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=Division of the 5/1 into 17 tones=
{{Infobox ET}}
'''[[Ed5|Division of the 5th harmonic]] into 17 equal parts''' (17ED5) is a good [[hyperpyth]] tuning. The step size is about 163.9008 cents, corresponding to 7.3215 [[EDO]].


A hyperpyth tuning, 17ed5 offers a good compromise between 13/5 and 17/5, but the 9/5 of 983 cents is a little bit flat. However, in hyperpyth, 21/5 isn't necessarily represented, at least not as well. In 17ed5, the 21/5 is represented about as well as the 9/5 is, although that's not too good. Luckily, 27, 29, and 39 do a fair job of it. Nevertheless it's the simplest equal hyperpyth after 5ed5, and quite consonant. I imagine it to be the traditional tonality of the tiny creatures living on subatomic particles.
== Division of the 5/1 into 17 tones ==
A hyperpyth tuning, 17ED5 offers a good compromise between 13/5 and 17/5, but the 9/5 of 983 cents is a little bit flat. However, in hyperpyth, 21/5 isn't necessarily represented, at least not as well. In 17ED5, the 21/5 is represented about as well as the 9/5 is, although that's not too good. Luckily, 27, 29, and 39 do a fair job of it. Nevertheless it's the simplest equal hyperpyth after 5ED5, and quite consonant. I imagine it to be the traditional tonality of the tiny creatures living on subatomic particles.


But wait, an interesting pattern emerges:
But wait, an interesting pattern emerges:


22ed5 focuses on 9/5
[[22ed5|22ED5]] focuses on 9/5


27ed5 focuses on 13/5
[[27ed5|27ED5]] focuses on 13/5


29ed5 focuses on 17/5
[[29ed5|29ED5]] focuses on 17/5


(and 34=17*2)
(and 34=17*2)
Line 15: Line 17:
so: 22+27+29=78=39*2
so: 22+27+29=78=39*2


and behold, of the lot, 39ed5 offers the best balance between those intervals.
and behold, of the lot, [[39ed5|39ED5]] offers the best balance between those intervals.


{| class="wikitable"
{| class="wikitable"
|-
|-
| | 0: 0.000 cents
! | degree
| | 1/1
! | cents value
! | corresponding <br>JI intervals
! | comments
|-
| | 0
| | 0.000
| | '''exact [[1/1]]'''
| |  
| |  
|-
|-
| | 1: 163.901
| | 1
| | 163.901
| | [[11/10]]
| |  
| |  
|-
| | 2
| | 327.802
| | [[6/5]]
| |  
| |  
|-
|-
| | 2: 327.802
| | 3
| | 491.702
| | [[4/3]]
| |  
| |  
|-
| | 4
| | 655.603
| | [[16/11]], [[19/13]], <br>[[22/15]]
| |  
| |  
|-
|-
| | 3: 491.702
| | 5
| | 819.504
| | [[8/5]]
| |  
| |  
|-
| | 6
| | 983.405
| | [[7/4]], [[9/5]], [[16/9]]
| |  
| |  
|-
|-
| | 4: 655.603
| | 7
| | 1147.306
| | [[25/13]], [[27/14]], <br>[[35/18]], [[64/33]]
| |  
| |  
|-
| | 8
| | 1311.206
| | [[16/15|32/15]]
| |  
| |  
|-
|-
| | 5: 819.504
| | 9
| |  
| | 1475.107
| | [[75/64|75/32]]
| |  
| |  
|-
|-
| | 6: 983.405
| | 10
| | 9/5, 16/9, 7/4
| | 1639.008
| | 1017
| | [[13/5]], [[9/7|18/7]]
|-
| | 7: 1147.306
| |
| |  
| |  
|-
|-
| | 8: 1311.206
| | 11
| |  
| | 1802.909
| | [[17/12|17/6]]
| |  
| |  
|-
|-
| | 9: 1475.107
| | 12
| |  
| | 1966.810
| | [[14/9|28/9]]
| |  
| |  
|-
|-
| | 10: 1639.008
| | 13
| | 13/5
| | 2130.710
| | 1654
| | [[17/10|17/5]], [[12/7|24/7]]
|-
| | 11: 1802.909
| |  
| |  
| |  
|-
|-
| | 12: 1966.810
| | 14
| |  
| | 2294.611
| | [[19/10|19/5]], [[32/17|64/17]]
| |  
| |  
|-
|-
| | 13: 2130.710
| | 15
| | 17/5
| | 2458.512
| | 2118
| | [[21/20|21/5]], [[25/24|25/6]], <br>[[33/32|33/8]]
|-
| | 14: 2294.611
| |  
| |  
| |  
|-
|-
| | 15: 2458.512
| | 16
| | (21/5)
| | 2622.413
| | 2486
| | [[17/15|68/15]]
|-
| | 16: 2622.413
| |  
| |  
| |  
|-
|-
| | 17: 2786.314
| | 17
| | 5/1
| | 2786.314
| |  
| | '''exact [[5/1]]'''
| | just major third plus two octaves
|}
|}
[[Category:Ed5]]
[[Category:Edonoi]]


