55edo: Difference between revisions
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→Intervals: sed -E "s#[0-9]+/[0-9]+#\[\[&\]\]#g" 55EDOintervals.txt > 55EDOintervals.new.txt |
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| Line 27: | Line 27: | ||
| 0 | | 0 | ||
| 0.0 | | 0.0 | ||
| 1/1 | | [[1/1]] | ||
| P1 | | P1 | ||
| perfect 1sn | | perfect 1sn | ||
| Line 34: | Line 34: | ||
| 1 | | 1 | ||
| 21.8 | | 21.8 | ||
| 65/64, 78/77, 99/98, ''128/125'' | | [[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, ''64/63'' | | [[36/35]], ''[[64/63]]'' | ||
| ^^1 | | ^^1 | ||
| dup 1sn | | dup 1sn | ||
| Line 48: | Line 48: | ||
| 3 | | 3 | ||
| 65.5 | | 65.5 | ||
| 28/27 | | [[28/27]] | ||
| vvm2 | | vvm2 | ||
| dudminor 2nd | | dudminor 2nd | ||
| Line 55: | Line 55: | ||
| 4 | | 4 | ||
| 87.3 | | 87.3 | ||
| 21/20, ''18/17'', ''25/24'' | | [[21/20]], ''[[18/17]]'', ''[[25/24]]'' | ||
| vm2 | | vm2 | ||
| downminor 2nd | | downminor 2nd | ||
| Line 62: | Line 62: | ||
| 5 | | 5 | ||
| 109.1 | | 109.1 | ||
| 16/15, 17/16 | | [[16/15]], [[17/16]] | ||
| m2 | | m2 | ||
| minor 2nd | | minor 2nd | ||
| Line 69: | Line 69: | ||
| 6 | | 6 | ||
| 130.9 | | 130.9 | ||
| 13/12, 14/13 | | [[13/12]], [[14/13]] | ||
| ^m2 | | ^m2 | ||
| upminor 2nd | | upminor 2nd | ||
| Line 76: | Line 76: | ||
| 7 | | 7 | ||
| 152.7 | | 152.7 | ||
| 12/11, ''11/10'' | | [[12/11]], ''[[11/10]]'' | ||
| ~2 | | ~2 | ||
| mid 2nd | | mid 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 97: | Line 97: | ||
| 10 | | 10 | ||
| 218.2 | | 218.2 | ||
| 17/15 | | [[17/15]] | ||
| ^M2 | | ^M2 | ||
| upmajor 2nd | | upmajor 2nd | ||
| Line 104: | Line 104: | ||
| 11 | | 11 | ||
| 240.0 | | 240.0 | ||
| 8/7 | | [[8/7]] | ||
| ^^M2 | | ^^M2 | ||
| dupmajor 2nd | | dupmajor 2nd | ||
| Line 111: | Line 111: | ||
| 12 | | 12 | ||
| 261.8 | | 261.8 | ||
| 7/6 | | [[7/6]] | ||
| vvm3 | | vvm3 | ||
| dudminor 3rd | | dudminor 3rd | ||
| Line 118: | Line 118: | ||
| 13 | | 13 | ||
| 283.6 | | 283.6 | ||
| 13/11 | | [[13/11]] | ||
| vm3 | | vm3 | ||
| downminor 3rd | | downminor 3rd | ||
| Line 125: | Line 125: | ||
| 14 | | 14 | ||
| 305.5 | | 305.5 | ||
| 6/5 | | [[6/5]] | ||
| m3 | | m3 | ||
| minor 3rd | | minor 3rd | ||
| Line 139: | Line 139: | ||
| 16 | | 16 | ||
| 349.1 | | 349.1 | ||
| 11/9, 27/22 | | [[11/9]], [[27/22]] | ||
| ~3 | | ~3 | ||
| mid 3rd | | mid 3rd | ||
| Line 146: | Line 146: | ||
| 17 | | 17 | ||
| 370.9 | | 370.9 | ||
| 26/21, ''16/13'' | | [[26/21]], ''[[16/13]]'' | ||
| vM3 | | vM3 | ||
| downmajor 3rd | | downmajor 3rd | ||
| Line 153: | Line 153: | ||
| 18 | | 18 | ||
| 392.7 | | 392.7 | ||
| 5/4 | | [[5/4]] | ||
| M3 | | M3 | ||
| major 3rd | | major 3rd | ||
| Line 160: | Line 160: | ||
| 19 | | 19 | ||
| 414.5 | | 414.5 | ||
| 14/11 | | [[14/11]] | ||
| ^M3 | | ^M3 | ||
| upmajor 3rd | | upmajor 3rd | ||
| Line 167: | Line 167: | ||
| 20 | | 20 | ||
| 436.4 | | 436.4 | ||
| 9/7 | | [[9/7]] | ||
| ^^M3 | | ^^M3 | ||
| dupmajor 3rd | | dupmajor 3rd | ||
| Line 174: | Line 174: | ||
| 21 | | 21 | ||
| 458.2 | | 458.2 | ||
| ''21/16'' | | ''[[21/16]]'' | ||
| vv4 | | vv4 | ||
| dud 4th | | dud 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 202: | Line 202: | ||
| 25 | | 25 | ||
| 545.5 | | 545.5 | ||
| 11/8, 15/11 | | [[11/8]], [[15/11]] | ||
