27ed4: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
Ayceman (talk | contribs)
autopatrolled, emailconfirmed
173 edits
Interval table and short MOS discussion.
Xenwolf (talk | contribs)
autopatrolled, Bureaucrats, confirmaccount-notify, Interface administrators, Sysops
8,705 edits
formatting: in tables, it helps to have a space between meta chars and the actual cell content. I suggest a different styling of lines 0, 4, 23, 27
Line 5: Line 5:
== Intervals ==
== Intervals ==
The following table of intervals uses the 7-note (6L 1s) [[MOS scale]] of nelindic for the naturals, using a simple A-G notation and standard sharps/flats for the [[chroma]]. Extended to the 13-note (7L 6s) scale, these would include all of the sharps except for F#. Due to the L/s ratio of 3:1, in the 13-note case, most former diminished intervals become minor, most former minor intervals become augmented, and most former augmented intervals become major.
The following table of intervals uses the 7-note (6L 1s) [[MOS scale]] of nelindic for the naturals, using a simple A-G notation and standard sharps/flats for the [[chroma]]. Extended to the 13-note (7L 6s) scale, these would include all of the sharps except for F#. Due to the L/s ratio of 3:1, in the 13-note case, most former diminished intervals become minor, most former minor intervals become augmented, and most former augmented intervals become major.
{| class="wikitable"
 
!'''Degree'''
{| class="wikitable center-all left-2 right-4"
!'''Note'''
! Steps
!'''Interval name'''
! Note
!'''Cents'''
! Interval name
!'''~ Ratios'''
! Cents
|-
! ~ Ratios
|'''0'''
|'''A'''
|'''unison'''
|'''0'''
|'''1/1'''
|-
|1
|A#
|aug unison
|88.89
|21/20
|-
|-
|2
| 0
|Bbb
| A
|ddim mos2nd
| unison
|177.78
| 0.00
|10/9
| 1/1
|-
|-
|3
| 1
|Bb
| A#
|dim mos2nd
| aug unison
|266.67
| 88.89
|7/6
| 21/20
|-
|-
|''4''
| 2
|''B''
| Bbb
|''perf mos2nd''
| ddim mos2nd
|''355.56''
| 177.78
|''16/13''
| 10/9
|-
|-
|5
| 3
|B#
| Bb
|aug mos2nd
| dim mos2nd
|444.44
| 266.67
|13/10, 9/7
| 7/6
|- style="background: #eee"
| 4
| B
| perf mos2nd
| 355.56
| 16/13
|-
|-
|6
| 5
|Cbb
| B#
|dim mos3rd
| aug mos2nd
|533.33
| 444.44
|27/20, 19/14
| 13/10, 9/7
|-
|-
|7
| 6
|Cb
| Cbb
|min mos3rd
| dim mos3rd
|622.22
| 533.33
|10/7, 13/9
| 27/20, 19/14
|-
|-
|8
| 7
|C
| Cb
|maj mos3rd
| min mos3rd
|711.11
| 622.22
|3/2
| 10/7, 13/9
|-
|-
|9
| 8
|C#
| C
|aug mos3rd
| maj mos3rd
|800.00
| 711.11
|8/5
| 3/2
|-
|-
|10
| 9
|Dbb
| C#
|dim mos4th
| aug mos3rd
|888.89
| 800.00
|5/3
| 8/5
|-
|-
|11
| 10
|Db
| Dbb
|min mos4th
| dim mos4th
|977.78
| 888.89
|7/4
| 5/3
|-
|-
|12
| 11
|D
| Db
|maj mos4th
| min mos4th
|1066.67
| 977.78
|13/7
| 7/4
|-
|-
|13
| 12
|D#
| D
|aug mos4th
| maj mos4th
|1155.56
| 1066.67
|39/20, 35/18
| 13/7
|-
|-
|14
| 13
|Ebb
| D#
|dim mos5th
| aug mos4th
|1244.44
| 1155.56
|80/39, 72/35
| 39/20, 35/18
|-
|-
|15
| 14
|Eb
| Ebb
|min mos5th
| dim mos5th
|1333.33
| 1244.44
|28/13
| 80/39, 72/35
|-
|-
|16
| 15
|E
| Eb
|maj mos5th
| min mos5th
|1422.22
| 1333.33
|16/7
| 28/13
|-
|-
|17
| 16
|E#
| E
|aug mos5th
| maj mos5th
|1511.11
| 1422.22
|12/5
| 16/7
|-
|-
|18
| 17
|Fbb
| E#
|dim mos6th
| aug mos5th
|1600.00
| 1511.11
|5/2
| 12/5
|-
|-
|19
| 18
|Fb
| Fbb
|min mos6th
| dim mos6th
|1688.89
| 1600.00
|8/3
| 5/2
|-
|-
|20
| 19
|F
| Fb
|maj mos6th
| min mos6th
|1777.78
| 1688.89
|14/5, 36/13
| 8/3
|-
|-
|21
| 20
|F#
| F
|aug mos6th
| maj mos6th
|1866.67
| 1777.78
|80/27, 38/13
| 14/5, 36/13
|-
|-
|22
| 21
|Gb
| F#
|dim mos7th
| aug mos6th
|1955.56
| 1866.67
|40/13, 28/9
| 80/27, 38/13
|-
|-
|''23''
| 22
|''G''
| Gb
|''perf mos7th''
| dim mos7th
|''2044.44''
| 1955.56
|''13/4''
| 40/13, 28/9
|- style="background: #eee"
| 23
| G
| perf mos7th
| 2044.44
| 13/4
|-
|-
|24
| 24
|G#
| G#
|aug mos7th
| aug mos7th
|2133.33
| 2133.33
|24/7
| 24/7
|-
|-
|25
| 25
|Abb
| Abb
|ddim tetratave
| ddim tetratave
|2222.22
| 2222.22
|18/5
| 18/5
|-
|-
|26
| 26
|Ab
| Ab
|dim tetratave
| dim tetratave
|2311.11
| 2311.11
|80/21
| 80/21
|-
|-
|'''27'''
| 27
|'''A'''
| A
|'''tetratave'''
| tetratave
|'''2400'''
| 2400.00
|'''4/1'''
| 4/1
|}
|}
The [[Generator|genchain]] for the nelindic scale is as follows:
The [[Generator|genchain]] for the nelindic scale is as follows:
{| class="wikitable"
{| class="wikitable"
|Abb
| Abb
|Bbb
| Bbb
|Cbb
| Cbb
|Dbb
| Dbb
|Ebb
| Ebb
|Fbb
| Fbb
|Gb
| Gb
|Ab
| Ab
|Bb
| Bb
|Cb
| Cb
|Db
| Db
|Eb
| Eb
|Fb
| Fb
|G
| G
|A
| A
|B
| B
|C
| C
|D
| D
|E
| E
|F
| F
|G#
| G#
|A#
| A#
|B#
| B#
|C#
| C#
|D#
| D#
|E#
| E#
|F#
| F#
|-
|-
|dd1
| dd1
|dd2
| dd2
|d3
| d3
|d4
| d4
|d5
| d5
|d6
| d6
|d7
| d7
|d1
| d1
|d2
| d2
|m3
| m3
|m4
| m4
|m5
| m5
|m6
| m6
|P7
| P7
|P1
| P1
|P2
| P2
|M3
| M3
|M4
| M4
|M5
| M5
|M6
| M6
|A7
| A7
|A1
| A1
|A2
| A2
|A3
| A3
|A4
| A4
|A5
| A5
|A6
| A6
|}
|}
[[Category:Nonoctave]]
[[Category:Nonoctave]]
[[Category:Edonoi]]
[[Category:Edonoi]]
[[Category:Ed4]]
[[Category:Ed4]]

