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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 [[support]]s the no-twos temperament tempering out 245/243 and 77/75, which in terms of tritave patent vals could be written b17&b21.
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 [[support]]s 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 [[The_Riemann_Zeta_Function_and_Tuning#Removing primes|zeta peak tritave division]].
17EDT is the sixth [[The Riemann zeta function and tuning#Removing primes|zeta peak tritave division]].


{{Harmonics in equal|17|3|1|intervals=prime|columns=15}}
{{Harmonics in equal|17|3|1|intervals=prime|columns=15}}


== Discussion ==
== Discussion ==
17edt is closely related to [[13edt]], the Bohlen-Pierce division, because they share the feature of tempering out 245/243, so that the interval of [[9/7]] stacked twice results in [[5/3]]. Therefore, like 13edt, 17edt's 9/7 generates an enneatonic Lambda ([[4L 5s (3/1-equivalent)|4L 5s]]) scale. If 13edt can be considered an analogue of [[12edo]] as the basic tuning of this scale, 17edt is an analogue of [[17edo]] as the hard 3:1 tuning. While the approximation of 5/3 and 7/3 is less good than that of 13edt, this scale has 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, as it is equated to [[25/21]] by virtue of tempering out 77/75.
17edt is closely related to [[13edt]], the Bohlen-Pierce division, because they share the feature of tempering out 245/243, so that the interval of [[9/7]] stacked twice results in [[5/3]]. Therefore, like 13edt, 17edt's 9/7 generates an enneatonic Lambda ([[4L 5s (3/1-equivalent)|4L 5s]]) scale. If 13edt can be considered an analogue of [[12edo]] as the basic tuning of this scale, 17edt is an analogue of [[17edo]] as the hard 3:1 tuning. While the approximation of 5/3 and 7/3 is less good than that of 13edt, this scale has 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, as it is equated to [[25/21]] by virtue of tempering out 77/75.


17edt's tuning of [[BPS]] (the temperament defined in 3.5.7 by tempering out 245/243) is also very notable in that its generator (9/7) spans 4 steps, meaning that it is divisible into 2; and the interval [[27/7]] spans 21 steps, meaning it is divisible into 3. These lead to weak extensions of BPS known as [[Dubhe]] (which splits the 9/7 generator into two intervals of [[17/15]], tempering out [[2025/2023]] as the additional comma in 3.5.7.17), and [[Mintra]]/Mintaka (which splits 27/7 into three intervals of 11/7, tempering out [[1331/1323]] as the additional comma in 3.5.7.11), respectively. 17edt in fact supports basic tunings of Dubhe[9] (which is [[8L 1s (3/1-equivalent)|8L 1s]]) and Mintaka[12] (which is [[5L 7s (3/1-equivalent)|5L 7s]], i.e. a macro-chromatic scale). Therefore 17edt is important as the smallest nontrivial tuning to support each, and it is remarkable for providing such an efficient intersection of temperaments in the 3.5.7.11.17 subgroup, despite being an extremal tuning of most of these (specifically since Mintaka asks for an 11/7 tuned flat, rather than close to just; and the 9/7 is highly overtempered for BPS) and losing much accuracy compared to more optimal tunings - its behavior in this subgroup is reminiscent of that of [[15edo]] in the [[11-limit]].
17edt's tuning of BPS ([[Bohlen–Pierce–Stearns]], the temperament defined in 3.5.7 by tempering out 245/243) is also very notable in that its generator (9/7) spans 4 steps, meaning that it is divisible into 2; and the interval [[27/7]] spans 21 steps, meaning it is divisible into 3. These lead to weak extensions of BPS known as [[Dubhe]] (which splits the 9/7 generator into two intervals of [[17/15]], tempering out [[2025/2023]] as the additional comma in 3.5.7.17), and [[Mintra]] (an extension of [[Mintaka]] which splits 27/7 into three intervals of 11/7, tempering out [[1331/1323]] as the additional comma in 3.5.7.11), respectively. 17edt in fact supports basic tunings of Dubhe[9] (which is [[8L 1s (3/1-equivalent)|8L 1s]]) and Mintaka[12] (which is [[5L 7s (3/1-equivalent)|5L 7s]], i.e. a macro-chromatic scale). Therefore 17edt is important as the smallest nontrivial tuning to support each, and it is remarkable for providing such an efficient intersection of temperaments in the 3.5.7.11.17 subgroup, despite being an extremal tuning of most of these (specifically since Mintaka asks for an 11/7 tuned flat, rather than close to just; and the 9/7 is highly overtempered for BPS) and losing much accuracy compared to more optimal tunings—its behavior in this subgroup is reminiscent of that of [[15edo]] in the [[11-limit]].