| ~4 | | ~4 | ||
| mid 4th | | mid 4th | ||
| Line 209: | Line 209: | ||
| 26 | | 26 | ||
| 567.3 | | 567.3 | ||
| [[7/5]], [[18/13]] | | [[[[7/5]]]], [[[[18/13]]]] | ||
| vA4 | | vA4 | ||
| downaug 4th | | downaug 4th | ||
| Line 216: | Line 216: | ||
| 27 | | 27 | ||
| 589.1 | | 589.1 | ||
| 24/17 | | [[24/17]] | ||
| A4, vd5 | | A4, vd5 | ||
| aug 4th, downdim 5th | | aug 4th, downdim 5th | ||
| Line 223: | Line 223: | ||
| 28 | | 28 | ||
| 610.9 | | 610.9 | ||
| 17/12 | | [[17/12]] | ||
| ^A4, d5 | | ^A4, d5 | ||
| upaug 4th, dim 5th | | upaug 4th, dim 5th | ||
| Line 230: | Line 230: | ||
| 29 | | 29 | ||
| 632.7 | | 632.7 | ||
| [[10/7]], [[13/9]] | | [[[[10/7]]]], [[[[13/9]]]] | ||
| ^d5 | | ^d5 | ||
| updim 5th | | updim 5th | ||
| Line 237: | Line 237: | ||
| 30 | | 30 | ||
| 654.5 | | 654.5 | ||
| 16/11, 22/15 | | [[16/11]], [[22/15]] | ||
| ~5 | | ~5 | ||
| mid 5th | | mid 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 265: | Line 265: | ||
| 34 | | 34 | ||
| 741.8 | | 741.8 | ||
| ''32/21'' | | ''[[32/21]]'' | ||
| ^^5 | | ^^5 | ||
| dup 5th | | dup 5th | ||
| Line 272: | Line 272: | ||
| 35 | | 35 | ||
| 763.6 | | 763.6 | ||
| 14/9 | | [[14/9]] | ||
| vvm6 | | vvm6 | ||
| dudminor 6th | | dudminor 6th | ||
| Line 279: | Line 279: | ||
| 36 | | 36 | ||
| 785.5 | | 785.5 | ||
| 11/7 | | [[11/7]] | ||
| vm6 | | vm6 | ||
| downminor 6th | | downminor 6th | ||
| Line 286: | Line 286: | ||
| 37 | | 37 | ||
| 807.3 | | 807.3 | ||
| 8/5 | | [[8/5]] | ||
| m6 | | m6 | ||
| minor 6th | | minor 6th | ||
| Line 293: | Line 293: | ||
| 38 | | 38 | ||
| 829.1 | | 829.1 | ||
| 21/13, ''13/8'' | | [[21/13]], ''[[13/8]]'' | ||
| ^m6 | | ^m6 | ||
| upminor 6th | | upminor 6th | ||
| Line 300: | Line 300: | ||
| 39 | | 39 | ||
| 850.9 | | 850.9 | ||
| 18/11, 44/27 | | [[18/11]], [[44/27]] | ||
| ~6 | | ~6 | ||
| mid 6th | | mid 6th | ||
| Line 314: | Line 314: | ||
| 41 | | 41 | ||
| 894.5 | | 894.5 | ||
| 5/3 | | [[5/3]] | ||
| M6 | | M6 | ||
| major 6th | | major 6th | ||
| Line 321: | Line 321: | ||
| 42 | | 42 | ||
| 916.4 | | 916.4 | ||
| 22/13 | | [[22/13]] | ||
| ^M6 | | ^M6 | ||
| upmajor 6th | | upmajor 6th | ||
| Line 328: | Line 328: | ||
| 43 | | 43 | ||
| 938.2 | | 938.2 | ||
| 12/7 | | [[12/7]] | ||
| ^^M6 | | ^^M6 | ||
| dupmajor 6th | | dupmajor 6th | ||
| Line 335: | Line 335: | ||
| 44 | | 44 | ||
| 960.0 | | 960.0 | ||
| 7/4 | | [[7/4]] | ||
| vvm7 | | vvm7 | ||
| dudminor 7th | | dudminor 7th | ||
| Line 342: | Line 342: | ||
| 45 | | 45 | ||
| 981.8 | | 981.8 | ||
| 30/17 | | [[30/17]] | ||
| vm7 | | vm7 | ||
| downminor 7th | | downminor 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 363: | Line 363: | ||
| 48 | | 48 | ||
| 1047.3 | | 1047.3 | ||
| 11/6, ''20/11'' | | [[11/6]], ''[[20/11]]'' | ||
| ~7 | | ~7 | ||
| mid 7th | | mid 7th | ||
| Line 370: | Line 370: | ||
| 49 | | 49 | ||
| 1069.1 | | 1069.1 | ||
| 13/7, 24/13 | | [[13/7]], [[24/13]] | ||
| vM7 | | vM7 | ||
| downmajor 7th | | downmajor 7th | ||
| Line 377: | Line 377: | ||
| 50 | | 50 | ||
| 1090.9 | | 1090.9 | ||
| 15/8, ''32/17'' | | [[15/8]], ''[[32/17]]'' | ||
| M7 | | M7 | ||
| major 7th | | major 7th | ||
| Line 384: | Line 384: | ||
| 51 | | 51 | ||
| 1112.7 | | 1112.7 | ||
| 40/21, ''17/9'', ''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 | | [[56/27]] | ||
