17edt: Difference between revisions

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__FORCETOC__
'''17EDT''' is [[EDT|equal division of the third harmonic]] into 17 parts of 111.880 cents each (corresponding to 10.726 [[EDO]]).
=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.
[[category:macrotonal]]


17edt is the sixth [[The_Riemann_Zeta_Function_and_Tuning#Removing primes|zeta peak tritave division]].
== Properties ==
In the no-twos subgroup, 17EDT 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 b17&b21.


=Discussion=
17EDT is the sixth [[The_Riemann_Zeta_Function_and_Tuning#Removing primes|zeta peak tritave division]].
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=
== 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 .15 cents sharp (in addition to equaling 256).


== Intervals ==
{| class="wikitable"
{| class="wikitable"
|-
|-
| | degree of 17edt
! | degree
| | note name
! | note name
| | cents value
! | cents value
!hekts
! | hekts
| | cents value octave reduced
! | cents value <br>octave reduced
|-
|-
| | 0
| | 0
| | C
| | C
! colspan="2" | 0
| colspan="2" | 0
| |  
| |  
|-
|-
Line 27: Line 26:
| | Db = B#
| | Db = B#
| | 111.9
| | 111.9
|76.5
| | 76.5
| |  
| |  
|-
|-
Line 33: Line 32:
| | Eb = C#
| | Eb = C#
| | 223.8
| | 223.8
|152.9
| | 152.9
| |  
| |  
|-
|-
Line 39: Line 38:
| | D
| | D
| | 335.6
| | 335.6
|229.4
| | 229.4
| |  
| |  
|-
|-
Line 45: Line 44:
| | E
| | E
| | 447.5
| | 447.5
|305.9
| | 305.9
| |  
| |  
|-
|-
Line 51: Line 50:
| | F = D#
| | F = D#
| | 559.4
| | 559.4
|382.35
| | 382.35
| |  
| |  
|-
|-
Line 57: Line 56:
| | Gb = E#
| | Gb = E#
| | 671.3
| | 671.3
|458.8
| | 458.8
| |  
| |  
|-
|-
Line 63: Line 62:
| | Hb = F#
| | Hb = F#
| | 783.2
| | 783.2
|535.3
| | 535.3
| |  
| |  
|-
|-
Line 69: Line 68:
| | G
| | G
| | 895.1
| | 895.1
|611.8
| | 611.8
| |  
| |  
|-
|-
Line 75: Line 74:
| | H
| | H
| | 1006.9
| | 1006.9
|688.2
| | 688.2
| |  
| |  
|-
|-
Line 81: Line 80:
| | Jb = G#
| | Jb = G#
| | 1118.8
| | 1118.8
|764.7
| | 764.7
| |  
| |  
|-
|-
Line 87: Line 86:
| | Ab = H#
| | Ab = H#
| | 1230.7
| | 1230.7
|841.2
| | 841.2
| | 30.7
| | 30.7
|-
|-
Line 93: Line 92:
| | J
| | J
| | 1342.6
| | 1342.6
|917.65
| | 917.65
| | 142.6
| | 142.6
|-
|-
Line 99: Line 98:
| | A
| | A
| | 1454.4
| | 1454.4
|994.1
| | 994.1
| | 254.4
| | 254.4
|-
|-
Line 105: Line 104:
| | Bb = J#
| | Bb = J#
| | 1566.3
| | 1566.3
|1070.6
| | 1070.6
| | 366.3
| | 366.3
|-
|-
Line 111: Line 110:
| | Cb = A#
| | Cb = A#
| | 1678.2
| | 1678.2
|1147.1
| | 1147.1
| | 478.2
| | 478.2
|-
|-
Line 117: Line 116:
| | B
| | B
| | 1790.1
| | 1790.1
|1223.5
| | 1223.5
| | 590.1
| | 590.1
|-
|-
Line 123: Line 122:
| | C
| | C
| | 1902
| | 1902
|1300
| | 1300
| | 702
| | 702
|-
|-
Line 129: Line 128:
| |  
| |  
| | 2013.8
| | 2013.8
|1376.5
| | 1376.5
| | 813.8
| | 813.8
|-
|-
Line 135: Line 134:
| |  
| |  
| | 2125.7
| | 2125.7
|1452.9
| | 1452.9
| | 925.7
| | 925.7
|-
|-
Line 141: Line 140:
| |  
| |  
| | 2237.6
| | 2237.6
|1529.4
| | 1529.4
| | 1037.6
| | 1037.6
|-
|-
Line 147: Line 146:
| |  
| |  
| | 2349.5
| | 2349.5
|1605.9
| | 1605.9
| | 1149.5
| | 1149.5
|-
|-
Line 153: Line 152:
| |  
| |  
| | 2461.35
| | 2461.35
|1682.35
| | 1682.35
| | 61.35
| | 61.35
|-
|-
Line 159: Line 158:
| |  
| |  
| | 2573.2
| | 2573.2
|1758.8
| | 1758.8
| | 173.2
| | 173.2
|-
|-
Line 165: Line 164:
| |  
| |  
| | 2685.1
| | 2685.1
|1835.3
| | 1835.3
| | 285.1
| | 285.1
|-
|-
Line 171: Line 170:
| |  
| |  
| | 2797
| | 2797
|1911.8
| | 1911.8
| | 397
| | 397
|-
|-
Line 177: Line 176:
| |  
| |  
| | 2908.9
| | 2908.9
|1988.2
| | 1988.2
| | 508.9
| | 508.9
|-
|-
Line 183: Line 182:
| |  
| |  
| | 3020.75
| | 3020.75
|2064.7
| | 2064.7
| | 620.75
| | 620.75
|-
|-
Line 189: Line 188:
| |  
| |  
| | 3132.6
| | 3132.6
|2141.2
| | 2141.2
| | 732.6
| | 732.6
|-
|-
Line 195: Line 194:
| |  
| |  
| | 3244.5
| | 3244.5
|2217.65
| | 2217.65
| | 844.5
| | 844.5
|-
|-
Line 201: Line 200:
| |  
| |  
| | 3356.4
| | 3356.4
|2294.1
| | 2294.1
| | 956.4
| | 956.4
|-
|-
Line 207: Line 206:
| |  
| |  
| | 3468.3
| | 3468.3
|2370.6
| | 2370.6
| | 1068.3
| | 1068.3
|-
|-
Line 213: Line 212:
| |  
| |  
| | 3580.15
| | 3580.15
|2447.1
| | 2447.1
| | 1180.15
| | 1180.15
|-
|-
Line 219: Line 218:
| |  
| |  
| | 3692
| | 3692
|2523.5
| | 2523.5
| | 92
| | 92
|-
|-
Line 225: Line 224:
| |  
| |  
| | 3803.9
| | 3803.9
|2600
| | 2600
| | 203.9
| | 203.9
|}
|}