Revision as of 19:00, 17 May 2021

27ed4 is an equal tuning that divides the 4/1 ratio (double-octave, tetratave, fifteenth) into steps of 88+(8/9) cents.

It serves as a good first approximation to nelindic temperament, and is in many respects a "3n+1 cousin" of 5-limit 12et (even though it takes every other step of the dissimilar 27et), with relatively high error but low complexity, similar step size, and even sharing a common comma (128/125). Note the latter means that 27ed4 divides 4/1 into three approximate 8/5's, just as 12ed2 divides 2/1 into three 5/4's, and thus it has a 5/2 equally sharp of rational as the 5/4 in 12ed2. Its 7 and 13 approximations are a bit sharp themselves, and overall it lends itself well to IoE compression: the TE tuning gives one of 2395.819236 cents.

Intervals

The following table of intervals uses the 7-note (6L 1s) MOS scale of nelindic for the naturals, using a simple A-G notation and standard sharps/flats for the chroma. Extended to the 13-note (7L 6s) scale, these would include all of the sharps except for F#. Due to the L/s ratio of 3:1, in the 13-note case, most former diminished intervals become minor, most former minor intervals become augmented, and most former augmented intervals become major.

Steps Note Interval name Cents ~ Ratios
0 A unison 0.00 1/1
1 A# aug unison 88.89 21/20
2 Bbb ddim mos2nd 177.78 10/9
3 Bb dim mos2nd 266.67 7/6
4 B perf mos2nd 355.56 16/13
5 B# aug mos2nd 444.44 13/10, 9/7
6 Cbb dim mos3rd 533.33 27/20, 19/14
7 Cb min mos3rd 622.22 10/7, 13/9
8 C maj mos3rd 711.11 3/2
9 C# aug mos3rd 800.00 8/5
10 Dbb dim mos4th 888.89 5/3
11 Db min mos4th 977.78 7/4
12 D maj mos4th 1066.67 13/7
13 D# aug mos4th 1155.56 39/20, 35/18
14 Ebb dim mos5th 1244.44 80/39, 72/35
15 Eb min mos5th 1333.33 28/13
16 E maj mos5th 1422.22 16/7
17 E# aug mos5th 1511.11 12/5
18 Fbb dim mos6th 1600.00 5/2
19 Fb min mos6th 1688.89 8/3
20 F maj mos6th 1777.78 14/5, 36/13
21 F# aug mos6th 1866.67 80/27, 38/13
22 Gb dim mos7th 1955.56 40/13, 28/9
23 G perf mos7th 2044.44 13/4
24 G# aug mos7th 2133.33 24/7
25 Abb ddim tetratave 2222.22 18/5
26 Ab dim tetratave 2311.11 80/21
27 A tetratave 2400.00 4/1

The genchain for the nelindic scale is as follows:

Abb Bbb Cbb Dbb Ebb Fbb Gb Ab Bb Cb Db Eb Fb G A B C D E F G# A# B# C# D# E# F#
dd1 dd2 d3 d4 d5 d6 d7 d1 d2 m3 m4 m5 m6 P7 P1 P2 M3 M4 M5 M6 A7 A1 A2 A3 A4 A5 A6
Retrieved from "https://en.xen.wiki/index.php?title=27ed4&oldid=69970"