55edo: Difference between revisions

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+subsets and supersets
Correct many entries in the interval table
Line 34: Line 34:
| 1
| 1
| 21.8
| 21.8
| 128/125, 64/63, 65/64, 78/77,<br>91/90, 99/98, ''81/80''
| 65/64, 78/77, 99/98, ''128/125''
| ^1
| ^1
| up 1sn
| up 1sn
Line 41: Line 41:
| 2
| 2
| 43.6
| 43.6
| 36/35
| 36/35, ''64/63''
| ^^1
| ^^1
| dup 1sn
| dup 1sn
Line 48: Line 48:
| 3
| 3
| 65.5
| 65.5
| 28/27, ''25/24''
| 28/27
| vvm2
| vvm2
| dudminor 2nd
| dudminor 2nd
Line 55: Line 55:
| 4
| 4
| 87.3
| 87.3
| 25/24, 21/20
| 21/20, ''18/17'', ''25/24''
| vm2
| vm2
| downminor 2nd
| downminor 2nd
Line 62: Line 62:
| 5
| 5
| 109.1
| 109.1
| 16/15
| 16/15, 17/16
| m2
| m2
| minor 2nd
| minor 2nd
Line 69: Line 69:
| 6
| 6
| 130.9
| 130.9
| 14/13, ''13/12''
| 13/12, 14/13
| ^m2
| ^m2
| upminor 2nd
| upminor 2nd
Line 76: Line 76:
| 7
| 7
| 152.7
| 152.7
| 13/12, 12/11, ''11/10''
| 12/11, ''11/10''
| ~2
| ~2
| mid 2nd
| mid 2nd
Line 83: Line 83:
| 8
| 8
| 174.5
| 174.5
| ''10/9''
|  
| vM2
| vM2
| downmajor 2nd
| downmajor 2nd
Line 90: Line 90:
| 9
| 9
| 196.4
| 196.4
| 9/8, 10/9
| 9/8, ''10/9''
| M2
| M2
| major 2nd
| major 2nd
Line 104: Line 104:
| 11
| 11
| 240.0
| 240.0
| 8/7, 15/13
| 8/7
| ^^M2
| ^^M2
| dupmajor 2nd
| dupmajor 2nd
Line 125: Line 125:
| 14
| 14
| 305.5
| 305.5
| 6/5-
| 6/5
| m3
| m3
| minor 3rd
| minor 3rd
Line 132: Line 132:
| 15
| 15
| 327.3
| 327.3
| 6/5+
|  
| ^m3
| ^m3
| upminor 3rd
| upminor 3rd
Line 146: Line 146:
| 17
| 17
| 370.9
| 370.9
| 16/13
| 26/21, ''16/13''
| vM3
| vM3
| downmajor 3rd
| downmajor 3rd
Line 174: Line 174:
| 21
| 21
| 458.2
| 458.2
| 13/10
| ''21/16''
| vv4
| vv4
| dud 4th
| dud 4th
Line 181: Line 181:
| 22
| 22
| 480.0
| 480.0
| 21/16
|  
| v4
| v4
| down 4th
| down 4th
Line 188: Line 188:
| 23
| 23
| 501.8
| 501.8
| 4/3, 27/20
| 4/3, ''27/20''
| P4
| P4
| perfect 4th
| perfect 4th
Line 195: Line 195:
| 24
| 24
| 523.6
| 523.6
| ''27/20''
|  
| ^4
| ^4
| up 4th
| up 4th
Line 202: Line 202:
| 25
| 25
| 545.5
| 545.5
| 11/8
| 11/8, 15/11
| ~4
| ~4
| mid 4th
| mid 4th
Line 209: Line 209:
| 26
| 26
| 567.3
| 567.3
| 18/13, 25/18
| 18/13
| vA4
| vA4
| downaug 4th
| downaug 4th
Line 216: Line 216:
| 27
| 27
| 589.1
| 589.1
| 7/5
| 7/5, 24/17
| A4, vd5
| A4, vd5
| aug 4th, downdim 5th
| aug 4th, downdim 5th
Line 223: Line 223:
| 28
| 28
| 610.9
| 610.9
| 10/7
| 10/7, 17/12
| ^A4, d5
| ^A4, d5
| upaug 4th, dim 5th
| upaug 4th, dim 5th
Line 230: Line 230:
| 29
| 29
| 632.7
| 632.7
| 13/9, 36/25
| 13/9
| ^d5
| ^d5
| updim 5th
| updim 5th
Line 237: Line 237:
| 30
| 30
| 654.5
| 654.5
| 16/11
| 16/11, 22/15
| ~5
| ~5
| mid 5th
| mid 5th
Line 244: Line 244:
| 31
| 31
| 676.4
| 676.4
| ''40/27''
|  
| v5
| v5
| down 5th
| down 5th
Line 251: Line 251:
| 32
| 32
| 698.2
| 698.2
| 3/2, 40/27
| 3/2, ''40/27''
| P5
| P5
| perfect 5th
| perfect 5th
Line 258: Line 258:
| 33
| 33
| 720.0
| 720.0
| 32/21
|  
| ^5
| ^5
| up 5th
| up 5th
Line 265: Line 265:
| 34
| 34
| 741.8
| 741.8
| 20/13
| ''32/21''
| ^^5
| ^^5
| dup 5th
| dup 5th
Line 293: Line 293:
| 38
| 38
| 829.1
| 829.1
| 13/8
| 21/13, ''13/8''
| ^m6
| ^m6
| upminor 6th
| upminor 6th
Line 307: Line 307:
| 40
| 40
| 872.7
| 872.7
| 5/3-
|  
| vM6
| vM6
| downmajor 6th
| downmajor 6th
Line 314: Line 314:
| 41
| 41
| 894.5
| 894.5
| 5/3+
| 5/3
| M6
| M6
| major 6th
| major 6th
Line 335: Line 335:
| 44
| 44
| 960.0
| 960.0
| 7/4, 26/15
| 7/4
| vvm7
| vvm7
| dudminor 7th
| dudminor 7th
Line 349: Line 349:
| 46
| 46
| 1003.6
| 1003.6
| 16/9, 9/5
| 16/9, ''9/5''
| m7
| m7
| minor 7th
| minor 7th
Line 356: Line 356:
| 47
| 47
| 1025.5
| 1025.5
| ''9/5''
|  
| ^m7
| ^m7
| upminor 7th
| upminor 7th
Line 363: Line 363:
| 48
| 48
| 1047.3
| 1047.3
| ''20/11'', 11/6, 24/13
| 11/6, ''20/11''
| ~7
| ~7
| mid 7th
| mid 7th
Line 370: Line 370:
| 49
| 49
| 1069.1
| 1069.1
| ''24/13'', 13/7
| 13/7, 24/13
| vM7
| vM7
| downmajor 7th
| downmajor 7th
Line 377: Line 377:
| 50
| 50
| 1090.9
| 1090.9
| 15/8
| 15/8, ''32/17''
| M7
| M7
| major 7th
| major 7th
Line 384: Line 384:
| 51
| 51
| 1112.7
| 1112.7
| 40/21, 48/25
| 40/21, ''17/9'', ''48/25''
| ^M7
| ^M7
| upmajor 7th
| upmajor 7th
Line 391: Line 391:
| 52
| 52
| 1134.5
| 1134.5
| 56/27, ''48/25''
| 56/27
| ^^M7
| ^^M7
| dupmajor 7th
| dupmajor 7th
Line 398: Line 398:
| 53
| 53
| 1156.4
| 1156.4
| 35/18
| 35/18, ''63/32''
| vv8
| vv8
| dud 8ve
| dud 8ve
Line 405: Line 405:
| 54
| 54
| 1178.2
| 1178.2
| 125/64, 63/32, 128/65, 77/39,<br>180/91, 196/99, ''160/81''
| 128/65, 77/39, 196/99, ''125/64''
| v8
| v8
| down 8ve
| down 8ve
Line 417: Line 417:
| D
| D
|}
|}
<nowiki>*</nowiki> 55f val (tending flat), inconsistent intervals labeled in ''italic''