== Intervals ==
== Intervals ==
The notation schemes below are based on the BPS-Lambda enneatonic scale presented in the symmetric (sLsLsLsLs, Cassiopeian) mode in J, and the Mintaka macrodiatonic scale presented in the macro-Phrygian (sLLLsLL) mode in E.  
The notation schemes below are based on the BPS ''Lambda'' enneatonic scale presented in the symmetric (sLsLsLsLs, Cassiopeian) mode in J, and the Mintaka macrodiatonic scale presented in the macro-Phrygian (sLLLsLL) mode in E.  


Interval approximations shown in the 3.5.7.11.17 [[subgroup]] column are those within the 77-throdd [[limit]] and less than 1/3 of a step off (all of these being [[consistent]]).
Interval approximations shown in the 3.5.7.11.17 [[subgroup]] column are those within the 77-throdd [[limit]] and less than 1/3 of a step off (all of these being [[consistent]]).
Line 24: Line 24:


{| class="wikitable center-2 center-3 center-4 mw-collapsible"
{| class="wikitable center-2 center-3 center-4 mw-collapsible"
|+ Intervals of 17edt
|+ style="font-size: 105%;" | Intervals of 17edt
|-
|-
! | Degree
! Degree
! | Note ([[4L 5s (3/1-equivalent)#Notation|BPS-Lambda notation]])
! Note ([[4L 5s (3/1-equivalent)#Notation|BPS Lambda notation]])
! | Note (Macrodiatonic notation)
! Note (Macrodiatonic notation)
! | Approximate 3.5.7.11.17 [[subgroup]] intervals
! Approximate 3.5.7.11.17 [[subgroup]] intervals
! | Additional 3.5.7.11''.14.16''.17 subgroup intervals^
! Additional 3.5.7.11''.14.16''.17 subgroup intervals^
! | Cents
! Cents
! | Hekts
! Hekts
|-
|-
| | 0
| 0
| | J
| J
| | E
| E
| | [[1/1]]
| [[1/1]]
| |
| |
| | 0
| 0
| | 0
| 0
|-
|-
| | 1
| 1
| | K
| K
| | F
| F
| | [[35/33]] (+10.0c); [[27/25]] (-21.4c); [[81/77]] (+24.2c); [[49/45]] (-35.5c)
| [[35/33]] (+10.0¢); [[27/25]] (−21.4¢); [[81/77]] (+24.2¢); [[49/45]] (−35.5¢)
| | [[16/15]] (+0.2c); [[17/16]] (+7.1c); [[15/14]] (-7.6c)
| [[16/15]] (+0.2¢); [[17/16]] (+7.1¢); [[15/14]] (−7.6¢)
| | 111.9
| 111.9
| | 76.5
| 76.5
|-
|-
| | 2
| 2
| | K#
| K♯
| | Gb
| G♭
| | [[25/22]] (+2.5c); [[17/15]] (+7.1c); [[63/55]] (-11.3c)
| [[25/22]] (+2.5¢); [[17/15]] (+7.1¢); [[63/55]] (−11.3¢)
| | 16/14 ([[8/7]]) (-7.5c)
| 16/14 ([[8/7]]) (-7.5¢)
| | 223.8
| 223.8
| | 152.9
| 152.9
|-
|-
| | 3
| 3
| | Lb
| L♭
| | F#
| F♯
| | [[11/9]] (-11.8c); [[21/17]] (-30.2c); [[25/21]] (+33.8c)
| [[11/9]] (−11.8¢); [[21/17]] (−30.2¢); [[25/21]] (+33.8¢)
| | [[17/14]] (-0.5c)
| [[17/14]] (−0.5¢)
| | 335.6
| 335.6
| | 229.4
| 229.4
|-
|-
| | 4
| 4
| | L
| L
| | G
| G
| | [[35/27]] (-1.8c); [[9/7]] (+12.4c)
| [[35/27]] (−1.8¢); [[9/7]] (+12.4¢)
| |  
|  
| | 447.5
| 447.5