== Harmonics ==
{{Harmonics in equal
| steps = 17
| num = 5
| denom = 1
}}
{{Harmonics in equal
| steps = 17
| num = 5
| denom = 1
| start = 12
| collapsed = 1
}}
[[Category:Hyperpyth]]
[[Category:Todo:add sound example]]
[[Category:Todo:add sound example]]

Latest revision as of 19:20, 1 August 2025

← 16ed5 17ed5 18ed5 →
Prime factorization 17 (prime)
Step size 163.901 ¢ 
Octave 7\17ed5 (1147.31 ¢)
Twelfth 12\17ed5 (1966.81 ¢)
Consistency limit 2
Distinct consistency limit 2

Division of the 5th harmonic into 17 equal parts (17ED5) is a good hyperpyth tuning. The step size is about 163.9008 cents, corresponding to 7.3215 EDO.

Division of the 5/1 into 17 tones

A hyperpyth tuning, 17ED5 offers a good compromise between 13/5 and 17/5, but the 9/5 of 983 cents is a little bit flat. However, in hyperpyth, 21/5 isn't necessarily represented, at least not as well. In 17ED5, the 21/5 is represented about as well as the 9/5 is, although that's not too good. Luckily, 27, 29, and 39 do a fair job of it. Nevertheless it's the simplest equal hyperpyth after 5ED5, and quite consonant. I imagine it to be the traditional tonality of the tiny creatures living on subatomic particles.

But wait, an interesting pattern emerges:

22ED5 focuses on 9/5

27ED5 focuses on 13/5

29ED5 focuses on 17/5

(and 34=17*2)

so: 22+27+29=78=39*2

and behold, of the lot, 39ED5 offers the best balance between those intervals.

degree cents value corresponding
JI intervals 		comments
0 0.000 exact 1/1
1 163.901 11/10
2 327.802 6/5
3 491.702 4/3
4 655.603 16/11, 19/13,
22/15
5 819.504 8/5
6 983.405 7/4, 9/5, 16/9
7 1147.306 25/13, 27/14,
35/18, 64/33
8 1311.206 32/15
9 1475.107 75/32
10 1639.008 13/5, 18/7
11 1802.909 17/6
12 1966.810 28/9
13 2130.710 17/5, 24/7
14 2294.611 19/5, 64/17
15 2458.512 21/5, 25/6,
33/8
16 2622.413 68/15
17 2786.314 exact 5/1 just major third plus two octaves

Harmonics

Approximation of harmonics in 17ed5
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -52.7 +64.9 +58.5 +0.0 +12.2 +73.1 +5.8 -34.2 -52.7 -53.8 -40.5
Relative (%) -32.2 +39.6 +35.7 +0.0 +7.4 +44.6 +3.5 -20.9 -32.2 -32.8 -24.7
Steps
(reduced) 		7
(7) 		12
(12) 		15
(15) 		17
(0) 		19
(2) 		21
(4) 		22
(5) 		23
(6) 		24
(7) 		25
(8) 		26
(9)
Approximation of harmonics in 17ed5
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) -15.2 +20.4 +64.9 -46.9 +12.1 +77.0 -16.6 +58.5 -26.0 +57.4 -19.5
Relative (%) -9.3 +12.4 +39.6 -28.6 +7.4 +47.0 -10.1 +35.7 -15.8 +35.0 -11.9
Steps
(reduced) 		27
(10) 		28
(11) 		29
(12) 		29
(12) 		30
(13) 		31
(14) 		31
(14) 		32
(15) 		32
(15) 		33
(16) 		33
(16)
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