Xenharmonic Wiki

17edt: Difference between revisions

← Older editNewer edit →

Moremajorthanmajor (talk | contribs)

3,126 edits
VisualWikitext
No edit summary
Line 7: Line 7:


=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
|}
|}


Retrieved from "https://en.xen.wiki/w/17edt"