| | 305.9
| 305.9
|-
|-
| | 5
| 5
| | M
| M
| | Ab
| A♭
| | [[15/11]] (+22.4c); [[7/5]] (-23.1c)
| [[15/11]] (+22.4¢); [[7/5]] (−23.1¢)
| |  
|  
| | 559.4
| 559.4
| | 382.35
| 382.35
|-
|-
| | 6
| 6
| | M#
| M♯
| | G#
| G♯
| | [[81/55]] (+1.1c); [[25/17]] (+3.6c); [[49/33]] (-13.1c); [[51/35]] (+19.5c)
| [[81/55]] (+1.1¢); [[25/17]] (+3.6¢); [[49/33]] (−13.1¢); [[51/35]] (+19.5¢)
| |  
|  
| | 671.3
| 671.3
| | 458.8
| 458.8
|-
|-
| | 7
| 7
| | Nb
| N♭
| | A
| A
| | [[11/7]] (+0.7c); [[27/17]] (-17.8c); [[17/11]] (+29.5c)
| [[11/7]] (+0.7¢); [[27/17]] (−17.8¢); [[17/11]] (+29.5¢)
| |  
|  
| | 783.2
| 783.2
| | 535.3
| 535.3
|-
|-
| | 8
| 8
| | N
| N
| | Bb
| B♭
| | [[5/3]] (+10.7c)
| [[5/3]] (+10.7¢)
| | [[27/16]] (-10.8c)
| [[27/16]] (−10.8¢)
| | 895.1
| 895.1
| | 611.8
| 611.8
|-
|-
| | 9
| 9
| | O
| O
| | A#
| A♯
| | [[9/5]] (-10.7c)
| [[9/5]] (−10.7¢)
| | [[25/14]] (+3.1c); [[16/9]] (+10.8c)
| [[25/14]] (+3.1¢); [[16/9]] (+10.8¢)
| | 1006.9
| 1006.9
| | 688.2
| 688.2
|-
|-
| | 10
| 10
| | O#
| O♯
| | B
| B
| | [[21/11]] (-0.7c); [[17/9]] (+17.8c); [[33/17]] (-29.5c)
| [[21/11]] (−0.7¢); [[17/9]] (+17.8¢); [[33/17]] (−29.5¢)
| |  
|  
| | 1118.8
| 1118.8
| | 764.7
| 764.7
|-
|-
| | 11
| 11
| | Pb
| P♭
| | C
| C
| | [[55/27]] (-1.1c); [[51/25]] (-3.6c); [[99/49]] (+13.1c); [[35/17]] (-19.5c)
| [[55/27]] (−1.1¢); [[51/25]] (−3.6¢); [[99/49]] (+13.1¢); [[35/17]] (−19.5¢)
| |  
|  
| | 1230.7
| 1230.7
| | 841.2
| 841.2
|-
|-
| | 12
| 12
| | P
| P
| | Db
| D♭
| | [[11/5]] (-22.4c); [[15/7]] (+23.1c)
| [[11/5]] (−22.4¢); [[15/7]] (+23.1¢)
| |  
|  
| | 1342.6
| 1342.6
| | 917.65
| 917.65
|-
|-
| | 13
| 13
| | Q
| Q
| | C#
| C♯
| | [[81/35]] (+1.8c); [[7/3]] (-12.4c)
| [[81/35]] (+1.8¢); [[7/3]] (−12.4¢)
| |  
|  
| | 1454.4
| 1454.4
| | 994.1
| 994.1
|-
|-
| | 14
| 14
| | Q#
| Q♯
| | D
| D
| | [[27/11]] (+11.8c); [[17/7]] (+30.2c); [[63/25]] (-33.8c)
| [[27/11]] (+11.8¢); [[17/7]] (+30.2¢); [[63/25]] (−33.8¢)
| |  
|  
| | 1566.3
| 1566.3
| | 1070.6
| 1070.6
|-
|-
| | 15
| 15
| | Rb
| R♭
| | Eb
| E♭
| | [[45/17]] (-7.1c); [[55/21]] (+11.3c)
| [[45/17]] (−7.1¢); [[55/21]] (+11.3¢)
| |  
|  
| | 1678.2
| 1678.2
| | 1147.1
| 1147.1
|-
|-
| | 16
| 16
| | R
| R
| | D#
| D♯
| | [[99/35]] (-10.0c); [[25/9]] (+21.4c); [[77/27]] (-24.2c); [[135/49]] (+35.5c)
| [[99/35]] (−10.0¢); [[25/9]] (+21.4¢); [[77/27]] (−24.2¢); [[135/49]] (+35.5¢)
| | [[45/16]] (-0.1c); [[14/5]] (+7.6c)
| [[45/16]] (−0.1¢); [[14/5]] (+7.6¢)
| | 1790.1
| 1790.1
| | 1223.5
| 1223.5
|-
|-
| | 17
| 17
| | J
| J
| | E
| E
| | [[3/1]]
| [[3/1]]
| |  
|  
| | 1901.955
| 1901.955
| | 1300.
| 1300.
|}
|}


