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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]]
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