55edo: Difference between revisions

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== Theory ==
== 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.) {{w|Georg Philipp Telemann|Telemann}} suggested it as a theoretical basis for analyzing the [[meantone intervals|intervals of meantone]], in which he was followed by {{w|Leopold Mozart|Leopold}} and {{w|Wolfgang Amadeus Mozart|Wolfgang Mozart}}. It can also be used for [[Meantone family|mohajira and liese]] temperaments. It also supports an extremely sharp tuning of [[huygens|Huygens/undecimal meantone]] using the 55de [[val]], meaning that primes 7 and 11 are mapped very sharply to their second-best mapping.
55edo can be used for a [[meantone]] tuning, and is close to [[1/6-comma meantone]] (and is almost exactly 10/57-comma meantone). {{w|Georg Philipp Telemann|Telemann}} suggested it as a theoretical basis for analyzing the [[meantone intervals|intervals of meantone]], in which he was followed by {{w|Leopold Mozart|Leopold}} and {{w|Wolfgang Amadeus Mozart|Wolfgang Mozart}}. It can also be used for [[Meantone family|mohajira and liese]] temperaments. It also supports an extremely sharp tuning of [[huygens|Huygens/undecimal meantone]] using the 55de [[val]], meaning that primes 7 and 11 are mapped very sharply to their second-best mapping.


=== Odd harmonics ===
=== Odd harmonics ===
Line 20: Line 20:
{| class="wikitable center-1 right-2 left-3"
{| class="wikitable center-1 right-2 left-3"
|-
|-
! [[Degree|#]]
! [[Degree|#]]
! [[Cent]]s
! [[Cent]]s
! Approximate Ratios
! Approximate Ratios
! colspan="3" | [[Ups and downs notation|Ups and Downs Notation]]
! colspan="3" | [[Ups and downs notation]]
|-
|-
| 0
| 0
Line 417: Line 417:
| D
| D
|}
|}
<nowiki>*</nowiki> 55f val (tending flat), inconsistent intervals labeled in ''italic''
<nowiki />* 55f val (tending flat), inconsistent intervals labeled in ''italic''


== Approximation to JI ==
== Approximation to JI ==
Line 430: Line 430:
! 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
|-
|-
Line 462: Line 462:
=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per 8ve
|-
! Periods<br />per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio*
! Associated<br />Ratio*
! Temperaments
! Temperaments
|-
|-
Line 500: Line 501:
|-
|-
| 5
| 5
| 17\55<br>(5\55)
| 17\55<br />(5\55)
| 370.9<br>(109.1)
| 370.9<br />(109.1)
| 99/80<br>(16/15)
| 99/80<br />(16/15)
| [[Quintosec]]
| [[Quintosec]]
|-
|-
| 11
| 11
| 23\55<br>(3\55)
| 23\55<br />(3\55)
| 501.8<br>(65.5)
| 501.8<br />(65.5)
| 4/3<br>(36/35)
| 4/3<br />(36/35)
| [[Hendecatonic]] (55)
| [[Hendecatonic]] (55)
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]]


== Instruments ==
== Instruments ==
Line 518: Line 520:
=== Modern renderings ===
=== Modern renderings ===
; {{W|Johann Sebastian Bach}}
; {{W|Johann Sebastian Bach}}
* [https://www.youtube.com/watch?v=oymJKnYzzOw "Jesus bleibet meine Freude" from ''Herz und Mund und Tat und Leben'', BWV 147] (1723) – arranged for two organs, rendered by Claudi Meneghin (2021)
* [https://www.youtube.com/watch?v=oymJKnYzzOw "Jesus bleibet meine Freude" from ''Herz und Mund und Tat und Leben'', BWV 147] (1723) &ndash; arranged for two organs, rendered by Claudi Meneghin (2021)
* [https://www.youtube.com/watch?v=Y5sIjh_Te40 "Contrapunctus 4" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
* [https://www.youtube.com/watch?v=Y5sIjh_Te40 "Contrapunctus 4" from ''The Art of Fugue'', BWV 1080] (1742&ndash;1749) &ndash; rendered by Claudi Meneghin (2024)
* [https://www.youtube.com/watch?v=QOPxqNgkVWM "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
* [https://www.youtube.com/watch?v=QOPxqNgkVWM "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742&ndash;1749) &ndash; rendered by Claudi Meneghin (2024)


; {{W|Nicolaus Bruhns}}
; {{W|Nicolaus Bruhns}}
* [https://www.youtube.com/watch?v=OfOt3nOp-f8 ''Prelude in E Minor "The Great"''] – rendered by Claudi Meneghin (2023)
* [https://www.youtube.com/watch?v=OfOt3nOp-f8 ''Prelude in E Minor "The Great"''] &ndash; rendered by Claudi Meneghin (2023)
* [https://www.youtube.com/watch?v=tuIPIhSxUPs ''Prelude in E Minor "The Little"''] – rendered by Claudi Meneghin (2024)
* [https://www.youtube.com/watch?v=tuIPIhSxUPs ''Prelude in E Minor "The Little"''] &ndash; rendered by Claudi Meneghin (2024)


