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Note: if the pitches in the first few seconds of the MIDI files sound wrong, try playing this GM reset file first.
*The smoothest 30-TET tunings of this sequence are identical to 15-TET and 10-TET. This is an "extra-spicy" discordant version unique to 30-TET.
**This new alternative version is based on a scale with two large and five small steps.
More notes on these tunings on this page.
*Tuning values for these scales are approximate. In actual use, each octave would vary slightly in tuning.
B F# C#
(C) G D A E
This structure can be generated automatically by defining two constants: the size of the perfect fifth and the major third. (The minor third is the difference between the two.) A third constant, the size of the octave, is used in non-octave scales and scales with tempered octaves, but most of the tunings here have exact 2/1 octaves. All the other intervals in the scale are combinations of the basic intervals multiplied together.
In the 5-limit JI (just intonation) tuning, the fifth is tuned to an exact 3/2 (702.0 cents) as nearly as possible, and the major third is tuned to 5/4 (386.3 cents). The interval between E and B, 40/27 (680.4 cents), is a discordant "wolf" fifth, which differs from the perfect fifth by a syntonic comma (81/80 or 21.5 cents). This scale has two different sizes of whole steps -- a somewhat larger one from D to E, G to A, or B to C# (9/8 or 203.9 cents), and a slightly smaller one from E to F# or A to B (10/9 or 182.4 cents). Two diatonic semitones (16/15 or 111.7 cents) complete the octave. Other tunings can be categorized by the sizes of their fifths and thirds in comparison to JI.
Note: defining the constant for the size of the third as a minor third instead of a major third results in a minor key version of the canon! Other substitutions have even more unusual results. See above under "Just Intonation" for a set of tunings based on this harmonic structure with different sizes of intervals.
Meantone temperaments divide the major third into two equal steps, intermediate in size between 10/9 and 9/8. A sequence of four ascending fifths ends up at the same note as a major third plus two octaves. Sharps are lower in pitch than the enharmonic flats (except in 12-TET, where they are equivalent). The fifths of meantone are tempered slightly flat to improve the accuracy of the thirds. Additionally, the octaves may be slightly sharpened to improve the thirds while keeping the fifths closer to just. Meantone scales may be classified by the size of their fifths.
tuning fifth scale 12-TET (24, 36) 700.0 0 2 4 5 7 9 11 12 67-TET* 698.5 1/6-comma 698.4 55-TET 698.2 0 9 18 23 32 41 50 55 98-TET* 698.0 43-TET (86*) 697.7 0 7 14 18 25 32 39 43 1/5-comma 697.7 Tenney-Optimal 697.5644 (with 1201.6985 cent octaves) 74-TET 697.3 0 12 24 31 43 55 67 74 31-TET (62, 93) 696.8 0 5 10 13 18 23 28 31 1/4-comma 696.6 81-TET 696.3 0 13 26 34 47 60 73 81 golden 696.2 7/26-comma 696.2 50-TET (100) 696.0 0 8 16 21 29 37 45 50 2/7-comma 695.8 69-TET 695.7 0 11 22 29 40 51 62 69 LucyTuning 695.5 88-TET 695.5 0 14 28 37 51 65 79 88 1/3-comma 694.8 19-TET (38, 57, 76) 694.7 0 3 6 8 11 14 17 19 45-TET 693.3 0 7 14 19 26 33 40 45 26-TET 692.3 0 4 8 11 15 19 23 26 33-TET* 690.9 0 5 10 14 19 24 29 33*33, 67, 86, and 98-TET tunings include major thirds that are closer to just than their meantone thirds.
Abb Ebb Bbb Fb Cb Gb Db Ab
Eb Bb F C G D A E
B F# C# G# D# A# E# B#
tuning fifth scale 29-TET 703.4 0 5 9 12 17 21 26 29 41-TET (82) 702.4 0 7 13 17 24 30 37 41 94-TET 702.1 0 16 30 39 55 69 85 94 3-limit JI* 702.0 53-TET* 701.9 0 9 17 22 31 39 48 53 171-TET* 701.8 0 29 55 71 100 126 155 171 1/8-schisma* 701.7 65-TET* 701.5 0 11 21 27 38 48 59 65 77-TET* 701.3 0 13 25 32 45 57 70 77 89-TET 701.1 0 15 29 37 52 66 81 89 12-TET (24, 36) 700.0 0 2 4 5 7 9 11 12*these tunings are near-just; fourths and thirds are within 5.0 cents of just.
