26edt: Difference between revisions

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== Intervals ==
== Intervals ==


{| class="wikitable center-1 right-2 right-3"
{| class="wikitable center-1 right-2 right-3"
Line 257: Line 258:
|}
|}


It is a weird coincidence how 26edt intones any [[26edo]] intervals within plus or minus 6.5 cents when it is supposed to have nothing to do with this other tuning:
{| class="wikitable center-1 right-2 right-3"
|-
! Steps
! [[Cent]]s
! [[Hekt]]s
! BP enneatonic degree
! Corresponding 3.5.7 subgroup intervals
! Comments
! Generator for...
! [[4L 5s (3/1-equivalent)#Notation|BPS enneatonic notation]] (J = 1/1)
|-
| 0
| 0
| 0
| P1
| 1/1
|
|
| J
|-
| 1
| 146.3
| 100
| A1/m2
| [[27/25]], [[49/45]]
| BP "semitone"
| [[Procyon]]
| K
|-
| 2
| 292.6
| 200
| M2/d3
| [[25/21]]
| BP "wholetone"
| [[Sirius]]
| K#, Lb
|-
| 3
| 438.9
| 300
| A2/P3/d4
| [[9/7]]
|
| [[Bohlen-Pierce-Stearns|Linear BP]]
| L
|-
| 4
| 585.2
| 400
| A3/m4/d5
| [[7/5]]
|
| [[Canopus]]
| M
|-
| 5
| 731.5
| 500
| M4/m5
| [[75/49]]
| false 3/2
| false Father
| M#, Nb
|-
| 6
| 877.8
| 600
| A4/M5
| [[5/3]]
|
| [[Arcturus]]
| N
|-
| 7
| 1024.1
| 700
| A5/m6/d7
| [[9/5]]
|
| Arcturus
| O
|-
| 8
| 1170.4
| 800
| M6/m7
| [[49/25]]
| false 2/1
| false Father
| O#, Pb
|-
| 9
| 1316.7
| 900
| A6/M7/d8
| [[15/7]]
|
| Canopus
| P
|-
| 10
| 1463.0
| 1000
| P8/d9
| [[7/3]]
|
| Linear BP
| Q
|-
| 11
| 1609.3
| 1100
| A8/m9
| [[63/25]]
|
| Sirius
| Q#, Rb
|-
| 12
| 1755.7
| 1200
| M9/d10
| [[25/9]], [[135/49]]
|
| Procyon
| R
|-
| 13
| 1902.0
| 1300
| A9/P10
| [[3/1]]
| Tritave
|
| J
|}
 
=== Connection to 26edo ===
 
It is a weird coincidence how 26edt intones many [[26edo]] intervals within plus or minus 6.5 cents when it is supposed to have nothing to do with this other tuning:


{| class="wikitable right-all"
{| class="wikitable right-all"

Revision as of 18:39, 2 October 2024

← 25edt 26edt 27edt →
Prime factorization 2 × 13
Step size 73.1521 ¢ 
Octave 16\26edt (1170.43 ¢) (→ 8\13edt)
Consistency limit 3
Distinct consistency limit 3

26edt divides the tritave (3/1) into 26 equal parts of 73.152 cents each, corresponding to 16.404edo. It is contorted in the 7-limit, tempering out the same commas, 245/243 and 3125/3087, as 13edt. In the 11-limit it tempers out 125/121 and 3087/3025, in the 13-limit 175/169, 147/143, and 847/845, and in the 17-limit 119/117. It is the seventh zeta peak tritave division.

A reason to double 13edt to 26edt is to approximate the 8th, 13th, 17th, 20th, and 22nd harmonics particularly well [citation needed]. Moreover, it has an exaggerated diatonic scale with 11:16:21 supermajor triads, though only the 16:11 is particularly just due to its best 16 still being 28.04 cents sharp, or just about as bad as the 25 of 12edo (which is 27.373 cents sharp, an essentially just 100:63).

Theory

While retaining 13edt's mapping of primes 3, 5, and 7, 26edt adds an accurate prime 17 to the mix, tempering out 2025/2023 to split the BPS generator of 9/7 into two intervals of 17/15. This 17/15 generates Dubhe temperament and a 8L 1s MOS scale that can be used as a simple traversal of 26edt.