== Selected just intervals by error ==
== Approximation to JI ==
=== Selected just intervals by error ===
The following table shows how [[15-odd-limit]] just intervals are represented in 55edo (ordered by absolute error).
The following table shows how [[15-odd-limit]] just intervals are represented in 55edo (ordered by absolute error).
{{15-odd-limit|55}}
{{15-odd-limit|55}}
Line 424: Line 426:
== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
! colspan="2" | Tuning Error
|-
|-
![[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
|-
|2.3
| 2.3
|{{monzo|-87 55}}
| {{monzo| -87 55 }}
|{{mapping|55 87}}
| {{mapping| 55 87 }}
| +1.1903
| +1.1903
| 1.1915
| 1.1915
| 5.46
| 5.46
|-
|-
|2.3.5
| 2.3.5
|81/80, 6442450944/6103515625
| 81/80, 6442450944/6103515625
|{{mapping|55 87 128}}
| {{mapping| 55 87 128 }}
| -0.1309
| -0.1309
| 2.1012
| 2.1012

Revision as of 07:42, 8 March 2024

← 54edo 55edo 56edo →
Prime factorization 5 × 11
Step size 21.8182 ¢ 
Fifth 32\55 (698.182 ¢)
Semitones (A1:m2) 4:5 (87.27 ¢ : 109.1 ¢)
Consistency limit 5
Distinct consistency limit 5

Template:EDO intro

Theory

55edo can be used for a meantone tuning, and is close to 1/6-comma meantone (and is almost exactly 10/57-comma meantone.) Telemann suggested it as a theoretical basis for analyzing the intervals of meantone, in which he was followed by Leopold and Wolfgang Mozart. It can also be used for mohajira and liese temperaments.

Odd harmonics

Approximation of odd harmonics in 55edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -3.77 +6.41 -8.83 -7.55 -5.86 +10.38 +2.64 +4.14 +7.94 +9.22 +4.45
Relative (%) -17.3 +29.4 -40.5 -34.6 -26.9 +47.6 +12.1 +19.0 +36.4 +42.3 +20.4
Steps
(reduced) 		87
(32) 		128
(18) 		154
(44) 		174
(9) 		190
(25) 		204
(39) 		215
(50) 		225
(5) 		234
(14) 		242
(22) 		249
(29)

Subsets and supersets

Since 55 factors into 5 × 11, 55edo contains 5edo and 11edo as its subsets.

Intervals

# Cents Approximate Ratios Ups and Downs Notation
0 0.0 1/1 P1 perfect 1sn D
1 21.8 65/64, 78/77, 99/98, 128/125 ^1 up 1sn ^D
2 43.6 36/35, 64/63 ^^1 dup 1sn ^^D
3 65.5 28/27 vvm2 dudminor 2nd vvEb
4 87.3 21/20, 18/17, 25/24 vm2 downminor 2nd vEb
5 109.1 16/15, 17/16 m2 minor 2nd Eb
6 130.9 13/12, 14/13 ^m2 upminor 2nd ^Eb
7 152.7 12/11, 11/10 ~2 mid 2nd vvE
8 174.5 vM2 downmajor 2nd vE
9 196.4 9/8, 10/9 M2 major 2nd E
10 218.2 17/15 ^M2 upmajor 2nd ^E
11 240.0 8/7 ^^M2 dupmajor 2nd ^^E
12 261.8 7/6 vvm3 dudminor 3rd vvF
13 283.6 13/11 vm3 downminor 3rd vF
14 305.5 6/5 m3 minor 3rd F
15 327.3 ^m3 upminor 3rd ^F
16 349.1 11/9, 27/22 ~3 mid 3rd ^^F
17 370.9 26/21, 16/13 vM3 downmajor 3rd vF#
18 392.7 5/4 M3 major 3rd F#
19 414.5 14/11 ^M3 upmajor 3rd ^F#
20 436.4 9/7 ^^M3 dupmajor 3rd ^^F#
21 458.2 21/16 vv4 dud 4th vvG
22 480.0 v4 down 4th vG
23 501.8 4/3, 27/20 P4 perfect 4th G
24 523.6 ^4 up 4th ^G
25 545.5 11/8, 15/11 ~4 mid 4th ^^G
26 567.3 18/13 vA4 downaug 4th vG#
27 589.1 7/5, 24/17 A4, vd5 aug 4th, downdim 5th G#, vAb
28 610.9 10/7, 17/12 ^A4, d5 upaug 4th, dim 5th ^G#, Ab
29 632.7 13/9 ^d5 updim 5th ^Ab
30 654.5 16/11, 22/15 ~5 mid 5th vvA
31 676.4 v5 down 5th vA
32 698.2 3/2, 40/27 P5 perfect 5th A
33 720.0 ^5 up 5th ^A
34 741.8 32/21 ^^5 dup 5th ^^A
35 763.6 14/9 vvm6 dudminor 6th vvBb
36 785.5 11/7 vm6 downminor 6th vBb
37 807.3 8/5 m6 minor 6th Bb
38 829.1 21/13, 13/8 ^m6 upminor 6th ^Bb
39 850.9 18/11, 44/27 ~6 mid 6th vvB
40 872.7 vM6 downmajor 6th vB
41 894.5 5/3 M6 major 6th B
42 916.4 22/13 ^M6 upmajor 6th ^B
43 938.2 12/7 ^^M6 dupmajor 6th ^^B
44 960.0 7/4 vvm7 dudminor 7th vvC
45 981.8 30/17 vm7 downminor 7th vC
46 1003.6 16/9, 9/5 m7 minor 7th C
47 1025.5 ^m7 upminor 7th ^C
48 1047.3 11/6, 20/11 ~7 mid 7th ^^C
49 1069.1 13/7, 24/13 vM7 downmajor 7th vC#
50 1090.9 15/8, 32/17 M7 major 7th C#
51 1112.7 40/21, 17/9, 48/25 ^M7 upmajor 7th ^C#
52 1134.5 56/27 ^^M7 dupmajor 7th ^^C#
53 1156.4 35/18, 63/32 vv8 dud 8ve vvD
54 1178.2 128/65, 77/39, 196/99, 125/64 v8 down 8ve vD
55 1200.0 2/1 P8 perfect 8ve D

