17edt: Difference between revisions

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=Discussion=
=Discussion=
17edt is closely related to [[13edt|13edt]], the Bohlen-Pierce division, because they share the feature of tempering out 245/243. Both 13edt and 17edt have 4L+5s nonatonic modes, but whereas the ratio of large to small steps in 13edt is a calm 2:1, in 17edt it is a hard 3:1. Thus, the approximation of 5/3 and 7/3 suffers slightly in return for gaining a good approximation of 11/9 (given the context of the weak 5/3 and 7/3), which is in fact the size of the large step. However, by the coincidence of the 11-limit commas 17edt tempers out, 5/3 and 11/9 are off by practically the same amount in opposite directions (+10.7 cents and -11.8 cents), leading to an excellent approximation of 55/27 (only 1.1 cents flat), as are 11/9 and 9/7 (-11.8 cents and +12.4 cents), leading to an excellent approximation of 11/7 (only .6 cents flat) and these sum to 605/189-1.7 cents, which is also a 16/5 which is only .3 cents flat (in addition to equaling 256).
17edt is closely related to [[13edt|13edt]], the Bohlen-Pierce division, because they share the feature of tempering out 245/243. Both 13edt and 17edt have 4L+5s nonatonic modes, but whereas the ratio of large to small steps in 13edt is a calm 2:1, in 17edt it is a hard 3:1. Thus, the approximation of 5/3 and 7/3 suffers slightly in return for gaining a good approximation of 11/9 (given the context of the weak 5/3 and 7/3), which is in fact the size of the large step. However, by the coincidence of the 11-limit commas 17edt tempers out, 5/3 and 11/9 are off by practically the same amount in opposite directions (+10.7 cents and -11.8 cents), leading to an excellent approximation of 55/27 (only 1.1 cents flat), as are 11/9 and 9/7 (-11.8 cents and +12.4 cents), leading to an excellent approximation of 11/7 (only .6 cents flat) and these sum to 605/189-1.7 cents, which is also a 16/5 which is only .15 cents sharp (in addition to equaling 256).


=Intervals=
=Intervals=
Line 16: Line 16:
| | note name
| | note name
| | cents value
| | cents value
|hekts
| | cents value octave reduced
| | cents value octave reduced
|-
|-
| | 0
| | 0
| | C
| | C
| | 0
| colspan="2"| 0
| |  
| |  
|-
|-
Line 26: Line 27:
| | Db = B#
| | Db = B#
| | 111.9
| | 111.9
|76.5
| |  
| |  
|-
|-
Line 31: Line 33:
| | Eb = C#
| | Eb = C#
| | 223.8
| | 223.8
|152.9
| |  
| |  
|-
|-
Line 36: Line 39:
| | D
| | D
| | 335.6
| | 335.6
|229.4
| |  
| |  
|-
|-
Line 41: Line 45:
| | E
| | E
| | 447.5
| | 447.5
|305.9
| |  
| |  
|-
|-
Line 46: Line 51:
| | F = D#
| | F = D#
| | 559.4
| | 559.4
|382.35
| |  
| |  
|-
|-
Line 51: Line 57:
| | Gb = E#
| | Gb = E#
| | 671.3
| | 671.3
|458.8
| |  
| |  
|-
|-
Line 56: Line 63:
| | Hb = F#
| | Hb = F#
| | 783.2
| | 783.2
|535.3
| |  
| |  
|-
|-
Line 61: Line 69:
| | G
| | G
| | 895.1
| | 895.1
|611.8
| |  
| |  
|-
|-
Line 66: Line 75:
| | H
| | H
| | 1006.9
| | 1006.9
|688.2
| |  
| |  
|-
|-
Line 71: Line 81:
| | Jb = G#
| | Jb = G#
| | 1118.8
| | 1118.8
|764.7
| |  
| |  
|-
|-
Line 76: Line 87:
| | Ab = H#
| | Ab = H#
| | 1230.7
| | 1230.7
|841.2
| | 30.7
| | 30.7
|-
|-
Line 81: Line 93:
| | J
| | J
| | 1342.6
| | 1342.6
|917.65
| | 142.6
| | 142.6
|-
|-
| | 13
| | 13
| | A
| | A
| | 1454.5
| | 1454.4
| | 254.5
|994.1
| | 254.4
|-
|-
| | 14
| | 14
| | Bb = J#
| | Bb = J#
| | 1566.3
| | 1566.3
|1070.6
| | 366.3
| | 366.3
|-
|-
Line 96: Line 111:
| | Cb = A#
| | Cb = A#
| | 1678.2
| | 1678.2
|1147.1
| | 478.2
| | 478.2
|-
|-
Line 101: Line 117:
| | B
| | B
| | 1790.1
| | 1790.1
|1223.5
| | 590.1
| | 590.1
|-
|-
| | 17
| | 17
| | C
| | C
| | 1902.0
| | 1902
| | 702.0
|1300
| | 702
|-
|-
| | 18
| | 18
| |  
| |  
| | 2013.9
| | 2013.8
| | 813.9
|1376.5
| | 813.8
|-
|-
| | 19
| | 19
| |  
| |  
| | 2125.8
| | 2125.7
| | 925.8
|1452.9
| | 925.7
|-
|-
| | 20
| | 20
| |  
| |  
| | 2237.6
| | 2237.6
|1529.4
| | 1037.6
| | 1037.6
|-
|-
Line 126: Line 147:
| |  
| |  
| | 2349.5
| | 2349.5
|1605.9
| | 1149.5
| | 1149.5
|-
|-
| | 22
| | 22
| |  
| |  
| | 2461.4
| | 2461.35
| | 61.4
|1682.35
| | 61.35
|-
|-
| | 23
| | 23
| |  
| |  
| | 2573.2
| | 2573.2
|1758.8
| | 173.2
| | 173.2
|-
|-
| | 24
| | 24
| |  
| |  
| | 2685.2
| | 2685.1
| | 285.2
|1835.3
| | 285.1
|-
|-
| | 25
| | 25
| |  
| |  
| | 2797.1
| | 2797
| | 397.1
|1911.8
| | 397
|-
|-
| | 26
| | 26
| |  
| |  
| | 2908.9
| | 2908.9
|1988.2
| | 508.9
| | 508.9
|-
|-
| | 27
| | 27
| |  
| |  
| | 3020.8
| | 3020.75
| | 620.8
|2064.7
| | 620.75
|-
|-
| | 28
| | 28
| |  
| |  
| | 3132.7
| | 3132.6
| | 732.7
|2141.2
| | 732.6
|-
|-
| | 29
| | 29
| |  
| |  
| | 3244.6
| | 3244.5
| | 844.6
|2217.65
| | 844.5
|-
|-
| | 30
| | 30
| |  
| |  
| | 3356.5
| | 3356.4
| | 956.5
|2294.1
| | 956.4
|-
|-
| | 31
| | 31
| |  
| |  
| | 3468.3
| | 3468.3
|2370.6
| | 1068.3
| | 1068.3
|-
|-
| | 32
| | 32
| |  
| |  
| | 3580.2
| | 3580.15
| | 1180.2
|2447.1
| | 1180.15
|-
|-
| | 33
| | 33
| |  
| |  
| | 3692.1
| | 3692
| | 92.1
|2523.5
| | 92
|-
|-
| | 34
| | 34
| |  
| |  
| | 3804.0
| | 3803.9
| | 204.0
|2600
| | 203.9
|}
|}


