55edo
| ← 54edo | 55edo | 56edo → |
55 equal divisions of the octave (abbreviated 55edo or 55ed2), also called 55-tone equal temperament (55tet) or 55 equal temperament (55et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 55 equal parts of about 21.8 ¢ each. Each step represents a frequency ratio of 21/55, or the 55th root of 2.
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. Leopold and Wolfgang Mozart recommended 55edo or something close to it, with a subset and further approximation used for keyboard instruments which (apart from an experimental instrument) did not have enough notes per octave to accommodate it in full.[1] 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
| 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
Notation
Ups and downs notation
Using Helmholtz–Ellis accidentals, 55edo can also be notated using ups and downs notation along with Stein–Zimmerman quarter-tone accidentals:
| Step offset | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|---|
| Sharp symbol |  |
 |
 |
 |
 |
 |
 |
 |
 |
 |
| Flat symbol |  |
 |
 |
 |
 |
 |
 |
 |
 |
Here, a sharp raises by four steps, and a flat lowers by four steps, so arrows can be used to fill in the gap.
Sagittal notation
Evo flavor
Revo flavor
Evo-SZ flavor
Approximation to JI
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.
| 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 |
| 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 |
| 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 | 1.19 | 7.21 |
| 2.3.5 | 81/80, [31 1 -14⟩ | [⟨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
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperament |
|---|---|---|---|---|
| 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, reduced to the first half-octave, and minimal form in parentheses if distinct
Scales
- Subsets of twothirdtonic[37]
- Undecimal otonal-like pentatonic: 17 8 7 12 11
- Subsets of hendecatonic[33]
- Septimal pentatonic-like: 10 13 9 13 10
- Septimal minor blues-like: 13 10 4 5 13 10
- Septimal heptatonic blues-like: 13 10 4 5 8 5 10
- Others
- Sakura-like scale containing phi: 9 6 18 5 17
- Quasi-equiheptatonic scale: 8 8 7 9 7 9 7
Instruments
Music
Modern renderings
- "Jesus bleibet meine Freude" from Herz und Mund und Tat und Leben, BWV 147 (1723) – arranged for two organs, rendered by Claudi Meneghin (2021)
- "Ricercar a 3" from The Musical Offering, BWV 1079 (1747) – rendered by Claudi Meneghin (2024)
- "Contrapunctus 4" from The Art of Fugue, BWV 1080 (1742–1749) – rendered by Claudi Meneghin (2024)
- "Contrapunctus 11" from The Art of Fugue, BWV 1080 (1742–1749) – rendered by Claudi Meneghin (2024)
- Prelude in E Minor "The Great" – rendered by Claudi Meneghin (2023)
- Prelude in E Minor "The Little" – rendered by Claudi Meneghin (2024)
- Maple Leaf Rag (1899) – arranged for harpsichord and rendered by Claudi Meneghin (2024)
- Rondo alla Turca from the Piano Sonata No. 11, KV 331 (1778) – rendered by Francium (2023)
- Fugue in G minor, KV 401 (1782) – rendered by Francium (2023)
- Adagio in B minor, KV 540 (1788) – rendered by Carlo Serafini (2011) (blog entry)
- Allegro from the Piano Sonata No. 16, KV 545 (1788) – rendered by Francium (2023)
- Yuutsu no Yuutsu (2006) – rendered by MortisTheneRd (2024)
21st century
- 55edo Melted Syntonic (2025)
- Improvisation One in 55edo (2025)
- Improvisation Two in 55edo (2025)
- Double Fugue on "We Wish You a Merry Christmas" for String Quartet (2020)
- Canon at the Diatonic Semitone on an Ancient Lombard Theme (2021)
- Chacony "Lament & Deception" for Two Violins and Cello (2021), for Baroque Wind Ensemble (2023)
- Fantasy "Almost a Fugue" on a Theme by Giuliani, for String Quartet (2021)
- Double Fugue on "Old McDonald" + "Shave & a Haircut" (2024)
- Road Trip to Nowhere (2021)
External links
- Mozart's tuning: 55-edo and its close relative, 1/6-comma meantone (containing another listening example) on Tonalsoft Encyclopedia
References
- ↑ Chesnut, John (1977) Mozart's Teaching of Intonation, Journal of the American Musicological Society Vol. 30, No. 2 (Summer, 1977), pp. 254-271 (Published By: University of California Press) doi.org/10.2307/831219, https://www.jstor.org/stable/831219