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| == Aureus Pseudo-Meantone == | | == Aureus Pseudo-Meantone == |
| [[Bryan Deister]] has demonstrated a [[11L 5s]] (7:2 step ratio) mapping for [[87edo]] in [https://www.youtube.com/shorts/rINJKiMQE78 ''Circuit Bent - Stomach Book (microtonal cover in 87edo)''] (2025). The generator 14\87, which is two keys right from the root note, is the quasi-meantone ~[[19/17]], which is composed of highly inaccurate harmonics whose errors nearly cancel out, rendering it near-just; two of them make an essentially just classic major third ~[[5/4]]. (In contrast to actual [[meantone]] temperament, 87edo represents ~19/17, ~[[10/9]], and ~[[9/8]] as distinct intervals — the syntonic comma [[81/80]] is not tempered out, and instead the aureusma [[1445/1444]] equates two quasi-meantones to a classic major third.) Since this interval is two keys right instead of just one (as needed to avoid missing notes), this version of the temperament can be viewed as a split generator version, in which each ~19/17 is split into two equal intervals that function as both a small septendecimal semitone ~[[18/17]] and a large undevicesimal semitone ~[[19/18]], both being close to just (and the photisma [[324/323]] is tempered out). Although no second generator is needed to reach all of the notes, for convenience, down-right by one key (2\87) can be used for convenience as a second generator that functions as ~[[49/48]] (inconsistently mapped), ~[[55/54]], ~[[64/63]], ~[[65/64]], and ~[[81/80]], of which the septimal chroma ~64/63 is near-just, and four of these make a near-just classic diatonic semitone ~[[16/15]] (the currently unnamed unnoticeable comma [[5250987/5242880]] is tempered out). The range is somewhat over two octaves (which slant very slightly downwards) with no missed notes and some repeated notes to mitigate vertical wraparounds.
| | It is possible to lay out [[87edo]] using the generator 14\87, which quasi-meantone ~[[19/17]], which is composed of highly inaccurate harmonics whose errors nearly cancel out, rendering it near-just; two of them make an essentially just classic major third ~[[5/4]]. (In contrast to actual [[meantone]] temperament, 87edo represents ~19/17, ~[[10/9]], and ~[[9/8]] as distinct intervals — the syntonic comma [[81/80]] is not tempered out, and instead the aureusma [[1445/1444]] equates two quasi-meantones to a classic major third.) |
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| | === Flipped and Rotated === |
| | In the flipped and rotated verions with a [[3L 5s]] scale (14:9 step ratio)the above-mentioned 14\87 generator is one key down-right, while key right is 9\87, which functions as a flat tridecimal 2/3-tone ~[[14/13]] and a sharp septimal diatonic semitone ~[[15/14]]. This mapping makes it easy to play melodies that sound very diatonic despite the scale being non-diatonic, with octaves sloping moderately down, and a range of over three octaves (although with large partial octaves chopped off by the upper left and lower right corners), which is a bit too much for 87edo, resulting in some missed notes even in the middle octaves. Despite the missing notes, [[Bryan Deister]] has used this mapping in [https://www.youtube.com/shorts/8mHBYBfRjy4 ''87edo improv''] (2026). |
| | {{Lumatone EDO mapping|n=87|start=0|xstep=9|ystep=5}} |
| | |
| | === Split Generator === |
| | A split generator version of the above mapping returns to mostly horizontal progression with a [[11L 5s]] scale (7:2 step ratio). As in the previous mapping, the generator 14\87 is the quasi-meantone ~[[19/17]], but in this case, it is two keys right, making this a split generator version of the mapping, in which each ~19/17 is split into two equal intervals that function as both a small septendecimal semitone ~[[18/17]] and a large undevicesimal semitone ~[[19/18]], both being close to just (and the photisma [[324/323]] is tempered out). Although no second generator is needed to reach all of the notes, for convenience, down-right by one key (2\87) can be used for convenience as a second generator that functions as ~[[49/48]] (inconsistently mapped), ~[[55/54]], ~[[64/63]], ~[[65/64]], and ~[[81/80]], of which the septimal chroma ~64/63 is near-just, and four of these make a near-just classic diatonic semitone ~[[16/15]] (the currently unnamed unnoticeable comma [[5250987/5242880]] is tempered out). The range is somewhat over two octaves (which slant very slightly downwards) with no missed notes and some repeated notes to mitigate vertical wraparounds. [[Bryan Deister]] has demonstrated this mapping in [https://www.youtube.com/shorts/rINJKiMQE78 ''Circuit Bent - Stomach Book (microtonal cover in 87edo)''] (2025). |
| {{Lumatone EDO mapping|n=87|start=6|xstep=7|ystep=-5}} | | {{Lumatone EDO mapping|n=87|start=6|xstep=7|ystep=-5}} |
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There are many conceivable ways to map 87edo onto the onto the Lumatone keyboard. However, it has 3 mutually-exclusive rings of fifths, so the Standard Lumatone mapping for Pythagorean is not one of them. Due to the edos size, it would not cover the whole gamut even if it was.