Approximation of prime harmonics in 26edt
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -29.6 +0.0 -6.5 -3.8 +18.4 +21.8 -3.8 +23.1 -15.0 +22.6 -19.7
Relative (%) -40.4 +0.0 -8.9 -5.2 +25.1 +29.7 -5.1 +31.6 -20.5 +30.9 -26.9
Steps
(reduced) 		16
(16) 		26
(0) 		38
(12) 		46
(20) 		57
(5) 		61
(9) 		67
(15) 		70
(18) 		74
(22) 		80
(2) 		81
(3)

Intervals

Steps Cents Hekts BP nonatonic degree Diatonic degree Corresponding JI intervals Comments Generator for...
1 73.2 50 Sa1/sd2 A1/dd2 25/24
2 146.3 100 A1/m2 AA1/sm2 27/25~49/45
3 219.5 150 N2 m2 9/8~312/275
4 292.6 200 M2/d3 M2 25/21~13/11
5 365.8 250 Sa2/sd3 SM2/dd3 5/4~243/196 False 11/9
6 438.9 300 A2/P3/d4 AA2/sm3 9/7
7 512.1 350 Sa3/sd4 m3 27/20 False 21/16
8 585.2 400 A3/m4/d5 M3 7/5
9 658.4 450 N4/sd5 SM3/dd4 16/11 False 13/9
10 731.5 500 M4/m5 AA3/d4 75/49 False 3/2
11 804.7 550 Sa4/N5 P4 8/5 False 11/7
12 877.8 600 A4/M5/d6 A4 5/3 False 27/16
13 951.0 650 Sa5/sd6 AA4/dd5 125/72
14 1024.1 700 A5/m6/d7 d5 9/5 False 16/9
15 1097.3 750 N6/sd7 P5 15/8 False 21/11
16 1170.4 800 M6/m7 A5/dd6 49/25 False 2/1
17 1243.6 850 Sa6/N7 AA5/sm6 33/16 False 27/13
18 1316.7 900 A6/M7/d8 m6 15/7
19 1389.9 950 Sa7/sd8 M6 20/9 False 16/7
20 1463.0 1000 A7/P8/d9 SM6/dd7 7/3
21 1536.2 1050 Sa8/sd9 AA6/sm7 12/5~196/81 False 27/11
22 1609.3 1100 A8/m9 m7 63/25~33/13
23 1682.5 1150 N9 M7 8/3~275/104
24 1755.7 1200 M9/d10 SM7/dd8 25/9~135/49
25 1828.8 1250 Sa9/sd10 A7/d8 72/25
26 1902.0 1300 A9/P10 P8 3/1 Tritave
Steps Cents Hekts BP enneatonic degree Corresponding 3.5.7 subgroup intervals Comments Generator for... BPS enneatonic notation (J = 1/1)
0 0 0 P1 1/1 J
1 146.3 100 A1/m2 27/25, 49/45 BP "semitone" Procyon K
2 292.6 200 M2/d3 25/21 BP "wholetone" Sirius K#, Lb
3 438.9 300 A2/P3/d4 9/7 Linear BP L
4 585.2 400 A3/m4/d5 7/5 Canopus M
5 731.5 500 M4/m5 75/49 false 3/2 false Father M#, Nb
6 877.8 600 A4/M5 5/3 Arcturus N
7 1024.1 700 A5/m6/d7 9/5 Arcturus O
8 1170.4 800 M6/m7 49/25 false 2/1 false Father O#, Pb
9 1316.7 900 A6/M7/d8 15/7 Canopus P
10 1463.0 1000 P8/d9 7/3 Linear BP Q
11 1609.3 1100 A8/m9 63/25 Sirius Q#, Rb
12 1755.7 1200 M9/d10 25/9, 135/49 Procyon R
13 1902.0 1300 A9/P10 3/1 Tritave J

Connection to 26edo

It is a weird coincidence how 26edt intones many 26edo intervals within plus or minus 6.5 cents when it is supposed to have nothing to do with this other tuning:

26edt 26edo Delta
365.761 369.231 -3.470
512.065 507.692 +4.373
877.825 876.923 +0.902
1243.586 1246.154 -2.168
1389.890 1384.615 +5.275
1755.651 1753.846 +1.805
2121.411 2123.077 -1.666
2633.476 2630.769 +2.647

etc.

Music

  • The Eel And Loach To Attack In Lasciviousness Are Insane: video | MP3 by Omega9
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