tuning fifth scale 10-TET (20) 720.0 0 2 3 4 6 7 9 10 32-TET 712.5 0 6 10 13 19 23 29 32 54-TET 711.1 0 10 17 22 32 39 49 54 22-TET (44, 66) 709.1 0 4 7 9 13 16 20 22 78-TET 707.7 0 14 25 32 46 57 71 78 56-TET 707.1 0 10 18 23 33 41 51 56 90-TET 706.7 0 16 29 37 53 66 82 90 34-TET (68) 705.9 0 6 11 14 20 25 31 34 80-TET 705.0 0 14 26 33 47 59 73 80 46-TET (92) 704.3 0 8 15 19 27 34 42 46 58-TET 703.4 0 10 19 24 34 43 53 58 70-TET 702.9 0 12 23 29 41 52 64 70
tuning third scale 25-TET 384.0 0 5 8 10 15 18 23 25 22-TET (44, 66) 381.8 0 4 7 9 13 16 20 22 85-TET 381.2 0 15 27 35 50 62 77 85 63-TET 381.0 0 11 20 26 37 46 57 63 41-TET (82) 380.5 0 7 13 17 24 30 37 41 60-TET 380.0 0 10 19 25 35 44 54 60 79-TET 379.7 0 13 25 33 46 58 71 79 19-TET (38, 57, 76) 378.9 0 3 6 8 11 14 17 19 35-TET 377.1 0 5 11 15 20 26 31 35
G# D#
E B F# C#
C G D A E
Bb F C
Ab
Note that the two pairs of notes in the scale diagram labeled "C" and "E" actually represent different pitches: Porcupine temperament is not represented well by standard diatonic notation. See the porcupine temperament page for more information about notating porcupine scales.
tuning third fifth scale (L M S) 15-TET (30): 400.0 720.0 0 3 5 6 9 11 14 15 3 2 1 37-TET: 389.2 713.5 0 7 12 15 22 27 34 37 7 5 3 59-TET: 386.4 711.9 0 11 19 24 35 43 54 59 11 8 5 22-TET (44, 66): 381.8 709.1 0 4 7 9 13 16 20 22 4 3 2 51-TET: 376.5 705.9 0 9 16 21 30 37 46 51 9 7 5 29-TET: 372.4 703.4 0 5 9 12 17 21 26 29 5 4 3
This is a small family of scales that shares the property of having relatively sharp fifths, which narrows the minor sevenths to just the right extent to sound like good approximations to the seventh harmonic (7/4, 968.8 cents). Again, 12-TET is the borderline case that just barely fits in this category (it actually has slightly narrow fifths, but it just happens that two fourths up is the best approximation of the 7th harmonic).
tuning fifth scale 15-TET: 720.0 0 3 5 6 9 11 14 15 42-TET: 714.3 0 8 14 17 25 31 39 42 37-TET: 713.5 0 7 12 15 22 27 34 37 59-TET: 711.9 0 11 19 24 35 43 54 59 27-TET: 711.1 0 5 9 11 16 20 25 27 49-TET: 710.2 0 9 16 20 29 36 45 49 22-TET: 709.1 0 4 7 9 13 16 20 22 12-TET: 700.0 0 2 4 5 7 9 11 12
tuning fifth third scale 90-TET 706.7 386.7 0 16 29 37 53 66 82 90 34-TET (68) 705.9 388.2 0 6 11 14 20 25 31 34 80-TET 705.0 390.0 0 14 26 33 47 59 73 80 46-TET (92) 704.3 391.3 0 8 15 19 27 34 42 46 75-TET 704.0 384.0 0 13 24 31 44 55 68 75 87-TET 703.4 386.2 0 15 28 36 51 64 79 87 99-TET 703.0 387.9 0 17 32 41 58 73 90 99 94-TET 702.1 383.0 0 16 30 39 55 69 85 94 3-limit JI schismic 702.0 384.4 53-TET 701.9 384.9 0 9 17 22 31 39 48 53 171-TET 701.8 386.0 0 29 55 71 100 126 155 171 1/8-schisma temp. 701.7 386.3 D E Gb G A Cb Db D 65-TET 701.5 387.7 0 11 21 27 38 48 59 65 77-TET 701.3 389.6 0 13 25 32 45 57 70 77 Starling temp. 700.0 388.0 72-TET 700.0 383.3 0 12 23 30 42 53 65 72 84-TET 700.0 385.7 0 14 27 35 49 62 76 84 96-TET 700.0 387.5 0 16 31 40 56 71 87 96 Lumma's temp. 700.0 384.4 91-TET 698.9 382.4 0 15 29 38 53 67 82 91Note: 41-TET (a schismic temperament) has major thirds that are 5.8 cents flat and minor thirds 6.3 cents sharp. However, it is often mentioned as a near-just scale because of its excellent approximations to higher-limit JI. 58-TET is almost as good at approximating 5-limit JI as 41-TET, and is consistent to the 17-limit.