* 55f val (tending flat), inconsistent intervals labeled in italic

Approximation to JI

Selected just intervals by error

The following table shows how 15-odd-limit just intervals are represented in 55edo (ordered by absolute error). The following tables show how 15-odd-limit intervals are represented in 55edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 55edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
9/7, 14/9 1.280 5.9
11/9, 18/11 1.683 7.7
11/6, 12/11 2.090 9.6
13/7, 14/13 2.611 12.0
15/8, 16/15 2.640 12.1
11/7, 14/11 2.963 13.6
3/2, 4/3 3.773 17.3
13/9, 18/13 3.890 17.8
13/10, 20/13 3.968 18.2
7/6, 12/7 5.053 23.2
13/11, 22/13 5.573 25.5
11/8, 16/11 5.863 26.9
5/4, 8/5 6.414 29.4
7/5, 10/7 6.579 30.2
9/8, 16/9 7.546 34.6
13/12, 24/13 7.664 35.1
15/13, 26/15 7.741 35.5
9/5, 10/9 7.858 36.0
15/11, 22/15 8.504 39.0
7/4, 8/7 8.826 40.5
11/10, 20/11 9.541 43.7
5/3, 6/5 10.187 46.7
15/14, 28/15 10.352 47.4
13/8, 16/13 10.381 47.6
15-odd-limit intervals in 55edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
9/7, 14/9 1.280 5.9
11/9, 18/11 1.683 7.7
11/6, 12/11 2.090 9.6
15/8, 16/15 2.640 12.1
11/7, 14/11 2.963 13.6
3/2, 4/3 3.773 17.3
13/10, 20/13 3.968 18.2
7/6, 12/7 5.053 23.2
11/8, 16/11 5.863 26.9
5/4, 8/5 6.414 29.4
9/8, 16/9 7.546 34.6
15/13, 26/15 7.741 35.5
15/11, 22/15 8.504 39.0
7/4, 8/7 8.826 40.5
5/3, 6/5 10.187 46.7
13/8, 16/13 10.381 47.6
15/14, 28/15 11.466 52.6
11/10, 20/11 12.277 56.3
9/5, 10/9 13.960 64.0
13/12, 24/13 14.155 64.9
7/5, 10/7 15.239 69.8
13/11, 22/13 16.245 74.5
13/9, 18/13 17.928 82.2
13/7, 14/13 19.207 88.0

