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'''4EDT''' is the [[Edt|equal division of the third harmonic]] into four parts of 475.4888 [[cent|cents]] each, corresponding to 2.5237 [[edo]].
{{Infobox ET}}
{{ED intro}}


The 4th root of 3, might be viewed alternately as a degenerate form or a fundamental building block of Bohlen-Pierce harmony, analogous to how 5edo relates to diatonic music. The situation is different however, as in this case both 5 and 7 are relatively well represented (opposed to just 3 in 5edo). While the approximations may seem excessively vague, and some might say impossible, they are nevertheless categorically important to the perception of the scale, and, may even be heard as those harmonies given the width of the "scale". It is doubtful however, that this scale could recieve much melodic treatment, and is more useful as a harmonic entity, either to demonstrate BP harmony, or as a component of scales like [[8edt]].
== Theory ==
4edt fails to approximate a lot of low prime harmonics well—the first prime harmonic that is approximated by 4edt within 10 cents is 47. (Compare this to [[3edt]], which has the 13th harmonic, and [[5edt]], which has the 5th harmonic.) Nevertheless, in terms of [[convergent]]s, 4edt manages to accurately approximate [[25/19]] with one step, and a less accurate [[16/7]] with three steps (hence [[21/16]] with one step). However, it is not until edts with further multiples of 4 (i.e. [[8edt]], [[12edt]], etc.) that these intervals see practical use.


{| class="wikitable"
4edt can be viewed as a "collapsed" version of the [[Bohlen–Pierce]] [[4L 5s (3/1-equivalent)|lambda scale]], analogous to how 5edo is a [[collapsed]] version of the [[diatonic]] scale.  While the approximation for the 5th and 7th harmonics by 4edt may seem excessively vague (or even impossibly vague, as some might say), they are nevertheless categorically important to the perception of the scale{{clarify}}. Given the width of the "scale", 4edt can even be perceived as within the modal logic of Bohlen-Pierce harmony. However, it is doubtful that this scale could receive much melodic treatment, and is more useful as an abstract harmonic entity, either to skeletonize BP harmony, or serving as a subset of scales like [[8edt]].
 
=== Harmonics ===
{{Harmonics in equal|4|3|1|columns=15}}
 
=== Approximation of intervals ===
{| class="wikitable right-all left-4 left-5"
|-
|-
! | degree
! Steps
! | cents value
! Cents
!hekts
! [[Hekt]]s
! | corresponding <br>JI intervals
! Corresponding<br>JI intervals
! | comments
! Comments
|-
|-
| | 0
| 0
|  colspan="2"| 0.0000
| 0.0000
| | '''exact [[1/1]]'''
| 0
| |
| [[1/1]]
| perfect unison
|-
|-
| | 1
| 1
| | 475.4888
| 475.4888
|325
| 325
| | [[17/13]], [[21/16]], 25/19, 33/25
| [[17/13]], [[21/16]], [[25/19]], 33/25
| |  
|  
|-
|-
| | 2
| 2
| | 950.9775
| 950.9775
|650
| 650
| | [[19/11]], 45/26, [[26/15]], (85/49), 33/19
| [[19/11]], 45/26, [[26/15]], (85/49), 33/19
| |
| one step of [[2edt]]
|-
|-
| | 3
| 3
| | 1426.4663
| 1426.4663
|975
| 975
| | 25/11, 57/25, [[16/7]], 39/17
| 25/11, 57/25, [[16/7]], 39/17
| |  
|
|-
|-
| | 4
| 4
| | 1901.9550
| 1901.9550
|1300
| 1300
| | '''exact [[3/1]]'''
| [[3/1]]
| | [[3/2|just perfect fifth]] plus an octave
| [[3/2|just perfect fifth]] plus an octave
|}
|}


==Related regular temperaments==
== Related regular temperaments ==
4EDT is a generator of the [[Vulture family|vulture temperament]], which tempers out 10485760000/10460353203 in the 5-limit.
4EDT is a generator of the [[Vulture family|vulture temperament]], which tempers out 10485760000/10460353203 in the 5-limit.


[[Category:Edt]]
[[Category:Edonoi]]
[[Category:Macrotonal]]
[[Category:Macrotonal]]

Latest revision as of 19:23, 1 August 2025

← 3edt 4edt 5edt →
Prime factorization 22
Step size 475.489 ¢ 
Octave 3\4edt (1426.47 ¢)
Consistency limit 3
Distinct consistency limit 3
Special properties

4 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 4edt or 4ed3), is a nonoctave tuning system that divides the interval of 3/1 into 4 equal parts of about 475 ¢ each. Each step represents a frequency ratio of 31/4, or the 4th root of 3.

Theory

4edt fails to approximate a lot of low prime harmonics well—the first prime harmonic that is approximated by 4edt within 10 cents is 47. (Compare this to 3edt, which has the 13th harmonic, and 5edt, which has the 5th harmonic.) Nevertheless, in terms of convergents, 4edt manages to accurately approximate 25/19 with one step, and a less accurate 16/7 with three steps (hence 21/16 with one step). However, it is not until edts with further multiples of 4 (i.e. 8edt, 12edt, etc.) that these intervals see practical use.

4edt can be viewed as a "collapsed" version of the Bohlen–Pierce lambda scale, analogous to how 5edo is a collapsed version of the diatonic scale. While the approximation for the 5th and 7th harmonics by 4edt may seem excessively vague (or even impossibly vague, as some might say), they are nevertheless categorically important to the perception of the scale[clarification needed]. Given the width of the "scale", 4edt can even be perceived as within the modal logic of Bohlen-Pierce harmony. However, it is doubtful that this scale could receive much melodic treatment, and is more useful as an abstract harmonic entity, either to skeletonize BP harmony, or serving as a subset of scales like 8edt.

Harmonics

Approximation of harmonics in 4edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Error Absolute (¢) +226 +0 -23 +67 +226 -40 +204 +0 -182 +128 -23 -161 +186 +67 -45
Relative (%) +47.6 +0.0 -4.7 +14.0 +47.6 -8.5 +42.9 +0.0 -38.4 +26.9 -4.7 -33.9 +39.1 +14.0 -9.5
Steps
(reduced) 		3
(3) 		4
(0) 		5
(1) 		6
(2) 		7
(3) 		7
(3) 		8
(0) 		8
(0) 		8
(0) 		9
(1) 		9
(1) 		9
(1) 		10
(2) 		10
(2) 		10
(2)

Approximation of intervals

Steps Cents Hekts Corresponding
JI intervals 		Comments
0 0.0000 0 1/1 perfect unison
1 475.4888 325 17/13, 21/16, 25/19, 33/25
2 950.9775 650 19/11, 45/26, 26/15, (85/49), 33/19 one step of 2edt
3 1426.4663 975 25/11, 57/25, 16/7, 39/17
4 1901.9550 1300 3/1 just perfect fifth plus an octave

Related regular temperaments

4EDT is a generator of the vulture temperament, which tempers out 10485760000/10460353203 in the 5-limit.

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