<ul><li>Notes named so that C D E F G H J A B C = Lambda mode</li></ul>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).
<ul><li>Notes named so that C D E F G H J A B C = Lambda mode</li></ul>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=
== Regular temperament ==
Below is a plot of the [[The_Riemann_Zeta_Function_and_Tuning#Removing primes|no-twos Z function]] in the vicinity of 17edt.
17EDT is also be thought of as a generator of the vavoom temperament. As one degree of 17EDT is very close to [[16/15]], an unnoticeable comma {{monzo|-68 18 17}} is tempered out in the vavoom temperament.
 
'''<font style="font-size: 1.35em">Vavoom (118&amp;783)</font>'''<br>
'''<font style="font-size: 1.2em">5-limit</font>'''<br>
Comma: {{monzo|-68 18 17}}<br>
Mapping: [{{val|1 0 4}}, {{val|0 17 -18}}]<br>
POTE generator: ~16/15 = 111.876<br>
Vals: 11, 32, 43, 75, 118, 429, 547, 665, 783, 901, 1684<br>
Badness: 0.098376<br><br>
 
== Z function ==
Below is a plot of the [[The_Riemann_Zeta_Function_and_Tuning#Removing primes|no-twos Z function]] in the vicinity of 17EDT.


[[File:17edt.png|alt=17edt.png|17edt.png]]       
[[File:17edt.png|alt=17edt.png|17edt.png]]       
[[Category:edt]]
[[Category:edt]]
[[Category:macrotonal]]
[[Category:tritave]]
[[Category:tritave]]

Revision as of 00:21, 9 August 2021

17EDT is equal division of the third harmonic into 17 parts of 111.880 cents each (corresponding to 10.726 EDO).

Properties

In the no-twos subgroup, 17EDT 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 b17&b21.

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 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).

Regular temperament

17EDT is also be thought of as a generator of the vavoom temperament. As one degree of 17EDT is very close to 16/15, an unnoticeable comma [-68 18 17⟩ is tempered out in the vavoom temperament.

Vavoom (118&783)
5-limit
Comma: [-68 18 17⟩
Mapping: [⟨1 0 4], ⟨0 17 -18]]
POTE generator: ~16/15 = 111.876
Vals: 11, 32, 43, 75, 118, 429, 547, 665, 783, 901, 1684
Badness: 0.098376

Z function

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

17edt.png

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