8ed6: Difference between revisions

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Created page with "{{Infobox ET}} {{ED intro}} == Intervals == {| class="wikitable" |+ !# !Cents !Approximate JI ratio(s) |- |0 |0.000 |exact 1/1 |- |1 |387.744 |5/4, 4/3, 6/5, 7/6, 9/7, 10/7, 9/8, 11/9, 11/10, 12/11 |- |2 |775.489 |3/2, 11/7 |- |3 |1163.233 |2/1 |- |4 |1550.978 |5/2, 7/3 |- |5 |1938.722 |3/1 |- |6 |2326.466 |4/1 |- |7 |2714.211 |5/1 |- |8 |3101.955 |exact 6/1 |} == Harmonics == {| class="wikitable" |+ !# !2 !3 !4 !5 !6 !7 !8 !9 !10 !11 !12 |- |Steps |3 |5 |6 |7 |8 |9..."
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{{Mathematical interest}}
{{Infobox ET}}
{{Infobox ET}}
{{ED intro}}


{{ED intro}}  
== Theory ==
8ed6 can be thought of as a subset (where the ~5/4 generator is stacked) of the 6/1-eigenmonzo tuning of [[würschmidt]].
 
=== Harmonics ===
{{Harmonics in equal|8|6|1}}


== Intervals ==
== Intervals ==
{| class="wikitable"
{| class="wikitable right-2"
|+
!#
!Cents
!Approximate JI ratio(s)
|-
|-
|0
! #
|0.000
! Cents
|exact 1/1
! Approximate JI ratio(s)
|-
|-
|1
| 0
|387.744
| 0
|5/4, 4/3, 6/5, 7/6, 9/7, 10/7, 9/8, 11/9, 11/10, 12/11
| 1/1
|-
|-
|2
| 1
|775.489
| 388
|3/2, 11/7
| 5/4
|-
|-
|3
| 2
|1163.233
| 775
|2/1
| 11/7, 25/16
|-
|-
|4
| 3
|1550.978
| 1163
|5/2, 7/3
|
|-
|-
|5
| 4
|1938.722
| 1551
|3/1
| 22/9
|-
|-
|6
| 5
|2326.466
| 1939
|4/1
| 49/16
|-
|-
|7
| 6
|2714.211
| 2326
|5/1
|
|-
|8
|3101.955
|exact 6/1
|}
 
== Harmonics ==
{| class="wikitable"
|+
!#
!2
!3
!4
!5
!6
!7
!8
!9
!10
!11
!12
|-
|-
|Steps
| 7
|3
| 2714
|5
| 24/5
|6
|7
|8
|9
|9
|10
|10
|11
|11
|-
|-
|Reduced
| 8
|3
| 3102
|5
| 6/1
|6
|7
|0
|1
|1
|2
|2
|3
|3
|}
|}

Latest revision as of 22:19, 10 August 2025

This page presents a topic of primarily mathematical interest.

While it is derived from sound mathematical principles, its applications in terms of utility for actual music may be limited, highly contrived, or as yet unknown.

← 7ed6 8ed6 9ed6 →
Prime factorization 23
Step size 387.744 ¢ 
Octave 3\8ed6 (1163.23 ¢)
(semiconvergent)
Twelfth 5\8ed6 (1938.72 ¢)
(semiconvergent)
Consistency limit 6
Distinct consistency limit 4

8 equal divisions of the 6th harmonic (abbreviated 8ed6) is a nonoctave tuning system that divides the interval of 6/1 into 8 equal parts of about 388 ¢ each. Each step represents a frequency ratio of 61/8, or the 8th root of 6.

Theory

8ed6 can be thought of as a subset (where the ~5/4 generator is stacked) of the 6/1-eigenmonzo tuning of würschmidt.

Harmonics

Approximation of harmonics in 8ed6
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -37 +37 -74 -72 +0 +121 -110 +74 -109 +114 -37
Relative (%) -9.5 +9.5 -19.0 -18.6 +0.0 +31.2 -28.4 +19.0 -28.1 +29.4 -9.5
Steps
(reduced) 		3
(3) 		5
(5) 		6
(6) 		7
(7) 		8
(0) 		9
(1) 		9
(1) 		10
(2) 		10
(2) 		11
(3) 		11
(3)

Intervals

# Cents Approximate JI ratio(s)
0 0 1/1
1 388 5/4
2 775 11/7, 25/16
3 1163
4 1551 22/9
5 1939 49/16
6 2326
7 2714 24/5
8 3102 6/1
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