; {{W|Scott Joplin}}
; {{W|Scott Joplin}}
* [https://www.youtube.com/watch?v=GbhpuoIJgxk ''Maple Leaf Rag''] (1899) – arranged for harpsichord and rendered by Claudi Meneghin (2024)
* [https://www.youtube.com/watch?v=GbhpuoIJgxk ''Maple Leaf Rag''] (1899) &ndash; arranged for harpsichord and rendered by Claudi Meneghin (2024)


; {{W|Wolfgang Amadeus Mozart}}
; {{W|Wolfgang Amadeus Mozart}}
* [https://www.youtube.com/watch?v=C_AML6XW-2g ''Rondo alla Turca'' from the Piano Sonata No. 11, KV 331] (1778) – rendered by Francium (2023)
* [https://www.youtube.com/watch?v=C_AML6XW-2g ''Rondo alla Turca'' from the Piano Sonata No. 11, KV 331] (1778) &ndash; rendered by Francium (2023)
* [https://www.youtube.com/watch?v=XgRksdk6zyQ ''Fugue in G minor'', KV 401] (1782) – rendered by Francium (2023)
* [https://www.youtube.com/watch?v=XgRksdk6zyQ ''Fugue in G minor'', KV 401] (1782) &ndash; rendered by Francium (2023)
* [http://www.seraph.it/dep/int/AdagioKV540.mp3 ''Adagio in B minor'', KV 540] (1788) – rendered by Carlo Serafini (2011) ([http://www.seraph.it/blog_files/706c4662272db7703def4d57edfcb955-119.html blog entry])
* [http://www.seraph.it/dep/int/AdagioKV540.mp3 ''Adagio in B minor'', KV 540] (1788) &ndash; rendered by Carlo Serafini (2011) ([http://www.seraph.it/blog_files/706c4662272db7703def4d57edfcb955-119.html blog entry])
* [https://www.youtube.com/watch?v=pFjJCj2MBTM ''Allegro'' from the Piano Sonata No. 16, KV 545] (1788) – rendered by Francium (2023)
* [https://www.youtube.com/watch?v=pFjJCj2MBTM ''Allegro'' from the Piano Sonata No. 16, KV 545] (1788) &ndash; rendered by Francium (2023)


; {{W|Keiichi Okabe}}
; {{W|Keiichi Okabe}}
* [https://www.youtube.com/watch?v=L24G4Y7tZgI ''Yuutsu no Yuutsu''] (2006) – rendered by MortisTheneRd (2024)
* [https://www.youtube.com/watch?v=L24G4Y7tZgI ''Yuutsu no Yuutsu''] (2006) &ndash; rendered by MortisTheneRd (2024)


=== 21st century ===
=== 21st century ===

Revision as of 23:32, 28 October 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. It also supports an extremely sharp tuning of Huygens/undecimal meantone using the 55de val, meaning that primes 7 and 11 are mapped very sharply to their second-best mapping.

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

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Selected 19-limit intervals approximated in 55edo

Selected just intervals by 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
15-odd-limit intervals by 55d val mapping
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
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
3/2, 4/3 3.773 17.3
13/10, 20/13 3.968 18.2
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
15/13, 26/15 7.741 35.5
15/11, 22/15 8.504 39.0
5/3, 6/5 10.187 46.7
15/14, 28/15 10.352 47.4
13/8, 16/13 10.381 47.6
11/10, 20/11 12.277 56.3
7/4, 8/7 12.992 59.5
9/5, 10/9 13.960 64.0
13/12, 24/13 14.155 64.9
13/11, 22/13 16.245 74.5
7/6, 12/7 16.765 76.8
13/9, 18/13 17.928 82.2
11/7, 14/11 18.856 86.4
9/7, 14/9 20.539 94.1

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [-87 55⟩ [⟨55 87]] +1.31 1.1915 7.21
2.3.5 81/80, 6442450944/6103515625 [⟨55 87 128]] −0.13 2.10 9.63

Commas

5-limit commas: 81/80, [47 -15 -10⟩, [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* Cents* Associated
Ratio* 		Temperaments
1 6\55 130.9 14/13 Twothirdtonic (55f)
1 16\55 349.1 11/9 Mohaha
1 23\55 501.8 4/3 Meantone (55d)
1 26\55 567.3 7/5 Liese (55)
1 27\55 589.1 45/32 Untriton (55d) / aufo (55)
5 17\55
(5\55) 		370.9
(109.1) 		99/80
(16/15) 		Quintosec
11 23\55
(3\55) 		501.8
(65.5) 		4/3
(36/35) 		Hendecatonic (55)

* Octave-reduced form

Instruments

Music

Modern renderings

Johann Sebastian Bach
Nicolaus Bruhns
Scott Joplin
  • Maple Leaf Rag (1899) – arranged for harpsichord and rendered by Claudi Meneghin (2024)
Wolfgang Amadeus Mozart
Keiichi Okabe

21st century

Claudi Meneghin

External links

and its close relative, 1/6-comma meantone] (containing another listening example) on Tonalsoft Encyclopedia

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