uncannily similar to meantone. | |||
== 7-limit, analogy with equal divisions of (3/2) == | == 7-limit, analogy with equal divisions of (3/2) == | ||
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ED4/3 tuning systems that accurately represent the intervals 8/7 and 7/6 include: [[13ed4/3]] (1.31 cent error), [[15ed4/3]] (1.25 cent error), and [[28ed4/3]] (0.06 cent error). | ED4/3 tuning systems that accurately represent the intervals 8/7 and 7/6 include: [[13ed4/3]] (1.31 cent error), [[15ed4/3]] (1.25 cent error), and [[28ed4/3]] (0.06 cent error). | ||
In this sense, [[13ed4/3]], [[15ed4/3]], and [[28ed4/3]] are to the division of the fourth what [[9edf|9ed3/2]], [[11edf|11ed3/2 | In this sense, [[13ed4/3]], [[15ed4/3]], and [[28ed4/3]] are to the division of the fourth what [[9edf|9ed3/2]], [[11edf|11ed3/2]], and [[20edf|20ed3/2]] are to the division of the fifth, and what [[5edo]], [[7edo]], and [[12edo]] are to the division of the octave. | ||
== Individual pages for ed4/3s == | == Individual pages for ed4/3s == | ||
{| class="wikitable center-all" | |||
|+ 0…9 | |||
|- | |||
! Standard name | |||
! Common name | |||
|- | |||
| [[3ed4/3]] | |||
| ED cube root of P4 | |||
|- | |||
| [[4ed4/3]] | |||
| | |||
|- | |||
| [[5ed4/3]] | |||
| Quintilipyth scale <br>{{citation needed|date=December 2021|reason=Who used that term?}} | |||
|- | |||
| [[6ed4/3]] | |||
| Sextilipyth scale <br>{{citation needed|date=December 2021|reason=Who used that term?}} | |||
|- | |||
| [[7ed4/3]] | |||
| | |||
|- | |||
| [[8ed4/3]] | |||
| | |||
|- | |||
| [[9ed4/3]] | |||
| Noleta scale | |||
|} | |||
{| class="wikitable center-all" | |||
|+ style=white-space:nowrap | 10…49 | |||
|- | |||
| [[10ed4/3|10]] | |||
| [[11ed4/3|11]] | |||
| [[12ed4/3|12]] | |||
| [[13ed4/3|13]] | |||
| [[14ed4/3|14]] | |||
| [[15ed4/3|15]] | |||
| [[16ed4/3|16]] | |||
| [[17ed4/3|17]] | |||
| [[18ed4/3|18]] | |||
| [[19ed4/3|19]] | |||
|- | |||
| [[20ed4/3|20]] | |||
| [[21ed4/3|21]] | |||
| [[22ed4/3|22]] | |||
| [[23ed4/3|23]] | |||
| [[24ed4/3|24]] | |||
| [[25ed4/3|25]] | |||
| [[26ed4/3|26]] | |||
| [[27ed4/3|27]] | |||
| [[28ed4/3|28]] | |||
| [[29ed4/3|29]] | |||
|- | |||
| [[30ed4/3|30]] | |||
| [[31ed4/3|31]] | |||
| [[32ed4/3|32]] | |||
| [[33ed4/3|33]] | |||
| [[34ed4/3|34]] | |||
| [[35ed4/3|35]] | |||
| [[36ed4/3|36]] | |||
| [[37ed4/3|37]] | |||
| [[38ed4/3|38]] | |||
| [[39ed4/3|39]] | |||
|- | |||
| [[40ed4/3|40]] | |||
| [[41ed4/3|41]] | |||
| [[42ed4/3|42]] | |||
| [[43ed4/3|43]] | |||
| [[44ed4/3|44]] | |||
| [[45ed4/3|45]] | |||
| [[46ed4/3|46]] | |||
| [[47ed4/3|47]] | |||
| [[48ed4/3|48]] | |||
| [[49ed4/3|49]] | |||
|} | |||
== See also == | == See also == | ||
* [[Square root of 13 over 10]] (previously listed here as an "edIV") | * [[Square root of 13 over 10]] (previously listed here as an "edIV") | ||
[[Category:Ed4/3| ]] <!-- main article --> | [[Category:Ed4/3's| ]] | ||
<!-- main article --> | |||
[[Category:Lists of scales]] | [[Category:Lists of scales]] | ||
{{todo|inline=1|cleanup|improve layout}} | |||
Latest revision as of 19:39, 1 August 2025
An equal division of the fourth (ed4/3) is an equal-step tuning in which the perfect fourth (4/3) is justly tuned and is divided in a given number of equal steps.
The expression equal division of the fourth could be interpreted as applying to other intervals in the region of the fourth (see Category: Fourth), such as 15/11. However, these should be named more specifically and be treated on other pages to avoid any confusion.
The utility of the fourth as structural scaffolding is apparent by being used at the base of so much Neo-Medieval harmony (see tetrachord). Division of 4/3 into equal parts does not necessarily imply directly using this interval as an equivalence. Many, though not all, ed4/3 scales have a perceptually important false octave, with various degrees of accuracy.
One approach to some ed4/3 tunings is the use of the 12:13:14:(16) 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 (an octave-reduced stack of) four 3/2 to get to 5/4, here it takes (a fourth-reduced stack of) eight 7/6 to get to 13/12 (tempering out the comma 5764801/5750784). So, doing this yields 13-, 15-, and 28-note mos scales for ed4/3's. While the notes are rather closer together, the scheme is uncannily similar to meantone.
7-limit, analogy with equal divisions of (3/2)
One of the key advantages of dividing the perfect fifth (3/2) into equal parts is that it creates scales where the interval between the unison (1/1) and the mapped minor third (6/5) is the same as the interval between the mapped major third (5/4) and the perfect fifth (3/2). This symmetry arises because the product of (6/5) and (5/4) equals (3/2). Consequently, the errors in approximating the minor third and the major third are of equal magnitude but in opposite directions. Similarly, when dividing the perfect fourth (4/3) into equal parts, the interval between the unison (1/1) and the mapped septimal major second (8/7) matches the interval between the mapped septimal minor third (7/6) and the perfect fourth (4/3), as (8/7) multiplied by (7/6) equals (4/3). Thus, the errors in approximating the septimal major second and the septimal minor third are also equal in size but opposite in direction. In essence, equal divisions of the perfect fourth (4/3) relate to 7-limit intervals in the same way that equal divisions of the perfect fifth (3/2) relate to 5-limit intervals.
ED4/3 tuning systems that accurately represent the intervals 8/7 and 7/6 include: 13ed4/3 (1.31 cent error), 15ed4/3 (1.25 cent error), and 28ed4/3 (0.06 cent error).
In this sense, 13ed4/3, 15ed4/3, and 28ed4/3 are to the division of the fourth what 9ed3/2, 11ed3/2, and 20ed3/2 are to the division of the fifth, and what 5edo, 7edo, and 12edo are to the division of the octave.
Individual pages for ed4/3s
| Standard name | Common name |
|---|---|
| 3ed4/3 | ED cube root of P4 |
| 4ed4/3 | |
| 5ed4/3 | Quintilipyth scale [citation needed] |
| 6ed4/3 | Sextilipyth scale [citation needed] |
| 7ed4/3 | |
| 8ed4/3 | |
| 9ed4/3 | Noleta scale |
| 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 |
See also
- Square root of 13 over 10 (previously listed here as an "edIV")
|
Todo: cleanup , improve layout |