tuning fifth scale 18-TET 733.3 0 4 6 7 11 13 17 18 15-TET (30) 720.0 0 3 5 6 9 11 14 15 42-TET 714.3 0 8 14 17 25 31 39 42 27-TET 711.1 0 5 9 11 16 20 25 27 39-TET 707.7 0 7 13 16 23 29 36 39 12-TET (24, 36) 700.0 0 2 4 5 7 9 11 12 33-TET 690.9 0 5 11 14 19 25 30 33 21-TET 685.7 0 3 7 9 12 16 19 21
7-TET: 0 1 2 3 4 5 6 7 14-TET: 0 2 5 6 8 11 13 14 21-TET: 0 3 7 9 12 16 19 21 28-TET: 0 4 9 12 16 21 25 28 35-TET: 0 5 11 15 20 26 31 35
10-TET: 0 2 3 4 6 7 9 10 13-TET: 0 3 4 5 8 9 12 13 17-TET: 0 3 5 7 10 12 15 17 24-TET: 0 4 7 10 14 17 21 24
9-TET: 0 1 3 4 5 7 8 9 11-TET: 0 1 4 5 6 9 10 11 16-TET: 0 2 5 7 9 12 14 16 23-TET: 0 3 7 10 13 17 20 23 30-TET: 0 4 9 13 17 22 26 30
6-TET works out to be 7-limit consistent, but the "scale" that results is reduced to a single augmented triad. To make a reasonable "translation", the equivalent of the major triad is defined as 0-3-4 (D-G#-A#). This results in the scale 0-2-3-2-4-5-7-6 (D-F#-G#-F#-A#-C-E-D).
11-TET is a difficult scale to write for; most intervals are dissonant. This retuning is based on the symmetrical 0-3-6 chord, which results in the scale 0-1-3-5-6-8-9-11. Symmetrical chords like this seem to work well in 11-TET.
The first version of 13-TET is also built from a symmetrical chord: 0-4-8 (scale: 0-3-4-5-8-9-12-13). The second version is built from the chord 0-4-7 (scale: 0-1-4-6-7-10-11-13).
The 17-TET version was tuned based on a chain of 17-TET fifths, resulting in a major chord of 0-6-10 and a symmetrical scale of 0-3-6-7-10-13-16-17. This scale, like the meantone scales, divides the "major third" (a very sharp one) into two equal parts, but unlike meantone tunings, sharps are higher in pitch than the enharmonic flats.
The 20-TET version is based on the major triad 0-7-12, resulting in a scale of 0-4-7-8-12-15-19-20.
24-TET, 30-TET, and 36-TET contain smaller ET scales within themselves (12-TET and 15-TET). The thirds in the 24-TET and 36-TET retunings are tuned one step smaller than normal, and the thirds in the 30-TET retuning are one step sharper.
The original 32-TET version was based on the major triad 0-11-19, resulting in a scale of 0-6-11-13-19-24-30-32.
Although Blackjack doesn't have enough fifths in a row for a traditional diatonic scale, it is rich in other harmonic structures. For the Blackjack retunings, I've tried a number of different ways to remap the harmony of the canon in a way that fits into the Blackjack lattice. Two of these tunings exploit the 7-limit harmonies that Blackjack is known for; there is an otonal version based on the harmonic series, approximating 4:5:7, and a utonal version approximating 1/(7:5:4). There is also a strange beating version, based on a discordant 1/1 : 11/9 : 32/21 triad, and a version that shows off the very small steps available in the Blackjack tuning, based on a 7:8:10 triad. The structure of the remapped scales works like this:
otonal utonal beating microstep
5> 3> 1> 7 5 3 7 3 9> 7 2 7>
2 0 8> 6> 2 0 8> 6> 4 0 6> 2> 5 0 5> 0>
original otonal utonal beating microstep
D 0 0 0 0
E 6> 6> 2> 0>
F# 3> 5 3 2
G 2 2 4 5
A 8> 8> 6> 5>
B 5> 7 7 7
C# 1> 3 9> 7>
Notice what this does to the melody!