Diatonic
You can use the b val, which is at least a near perfect tuning for deeptone.
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5
There is an alternate diatonic scale that makes 5-limit chords easy to play, but it has a slight octave stretch. Since every note is available in at least some of the octaves, this may actually be advantageous.
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86
Pseudo-Isomorphic Quasi-Diatonic
For those who do not want the octave stretch, Bryan Deister has demonstrated a pseudo-isomorphic mapping for 87edo in 87edo waltz (2025). This mapping retains a diatonic-style layout for ease of access to common consonant intervals, although it does not include all the notes of the tuning system. Each note 87 is actually another note 0, so as to reset octave pitch down by half of ~81/80 (moderately inflated).
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Aureus Pseudo-Meantone
It is possible to lay out 87edo using the generator 14\87, which quasi-meantone ~19/17, which is composed of highly inaccurate harmonics whose errors nearly cancel out, rendering it near-just; two of them make an essentially just classic major third ~5/4. (In contrast to actual meantone temperament, 87edo represents ~19/17, ~10/9, and ~9/8 as distinct intervals — the syntonic comma 81/80 is not tempered out, and instead the aureusma 1445/1444 equates two quasi-meantones to a classic major third.)
Flipped and Rotated
In the flipped and rotated verions with a 3L 5s scale (14:9 step ratio)the above-mentioned 14\87 generator is one key down-right, while key right is 9\87, which functions as a flat tridecimal 2/3-tone ~14/13 and a sharp septimal diatonic semitone ~15/14. This mapping makes it easy to play melodies that sound very diatonic despite the scale being non-diatonic, with octaves sloping moderately down, and a range of over three octaves (although with large partial octaves chopped off by the upper left and lower right corners), which is a bit too much for 87edo, resulting in some missed notes even in the middle octaves. Despite the missing notes, Bryan Deister has used this mapping in 87edo improv (2026).
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Split Generator
A split generator version of the above mapping returns to mostly horizontal progression with a 11L 5s scale (7:2 step ratio). As in the previous mapping, the generator 14\87 is the quasi-meantone ~19/17, but in this case, it is two keys right, making this a split generator version of the mapping, in which each ~19/17 is split into two equal intervals that function as both a small septendecimal semitone ~18/17 and a large undevicesimal semitone ~19/18, both being close to just (and the photisma 324/323 is tempered out). Although no second generator is needed to reach all of the notes, for convenience, down-right by one key (2\87) can be used for convenience as a second generator that functions as ~49/48 (inconsistently mapped), ~55/54, ~64/63, ~65/64, and ~81/80, of which the septimal chroma ~64/63 is near-just, and four of these make a near-just classic diatonic semitone ~16/15 (the currently unnamed unnoticeable comma 5250987/5242880 is tempered out). The range is somewhat over two octaves (which slant very slightly downwards) with no missed notes and some repeated notes to mitigate vertical wraparounds. Bryan Deister has demonstrated this mapping in Circuit Bent - Stomach Book (microtonal cover in 87edo) (2025).
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Rodan
The Rodan mapping is quite efficient and 87edo is an essentially perfect tuning for the temperament, which makes it a good option. However, the 1L 4s and 5L 1s mappings only cover about half the notes and the 5L 6s one has heavily sloped octaves whichever direction you put the chroma. The reverse chroma one is the only one of these that covers the full gamut in the central octave.