| ^^M7 | | ^^M7 | ||
| dupmajor 7th | | dupmajor 7th | ||
| Line 398: | Line 398: | ||
| 53 | | 53 | ||
| 1156.4 | | 1156.4 | ||
| 35/18, ''63/32'' | | [[35/18]], ''[[63/32]]'' | ||
| vv8 | | vv8 | ||
| dud 8ve | | dud 8ve | ||
| Line 405: | Line 405: | ||
| 54 | | 54 | ||
| 1178.2 | | 1178.2 | ||
| 128/65, 77/39, 196/99, ''125/64'' | | [[128/65]], [[77/39]], [[196/99]], ''[[125/64]]'' | ||
| v8 | | v8 | ||
| down 8ve | | down 8ve | ||
| Line 412: | Line 412: | ||
| 55 | | 55 | ||
| 1200.0 | | 1200.0 | ||
| 2/1 | | [[2/1]] | ||
| P8 | | P8 | ||
| perfect 8ve | | perfect 8ve | ||
Revision as of 08:47, 19 November 2025
| ← 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 is a zeta valley edo, so it does not approximate the harmonic series very well for its size. Despite this, it can be used as 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 | [[7/5]], [[18/13]] | vA4 | downaug 4th | vG# |
| 27 | 589.1 | 24/17 | A4, vd5 | aug 4th, downdim 5th | G#, vAb |
| 28 | 610.9 | 17/12 | ^A4, d5 | upaug 4th, dim 5th | ^G#, Ab |
| 29 | 632.7 | [[10/7]], [[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
55edo can be notated with ups and downs, spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat.
| Step offset | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|---|
| Sharp symbol | 
|
 |
|
  |
|
|
|
 |
|
|
| Flat symbol | |
|
 |
|
|
|
 |
|
|
Alternative symbols for ups and downs notation uses sharps, flats, half- and sesquisharps, and half- and sesquiflats with arrows, borrowed from extended Helmholtz–Ellis notation and Stein-Zimmerman accidental set:
| Step offset | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|---|
| Sharp symbol | |
 |
 |
 |
|
 |
 |
 |
|
 |
| Flat symbol |  |
 |
 |
|
 |
 |
 |
|
 |
Sagittal notation
Evo flavor
Revo flavor
Evo-SZ flavor
31-tone subset
The 31-out-of-55edo subset can be notated entirely with the standard notation of 7 each of naturals/sharps/flats, and 5 each of doublesharps/doubleflats, as a 31-tone chain-of-5ths from Gbb to Ax.
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 |
Uniform maps
| Min. size | Max. size | Wart notation | Map |
|---|---|---|---|
| 54.7778 | 54.9113 | 55cf | ⟨55 87 127 154 190 203] |
| 54.9113 | 54.9935 | 55f | ⟨55 87 128 154 190 203] |
| 54.9935 | 55.0340 | 55 | ⟨55 87 128 154 190 204] |
| 55.0340 | 55.0668 | 55d | ⟨55 87 128 155 190 204] |
| 55.0668 | 55.2064 | 55de | ⟨55 87 128 155 191 204] |
Commas
|
Todo: cleanup |
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 | 8\55 | 174.5 | 10/9~11/10 | Tetracot (55c) |
| 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)
- "Ricercar a 6" from The Musical Offering, BWV 1079 (1747) – rendered by Claudi Meneghin (2025)
- "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)
- Fugue from "Suite in E minor", HWV 429 (1720) – arranged for Baroque ensemble and drums, rendered by Claudi Meneghin (2025)
- 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)
- Mozart's Gigue KV 574, Arranged for Fortepiano (55-edo) – rendered by Claudi Meneghin (2025)
- Yuutsu no Yuutsu (2006) – rendered by MortisTheneRd (2024)
21st century
- 55edo improv (2025)
- Waltz in 55edo (2025)
- 55edo prelude (2025)
- 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)
- Migration (2025)
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