Original: F# E D C# B A B C# | D C# B A G F# G E otonal 3> 6> 0 1> 5> 8> 5> 1> | 0 1> 5> 8> 2 3> 2 6> utonal 5 6> 0 3 7 8> 7 3 | 0 3 7 8> 2 5 2 6> beating 3 2> 0 9> 7 6> 7 9> | 0 9> 7 6> 4 3 4 2> microstep 2 0> 0 7> 7 5> 7 7> | 0 7> 7 5> 5 2 5 0>
15 07 17 08 00 10 02 original 88CET cents 12-TET + cents D 0 0 D +0 E 2 176 E -24 F# 7 616 Ab+16 G 8 704 A +4 A 10 880 B -20 B 15 1320 Eb+20 C# 17 1496 F -4 D 18 1584 F#-16The alternate version is mapped to a chord that approximates 4:6:7:10.
The Bohlen-Pierce (BP) scale is a 7-limit JI tuning discovered by Heinz Bohlen, which is built from factors of 3, 5, and 7 without any octaves. The original BP scale is represented as a sequence of ratios from a base pitch: 1/1 27/25 25/21 9/7 7/5 75/49 5/3 9/5 49/25 15/7 7/3 63/25 25/9 3/1. The 3/1 interval serves as the equivalent of the octave for this scale. There is also a tempered version of the BP scale (independently discovered by Heinz Bohlen, John Robinson Pierce, and Kees van Prooijen), based on dividing the 3/1 into 13 equal steps of 146.3 cents each. Because this scale is based on ratios of odd integers, it works best with timbres containing odd harmonics, such as clarinets. For this retuning of the canon, the 5:7:9 triad (or its tempered equivalent) is used as a substitute for the major triad.
original Just BP cents 12-TET + cents Tempered BP cents 12-TET + cents D 1/1 0.0 D +0 0 0.0 D +0 E 27/25 133.2 Eb+33 1 146.3 Eb+46 F# 7/5 582.5 G#-17 4 585.2 G#-15 G 5/3 884.4 B -16 6 877.8 B -22 A 9/5 1017.6 C +18 7 1024.1 C +24 B 7/3 1466.9 E#-33 10 1463.0 E#-37 C# 63/25 1600.1 F#+0 11 1609.3 F#+9 D 3/1 1902.0 A +2 13 1902.0 A +2
The "Top" meantone temperament has a generator of 504.13 cents (a very slightly wide fourth) and a period of 1201.70 cents (just barely wider than an octave). It has a very smooth, pleasant sound, with fifths of 697.56 cents and major thirds of 386.86 cents, and the 1.7-cent wide octaves help to add a bit of realistic animation to electronic instruments with their unnaturally perfect octaves.
Mavila temperament (named in reference to Erv Wilson's "meta-mavila" tuning, which gets its name from the Chopi village of Mavila) is a tuning in which the generator is even wider than the fourths of meantone temperament. It has also been referred to as "pelogic" temperament, from its resemblance to the Indonesian Pelog scale, with five small steps and two large steps. The main difference is that in genuine Balinese and Javanese pelog scales, the steps are of all different sizes, but in mavila temperament, there are exactly two different sizes of steps. And although pelog is typically used with five-note modes, mavila temperament can be extended to 9-, 16-, and 23-note scales. An interesting feature of mavila temperament is that its "major" intervals correspond with the "minor" intervals of the diatonic scale, and vice versa. As with the genuine pelog scales, the mavila temperament works best with slightly enlarged octaves: the "Top" mavila temperament has a generator of 521.52 cents (a wide fourth) with an octave of 1206.55 cents.
"Father" is a temperament in which the closest approximation of a major third or a perfect fourth is the same interval, a quarter-tone-flattened fourth. This results in a bizarre "anti-pentatonic" scale with two small steps and three large steps (L-s-L-s-L). It's very dissonant with most ordinary timbres, but it sounds a little better with carefully selected timbres, as in my 8-TET and 13-TET "anti-pentatonic" versions of the canon. The "Top" father temperament has a generator of 447.39 cents (a quarter-tone-flattened fourth) and a period of 1185.87 cents (a very narrow octave, very noticeably flat).