It's a strange 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).
It is a strange 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 ==
== Regular temperament ==
Line 193: Line 193:


== Z function ==
== 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.
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]]       

Revision as of 13:39, 7 January 2025

← 16edt 17edt 18edt →
Prime factorization 17 (prime)
Step size 111.88 ¢ 
Octave 11\17edt (1230.68 ¢)
Consistency limit 3
Distinct consistency limit 3

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

Properties

Following 13edt, 17edt is the first EDT that can reasonably be described as a 3.5.7 subgroup temperament, though one that sacrifices much accuracy compared to 13edt, compensating for that by representing primes 11 and 17. 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). 17edt's step is also, notably, only 0.15 cents sharp of 16/15.

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.


Approximation of prime harmonics in 17edt
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Error Absolute (¢) +30.7 +0.0 +10.7 -12.4 -11.8 +34.7 +17.8 +49.0 +53.8 -11.8 -15.4 +13.9 -51.9 -22.5 +47.3
Relative (%) +27.4 +0.0 +9.5 -11.1 -10.5 +31.0 +15.9 +43.8 +48.1 -10.6 -13.8 +12.4 -46.4 -20.1 +42.3
Steps
(reduced) 		11
(11) 		17
(0) 		25
(8) 		30
(13) 		37
(3) 		40
(6) 		44
(10) 		46
(12) 		49
(15) 		52
(1) 		53
(2) 		56
(5) 		57
(6) 		58
(7) 		60
(9)

Discussion

17edt is closely related to 13edt, the Bohlen-Pierce division, because they share the feature of tempering out 245/243, so that the interval of 9/7 stacked twice results in 5/3. Therefore, like 13edt, 17edt's 9/7 generates an enneatonic Lambda (4L 5s) scale. If 13edt can be considered an analogue of 12edo as the basic tuning of this scale, 17edt is an analogue of 17edo as the hard 3:1 tuning. While the approximation of 5/3 and 7/3 is less good than that of 13edt, this scale has 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, as it is equated to 25/21 by virtue of tempering out 77/75.

17edt's tuning of BPS (Bohlen–Pierce–Stearns, the temperament defined in 3.5.7 by tempering out 245/243) is also very notable in that its generator (9/7) spans 4 steps, meaning that it is divisible into 2; and the interval 27/7 spans 21 steps, meaning it is divisible into 3. These lead to weak extensions of BPS known as Dubhe (which splits the 9/7 generator into two intervals of 17/15, tempering out 2025/2023 as the additional comma in 3.5.7.17), and Mintra (an extension of Mintaka which splits 27/7 into three intervals of 11/7, tempering out 1331/1323 as the additional comma in 3.5.7.11), respectively. 17edt in fact supports basic tunings of Dubhe[9] (which is 8L 1s) and Mintaka[12] (which is 5L 7s, i.e. a macro-chromatic scale). Therefore 17edt is important as the smallest nontrivial tuning to support each, and it is remarkable for providing such an efficient intersection of temperaments in the 3.5.7.11.17 subgroup, despite being an extremal tuning of most of these (specifically since Mintaka asks for an 11/7 tuned flat, rather than close to just; and the 9/7 is highly overtempered for BPS) and losing much accuracy compared to more optimal tunings—its behavior in this subgroup is reminiscent of that of 15edo in the 11-limit.

