46 Note Subset
72 Tone Equal Temperament
The ratios shown on the 6 (six) keyboards found at 72note.com suggest a 46 note subset. These intervals are the 17-limit otonalities and utonalities. This means that intervals can be paired as ascending and descending (i.e., m6 ascending is equivalent to a descending P3.) Here, ratios find their closest approximations in 72 tone equal temperament.
|
Num- ber |
Ratio
and Interval
Size
|
Approximation
in
72t-ET(Sounding
Error)
|
Suggested Interval Name
|
Suggested
Abbreviation
|
Helmholtz
|
Paul
Erlich etc.
|
||
|
1
|
1/1
(0.0¢)
|
0¢
|
0
|
perfect
prime*
|
PP*
|
Unison
|
|
tonic
|
|
2
|
3/2
(702.0¢)
|
700¢
(b2¢)
|
42
|
perfect
5
|
P5
|
5
|
perfect
fifth
|
|
|
3
|
4/3
(498¢)
|
500¢
(#2¢)
|
30
|
perfect
4
|
P4
|
4
|
||
|
4
|
5/3
(884.4¢)
|
883.3¢
(b1.0¢)
|
53
|
major
6
|
M6
|
Major
6
|
5-limit
major sixth
|
|
|
5
|
5/4
(386.3¢)
|
383.3¢
(b3.0¢)
|
23
|
perfect
3
|
P3
|
Major
3
|
5-limit
major third
|
|
|
6
|
6/5
(315.6¢)
|
316.6¢
(#1.0¢)
|
19
|
minor
3
|
m3
|
Minor
3
|
5-limit
minor third
|
|
|
7
|
7/4
(968.8¢)
|
966.6¢
(b2.2¢)
|
58
|
perfect
7
|
P7
|
Subminor
7
|
harmonic
minor 7
|
septimal
minor seventh
|
|
8
|
7/5
(582.5¢)
|
583.3¢
(#0.8¢)
|
35
|
diminished
augmented 4
|
DA4
|
Subminor
5
|
septimal
augmented 4
|
septimal
tritone
|
|
9
|
7/6
(266.9¢)
|
266.6¢
(b0.2¢)
|
16
|
narrow
minor 3
|
Nm3
|
Subminor
3
|
septimal
minor third
|
|
|
10
|
8/5
(813.7¢)
|
816.6¢
(#3.0¢)
|
49
|
minor
6
|
m6
|
Minor
6
|
5-limit
minor sixth
|
|
|
11
|
8/7
(231.2¢)
|
233.3¢
(#2.2¢)
|
14
|
small
minor 3
|
Sm3
|
Super
2
|
septimal
major 2
|
septimal
whole-tone
|
|
12
|
9/5
(1017.6¢)
|
1016.6¢
(b0.9¢)
|
61
|
wide
minor 7
|
Wm7
|
Minor
7
|
5-limit
large minor seventh
|
|
|
13
|
9/7
(435.1¢)
|
433.3¢
(b1.8¢)
|
26
|
diminished
4
|
D4
|
Supermajor
3
|
septimal
major third
|
|
|
14
|
9/8
(203.9¢)
|
200¢
(b3.9¢)
|
12
|
perfect
9
|
P9
|
2
|
major
tone
|
major
whole-tone
|
|
15
|
10/7
(617.5¢)
|
616.6¢
(b0.8¢)
|
37
|
wide
augmented 4
|
WA4
|
septimal
diminished 5
|
septimal
tritone
|
|
|
16
|
10/9
(182.4¢)
|
183.3¢
(#0.9¢)
|
11
|
diminished
9
|
D9
|
minor
tone
|
minor
whole-tone
|
|
|
17
|
11/6
(1049.4¢)
|
1050¢
(#0.6¢)
|
63
|
quarter-tone
15
|
Q15
|
|
||
|
18
|
11/7
(782.5¢)
|
783.3¢
(#0.8¢)
|
47
|
diminished
minor 6
|
Dm6
|
septimal
minor 6
|
undecimal
minor sixth
|
|
|
19
|
11/8
(551.3¢)
|
550¢
(b1.3¢)
|
33
|
perfect
11
|
P11
|
undecimal
tritone (11th harmonic)
|
||
|
20
|
11/9
(347.4¢)
|
350¢
(#2.6¢)
|
21
|
diminished
3
|
D3
|
undecimal
"median" third
|
||
|
21
|
11/10
(165.0¢)
|
166.6¢
(#1.7¢)
|
10
|
narrow
9
|
N9
|
|||
|
22
|
12/7
(933.1¢)
|
933.3¢
(#0.2¢)
|
56
|
small
7
|
S7
|
septimal
major sixth
|
||
|
23
|
12/11
(150.6¢)
|
150¢
(b0.6¢)
|
9
|