Revision as of 18:15, 13 April 2019

Properties

17edt divides 3, the tritave, into 17 equal parts of 111.880 cents each, corresponding to 10.726 edo. It tempers out 245/243 and 16807/15625 in the 7-limit, 77/75 and 1331/1323 in the 11-limit, and 175/169 and 121/117 in the 13-limit. It supports the no-twos temperament tempering out 245/243 and 77/75, which in terms of tritave patent vals could be written 17&21.

17edt is the sixth zeta peak tritave division.

Discussion

17edt is closely related to 13edt, the Bohlen-Pierce division, because they share the feature of tempering out 245/243. Both 13edt and 17edt have 4L+5s nonatonic modes, but whereas the ratio of large to small steps in 13edt is a calm 2:1, in 17edt it is a hard 3:1. Thus, the approximation of 5/3 and 7/3 suffers slightly in return for gaining a good approximation of 11/9 (given the context of the weak 5/3 and 7/3), which is in fact the size of the large step. However, by the coincidence of the 11-limit commas 17edt tempers out, 5/3 and 11/9 are off by practically the same amount in opposite directions (+10.7 cents and -11.8 cents), leading to an excellent approximation of 55/27 (only 1.1 cents flat), as are 11/9 and 9/7 (-11.8 cents and +12.4 cents), leading to an excellent approximation of 11/7 (only .6 cents flat) and these sum to 605/189-1.7 cents, which is also a 16/5 which is only .15 cents sharp (in addition to equaling 256).

Intervals

degree of 17edt note name cents value hekts cents value octave reduced
0 C 0
1 Db = B# 111.9 76.5
2 Eb = C# 223.8 152.9
3 D 335.6 229.4
4 E 447.5 305.9
5 F = D# 559.4 382.35
6 Gb = E# 671.3 458.8
7 Hb = F# 783.2 535.3
8 G 895.1 611.8
9 H 1006.9 688.2
10 Jb = G# 1118.8 764.7
11 Ab = H# 1230.7 841.2 30.7
12 J 1342.6 917.65 142.6
13 A 1454.4 994.1 254.4
14 Bb = J# 1566.3 1070.6 366.3
15 Cb = A# 1678.2 1147.1 478.2
16 B 1790.1 1223.5 590.1
17 C 1902 1300 702
18 2013.8 1376.5 813.8
19 2125.7 1452.9 925.7
20 2237.6 1529.4 1037.6
21 2349.5 1605.9 1149.5
22 2461.35 1682.35 61.35
23 2573.2 1758.8 173.2
24 2685.1 1835.3 285.1
25 2797 1911.8 397
26 2908.9 1988.2 508.9
27 3020.75 2064.7 620.75
28 3132.6 2141.2 732.6
29 3244.5 2217.65 844.5
30 3356.4 2294.1 956.4
31 3468.3 2370.6 1068.3
32 3580.15 2447.1 1180.15
33 3692 2523.5 92
34 3803.9 2600 203.9
  • Notes named so that C D E F G H J A B C = Lambda mode

It's a weird coincidence how the schemes of 17edo and 17edt diatonicism are so similar and how their approximations of 9/7 are off by such similar amounts in opposite directions (17edo -11.6 cents and 17edt +12.4 cents).

Z function

Below is a plot of the no-twos Z function in the vicinity of 17edt.

17edt.png

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