1L 4s
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4
21
46
63
80
10
27
44
61
78
8
25
42
59
76
6
12
29
46
63
80
10
27
44
61
78
8
48
65
82
12
29
46
63
80
14
31
48
65
82
50
67
5L 6s
11
26
13
28
43
58
73
0
15
30
45
60
75
3
18
2
17
32
47
62
77
5
20
35
50
65
76
4
19
34
49
64
79
7
22
37
52
67
82
10
78
6
21
36
51
66
81
9
24
39
54
69
84
12
27
42
57
65
80
8
23
38
53
68
83
11
26
41
56
71
86
14
29
44
59
74
2
67
82
10
25
40
55
70
85
13
28
43
58
73
1
16
31
46
61
76
4
19
34
49
54
69
84
12
27
42
57
72
0
15
30
45
60
75
3
18
33
48
63
78
6
21
36
51
66
81
71
86
14
29
44
59
74
2
17
32
47
62
77
5
20
35
50
65
80
8
23
38
53
68
83
11
26
41
16
31
46
61
76
4
19
34
49
64
79
7
22
37
52
67
82
10
25
40
55
70
85
13
28
43
63
78
6
21
36
51
66
81
9
24
39
54
69
84
12
27
42
57
72
0
15
30
45
8
23
38
53
68
83
11
26
41
56
71
86
14
29
44
59
74
2
17
32
55
70
85
13
28
43
58
73
1
16
31
46
61
76
4
19
34
0
15
30
45
60
75
3
18
33
48
63
78
6
21
47
62
77
5
20
35
50
65
80
8
23
79
7
22
37
52
67
82
10
39
54
69
84
12
71
86
31
33
46
48
50
52
54
59
61
63
65
67
69
71
73
74
76
78
80
82
84
86
1
3
5
7
0
2
4
6
8
10
12
14
16
18
20
22
24
26
15
17
19
21
23
25
27
29
31
33
35
37
39
41
43
45
47
28
30
32
34
36
38
40
42
44
46
48
50
52
54
56
58
60
62
64
66
43
45
47
49
51
53
55
57
59
61
63
65
67
69
71
73
75
77
79
81
83
85
0
56
58
60
62
64
66
68
70
72
74
76
78
80
82
84
86
1
3
5
7
9
11
13
15
17
19
73
75
77
79
81
83
85
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
5
7
9
11
13
15
17
19
21
23
25
27
29
31
33
35
37
39
41
43
45
47
49
51
53
55
26
28
30
32
34
36
38
40
42
44
46
48
50
52
54
56
58
60
62
64
66
68
70
45
47
49
51
53
55
57
59
61
63
65
67
69
71
73
75
77
79
81
83
66
68
70
72
74
76
78
80
82
84
86
1
3
5
7
9
11
85
0
2
4
6
8
10
12
14
16
18
20
22
24
19
21
23
25
27
29
31
33
35
37
39
38
40
42
44
46
48
50
52
59
61
63
65
67
78
80
Kleismic
The 4L 7s mapping covers the full gamut and keeps octaves fairly horizontal when the small step goes rightward.
53
58
66
71
76
81
86
74
79
84
2
7
12
17
22
0
5
10
15
20
25
30
35
40
45
50
8
13
18
23
28
33
38
43
48
53
58
63
68
73
21
26
31
36
41
46
51
56
61
66
71
76
81
86
4
9
14
29
34
39
44
49
54
59
64
69
74
79
84
2
7
12
17
22
27
32
37
42
47
52
57
62
67
72
77
82
0
5
10
15
20
25
30
35
40
45
50
55
60
65
50
55
60
65
70
75
80
85
3
8
13
18
23
28
33
38
43
48
53
58
63
68
73
78
83
1
68
73
78
83
1
6
11
16
21
26
31
36
41
46
51
56
61
66
71
76
81
86
4
9
14
19
24
29
4
9
14
19
24
29
34
39
44
49
54
59
64
69
74
79
84
2
7
12
17
22
27
32
37
42
32
37
42
47
52
57
62
67
72
77
82
0
5
10
15
20
25
30
35
40
45
50
55
55
60
65
70
75
80
85
3
8
13
18
23
28
33
38
43
48
53
58
63
83
1
6
11
16
21
26
31
36
41
46
51
56
61
66
71
76
19
24
29
34
39
44
49
54
59
64
69
74
79
84
47
52
57
62
67
72
77
82
0
5
10
70
75
80
85
3
8
13
18
11
16
21
26
31
34
39
Misty
The 3L 9s Misty mapping is quite efficient at making harmonics easy to play together, gives you a relatively 12edo-like experience and keeps octaves near horizontal, but has a slightly smaller range. Bryan Deister has demonstrated this mapping in microtonal improvisation in 87edo (2025).
54
61
62
69
76
83
3
63
70
77
84
4
11
18
25
71
78
85
5
12
19
26
33
40
47
54
72
79
86
6
13
20
27
34
41
48
55
62
69
76
80
0
7
14
21
28
35
42
49
56
63
70
77
84
4
11
18
81
1
8
15
22
29
36
43
50
57
64
71
78
85
5
12
19
26
33
40
2
9
16
23
30
37
44
51
58
65
72
79
86
6
13
20
27
34
41
48
55
62
69
3
10
17
24
31
38
45
52
59
66
73
80
0
7
14
21
28
35
42
49
56
63
70
77
84
4
18
25
32
39
46
53
60
67
74
81
1
8
15
22
29
36
43
50
57
64
71
78
85
5
12
19
26
33
40
47
54
61
68
75
82
2
9
16
23
30
37
44
51
58
65
72
79
86
6
13
20
27
34
41
69
76
83
3
10
17
24
31
38
45
52
59
66
73
80
0
7
14
21
28
35
42
49
4
11
18
25
32
39
46
53
60
67
74
81
1
8
15
22
29
36
43
50
33
40
47
54
61
68
75
82
2
9
16
23
30
37
44
51
58
55
62
69
76
83
3
10
17
24
31
38
45
52
59
84
4
11
18
25
32
39
46
53
60
67
19
26
33
40
47
54
61
68
48
55
62
69
76
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