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [-87 55⟩ [⟨55 87]] +1.1903 1.1915 5.46
2.3.5 81/80, 6442450944/6103515625 [⟨55 87 128]] -0.1309 2.1012 9.63

Commas

5-limit commas: 81/80, [31 1 -14⟩, [27 5 -15⟩

7-limit commas: 31104/30625, 6144/6125, 81648/78125, 16128/15625, 28672/28125, 33075/32768, 83349/80000, 1029/1000, 686/675, 10976/10935, 16807/16384, 84035/82944

11-limit commas: 59049/58564, 74088/73205, 46656/46585, 21609/21296, 12005/11979, 19683/19360, 243/242, 3087/3025, 5488/5445, 19683/19250, 1944/1925, 45927/45056, 2835/2816, 35721/34375, 7056/6875, 12544/12375, 7203/7040, 2401/2376, 24057/24010, 72171/70000, 891/875, 176/175, 2079/2048, 385/384, 3234/3125, 17248/16875, 26411/25600, 26411/2592, 26411/262404, 88209/87808, 30976/30625, 3267/3200, 121/120, 81312/78125, 41503/40000, 41503/40500, 35937/35000, 2662/2625, 42592/42525, 83853/81920, 9317/9216, 65219/62500, 43923/43904, 14641/14400, 14641/14580

13-limit commas: 59535/57122, 29400/28561, 29568/28561, 29645/28561, 24576/24167, 99225/96668, 24500/24167, 50421/48334, 45927/43940, 2268/2197, 2240/2197, 57624/54925, 61875/61516, 57024/54925, 11264/10985, 72765/70304, 13475/13182, 22869/21970, 6776/6591, 20736/20449, 20480/20449, 84035/81796, 91125/91091, 65536/65065, 15309/14872, 1890/1859, 5600/5577, 9604/9295, 59049/57967, 58320/57967, 4374/4225, 864/845, 512/507, 11025/10816, 6125/6084, 21952/21125, 16807/16224, 84035/82134, 66825/66248, 90112/88725, 56133/54080, 693/676, 1540/1521, 26411/25350, 58806/57967, 58080/57967, 88209/84500, 4356/4225, 7744/7605, 88935/86528, 33275/33124, 27951/27040, 9317/9126, 58564/57967, 43923/42250, 17496/17303, 87808/86515, 55296/55055, 25515/25168, 1575/1573, 64827/62920, 4802/4719, 98415/98098, 59049/57200, 729/715, 144/143, 18375/18304, 18522/17875, 10976/10725, 84035/82368, 59049/56875, 11664/11375, 2304/2275, 4096/4095, 1701/1664, 105/104, 42336/40625, 25088/24375, 21609/20800, 2401/2340, 9604/9477, 72171/71344, 2673/2600, 66/65, 352/351, 13475/13312, 33957/32500, 15092/14625, 81675/81536, 58806/56875, 11616/11375, 61952/61425, 68607/66560, 847/832, 4235/4212, 35937/35672, 1331/1300, 5324/5265, 58564/56875, 85293/85184, 13377/13310, 85293/84700, 15288/15125, 31213/30976, 67392/67375, 28431/28160, 34944/34375, 4459/4400, 4459/4455, 28431/28000, 351/350, 79872/78125, 66339/65536, 51597/50000, 637/625, 10192/10125, 31213/30720, 31213/31104, 30888/30625, 1287/1280, 81081/78125, 16016/15625, 49049/48000, 49049/48600, 14157/14000, 33033/32768, 77077/75000, 51909/51200, 17303/17280, 75712/75625, 8281/8250, 41067/40960, 31941/31250, 9464/9375, 57967/57600, 91091/90000, 61347/61250, 79092/78125

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator Temperaments
1 6\55 Twothirdtonic
1 16\55 Vicentino / mohajira
1 23\55 Meantone
1 26\55 Liese
1 27\55 Untriton / aufo
5 6\55 Qintosec
11 3\55 Hendecatonic

Instruments

Music

Modern renderings

Johann Sebastian Bach
Nicolaus Bruhns
Wolfgang Amadeus Mozart

21st century

Claudi Meneghin

External links

Retrieved from "https://en.xen.wiki/index.php?title=55edo&oldid=138579"