quarter-tone
9
|
Q9
|
neutral
2
|
undecimal
"median" 1/2-step
|
|
|
24
|
13/7
(1071.7¢)
|
1066.6¢
(b5¢)
|
64
|
diminished
15
|
D15
|
|
||
|
25
|
13/9
(636.6¢)
|
633.3¢
(b3.3¢)
|
38
|
small
5
|
S5
|
|||
|
26
|
13/11
(289.2¢)
|
283.3¢
(b5.9¢)
|
17
|
diminished
minor 3
|
Dm3
|
|||
|
27
|
13/12
(138.6¢)
|
133.3¢
(b5.2¢)
|
8
|
small
9
|
S9
|
3/4-tone
(Avicenna)
|
||
|
28
|
14/9
(764.9¢)
|
766.6¢
(#1.8¢)
|
46
|
narrow
minor 6
|
Nm6
|
septimal
minor sixth
|
||
|
29
|
14/11
(417.5¢)
|
416.6¢
(b0.8¢)
|
25
|
augmented
3
|
A3
|
|||
|
27
|
14/13
(128.3¢)
|
133.3¢
(#5¢)
|
8
|
small
9
|
S9
|
|||
|
30
|
15/8
(1088.3¢)
|
1083.3¢
(b4.9¢)
|
65
|
perfect
15
|
P15
|
|
5-limit
major seventh
|
|
|
31
|
15/11
(537.0¢)
|
533.3¢
(b3.6¢)
|
32
|
diminished
11
|
D11
|
|||
|
32
|
15/13
(247.7¢)
|
250¢
(#2.3¢)
|
15
|
quarter-tone
minor 3
|
Qm3
|
|||
|
33
|
15/14
(119.4¢)
|
116.6¢
(b2.8¢)
|
7
|
wide
minor 2
|
Wm2
|
Cowell
just half-step
|
||
|
34
|
16/9
(996.1¢)
|
1000¢
(#3.9¢)
|
60
|
minor
7
|
m7
|
Pythagorean
minor 7
|
Pythagorean
small min. seventh
|
|
|
35
|
16/11
(648.7)
|
650¢
(#1.3¢)
|
39
|
quarter-tone
5
|
Q5
|
|||
|
33
|
16/15
(111.7¢)
|
116.6¢
(#4.9¢)
|
7
|
wide
minor 2
|
Wm2
|
semitone just diatonic semitone |
major
5-limit half-step
|
|
|
36
|
17/9
(1101.0¢)
|
1100¢
(b1.0¢)
|
66
|
major
7
|
M7
|
|||
|
37
|
17/10
(918.6¢)
|
916.6¢
(b2.0¢)
|
55
|
wide
major 6
|
WM6
|
|||
|
38
|
17/12
(603.0¢)
|
600¢
(b3.0¢)
|
36
|
augmented
4
|
A4
|
|||
|
39
|
17/15
(216.7¢)
|
216.6¢
(0.0¢)
|
13
|
augmented
9
|
A9
|
|||
|
40
|
17/16
(105.0¢)
|
100¢
(b5.0¢)
|
6
|
minor
2
|
m2
|
overtone
half-step
|
||
|
41
|
18/11
(852.6¢)
|
850¢
(b2.6¢)
|
51
|
quarter-tone
major 6
|
QM6
|
undecimal
"median" sixth
|
||
|
42
|
18/13
(563.4¢)
|
566.6¢
(#3.3¢)
|
34
|
augmented
11
|
A11
|
|||
|
40
|
18/17
(99.0¢)
|
100¢
(#1.0¢)
|
6
|
minor
2
|
m2
|
E.T.
half-step approximation
|
||
|
43
|
20/11
(1035.0¢)
|
1033.3¢
(b1.7¢)
|
62
|
small
15
|
S15
|
|||
|
26
|
20/17
(281.4¢)
|
283.3¢
(#2.0¢)
|
17
|
diminished
minor 3
|
Dm3
|
|||
|
37
|
22/13
(910.8¢)
|
916.6¢
(#5.9¢)
|
55
|
wide
major 6
|
WM6
|
|||
|
44
|
22/15
(663.0¢)
|
666.6¢
(#3.6¢)
|
40
|
diminished
5
|
D5
|
|||
|
24
|
24/13
(1061.4¢)
|
1066.6¢
(#5.2¢)
|
64
|
diminished
15
|
D15
|
|
||
|
38
|
24/17
(597.0¢)
|
600¢
(#3.0¢)
|
36
|
augmented
4
|
A4
|
|||
|
45
|
26/15
(952.3¢)
|
950¢
(b2.3¢)
|
57
|
diminished
7
|
D7
|
|||
|
30
|
28/15
(1080.6¢)
|
1083.3¢
(#2.8¢)
|
65
|
perfect
15
|
P15
|
|
||
|
46
|
30/17
(983.3¢)
|
983.3¢
(0.0¢)
|
59
|
augmented
7
|
A7
|
|||
|
36
|
32/17
(1095.0¢)
|
1100¢
(#5.0¢)
|
66
|
major
7
|
M7
|
Helmholtz, Hermann, On the Sensations of Tone, Dover Publications, Inc., New York, 1954, p187
Musicians might confuse 15 with 15va "double octave."
December
18, 2003