Intervals

The notation schemes below are based on the BPS Lambda enneatonic scale presented in the symmetric (sLsLsLsLs, Cassiopeian) mode in J, and the Mintaka macrodiatonic scale presented in the macro-Phrygian (sLLLsLL) mode in E.

Interval approximations shown in the 3.5.7.11.17 subgroup column are those within the 77-throdd limit and less than 1/3 of a step off (all of these being consistent).

Interval approximations shown in the 3.5.7.11.14.16.17 subgroup column are those within the 77-integer limit and less than 15 cents off, consistency is not taken into account.

Intervals of 17edt
Degree Note (BPS Lambda notation) Note (Macrodiatonic notation) Approximate 3.5.7.11.17 subgroup intervals Additional 3.5.7.11.14.16.17 subgroup intervals^ Cents Hekts
0 J E 1/1 0 0
1 K F 35/33 (+10.0¢); 27/25 (−21.4¢); 81/77 (+24.2¢); 49/45 (−35.5¢) 16/15 (+0.2¢); 17/16 (+7.1¢); 15/14 (−7.6¢) 111.9 76.5
2 K♯ G♭ 25/22 (+2.5¢); 17/15 (+7.1¢); 63/55 (−11.3¢) 16/14 (8/7) (-7.5¢) 223.8 152.9
3 L♭ F♯ 11/9 (−11.8¢); 21/17 (−30.2¢); 25/21 (+33.8¢) 17/14 (−0.5¢) 335.6 229.4
4 L G 35/27 (−1.8¢); 9/7 (+12.4¢) 447.5 305.9
5 M A♭ 15/11 (+22.4¢); 7/5 (−23.1¢) 559.4 382.35
6 M♯ G♯ 81/55 (+1.1¢); 25/17 (+3.6¢); 49/33 (−13.1¢); 51/35 (+19.5¢) 671.3 458.8
7 N♭ A 11/7 (+0.7¢); 27/17 (−17.8¢); 17/11 (+29.5¢) 783.2 535.3
8 N B♭ 5/3 (+10.7¢) 27/16 (−10.8¢) 895.1 611.8
9 O A♯ 9/5 (−10.7¢) 25/14 (+3.1¢); 16/9 (+10.8¢) 1006.9 688.2
10 O♯ B 21/11 (−0.7¢); 17/9 (+17.8¢); 33/17 (−29.5¢) 1118.8 764.7
11 P♭ C 55/27 (−1.1¢); 51/25 (−3.6¢); 99/49 (+13.1¢); 35/17 (−19.5¢) 1230.7 841.2
12 P D♭ 11/5 (−22.4¢); 15/7 (+23.1¢) 1342.6 917.65
13 Q C♯ 81/35 (+1.8¢); 7/3 (−12.4¢) 1454.4 994.1
14 Q♯ D 27/11 (+11.8¢); 17/7 (+30.2¢); 63/25 (−33.8¢) 1566.3 1070.6
15 R♭ E♭ 45/17 (−7.1¢); 55/21 (+11.3¢) 1678.2 1147.1
16 R D♯ 99/35 (−10.0¢); 25/9 (+21.4¢); 77/27 (−24.2¢); 135/49 (+35.5¢) 45/16 (−0.1¢); 14/5 (+7.6¢) 1790.1 1223.5
17 J E 3/1 1901.955 1300.

It is a strange 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

One step of 17edt can also be thought of as a generator of the vavoom temperament, since it is very close to 16/15; interpreting it thus, 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
Optimal ET sequence: 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

Music

Togenom
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