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17edo: Difference between revisions

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== Theory ==
== Theory ==
17edo's perfect fifth is around 4 cents sharp of just, and around 6 cents sharp of [[12edo]]'s, lending itself to a [[5L 2s|diatonic]] scale with more constrasting large and small steps. Meanwhile, it approximates [[harmonic]]s [[7/1|7]], [[11/1|11]], [[13/1|13]], and [[23/1|23]] to reasonable degrees, despite completely missing harmonic [[5/1|5]]. Thus it can plausibly be treated as a 2.3.25.7.11.13.23 [[subgroup temperament]], for which it is quite accurate (though the 7-limit ratios are generally not as well-represented as those of the other integers). Because these harmonics are all tempered sharp, it adapts well to octave shrinking; [[27edt]] (a variant of 17edo in which the octaves are flattened by ~2.5 cents) is a good alternative. Another one is [[44ed6]].
17edo's [[3/2|perfect fifth]] is around 4 cents sharp of just, and around 6 cents sharp of [[12edo]]'s, lending itself to a [[5L 2s|diatonic]] scale with more constrasting large and small steps, so it can be seen as a tuning that emphasizes the hardness of [[Pythagorean tuning]] rather than mellowing it out as in [[meantone]]. Meanwhile, it approximates [[harmonic]]s [[7/1|7]], [[11/1|11]], [[13/1|13]], and [[23/1|23]] to reasonable degrees, despite completely missing harmonic [[5/1|5]]. Thus it can plausibly be treated as a temperament of the 2.3.25.7.11.13.23 [[subgroup]], for which it is quite accurate (though the 7-limit ratios are generally not as well-represented as those of the other integers).  


Because the 5th harmonic is not well-approximated, using timbres with attenuated 5th harmonics (and its multiples) may reduce audible beating.  
Because the 5th harmonic is not well-approximated, using timbres with attenuated 5th harmonics (and its multiples) may reduce audible beating.  


The standard major triad is quite dissonant as the major third is closer to [[9/7]] than the traditional [[5/4]]. Instead, the tonic chords of 17edo could be considered to be the tetrad 6:7:8:9 and its utonal inversion (representing 14:16:18:21 as [[64/63]] is tempered out), the former of which is a subminor chord with added fourth, and the latter a supermajor chord with added second (resembling the {{w|mu chord}} of Steely Dan fame). These are realized in 17edo as 0–4–7–10 and 0–3–6–10, respectively. Both of these have distinct moods, and are stable and consonant, if somewhat more sophisticated than their classic 5-limit counterparts. To this group we could also add the 0–3–7–10 (which is a sus4 with added second, or sus2 with added fourth). These three chords comprise the three ways to divide the 17edo perfect fifth into two whole tones and one subminor third. Chromatic alterations of them also exist, for example, the 0–3–7–10 chord may be altered to 0–2–7–10 (which approximates 12:13:16:18) or 0–3–8–10 (which approximates 8:9:11:12). The 0–3–8–10 chord is impressive-sounding, resembling a sus4 but with even more tension; it resolves quite nicely to 0–3–6–10.  
The standard major triad is quite [[dissonant]] as the major third is closer to [[9/7]] than the traditional [[5/4]]. Instead, the tonic chords of 17edo could be considered to be the tetrad [[6:7:8:9]] and its utonal inversion (representing [[14:16:18:21]] as [[64/63]] is tempered out), the former of which is a subminor chord with added fourth, and the latter a supermajor chord with added second (resembling the {{w|mu chord}} of {{w|Steely Dan}} fame). These are realized in 17edo as 0–4–7–10 and 0–3–6–10, respectively. Both of these have distinct moods, and are stable and consonant, if somewhat more sophisticated than their classic 5-limit counterparts. To this group we could also add the 0–3–7–10 (a sus2-4 chord). These three chords comprise the three ways to divide the 17edo perfect fifth into two whole tones and one subminor third. Chromatic alterations of them also exist, for example, the 0–3–7–10 chord may be altered to 0–2–7–10 (which approximates 12:13:16:18) or 0–3–8–10 (which approximates 8:9:11:12). The 0–3–8–10 chord is impressive-sounding, resembling a sus4 but with even more tension; it resolves quite nicely to 0–3–6–10.  


Another construction of septimal chords involves 4:7:12 and its inversion 7:12:21. These triads span a twelfth, realized in 17edo as 0–14–27 and 0–13–27, respectively. To this we may add 0–12–14–27, representing 8:13:14:24, or 0–13–15–27, representing 7:12:13:21.  
Another construction of septimal chords involves 4:7:12 and its inversion 7:12:21. These triads span a twelfth, realized in 17edo as 0–14–27 and 0–13–27, respectively. To this we may add 0–12–14–27, representing 8:13:14:24, or 0–13–15–27, representing 7:12:13:21.  
In the no-5 [[13-odd-limit]], 17edo maintains the smallest relative error{{clarify}} of any edo until [[166edo]]. <!-- explain relative error in an odd limit -->


=== Odd harmonics ===
=== Odd harmonics ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
17edo is the seventh [[prime edo]], following [[13edo]] and coming before [[19edo]]. [[34edo]], which doubles it, provides a good correction to harmonics 5 and 17; while [[68edo]], which quadruples it, provides additionally the primes 7, 19, and 31.
17edo is the seventh [[prime edo]], following [[13edo]] and coming before [[19edo]]. It does not contain any nontrivial subset edos, though it contains [[17ed4]] and [[17ed8]]. 17ed8, built by taking every third step of 17edo, is a system where all odd harmonics up to the 21st are mapped exactly as in 17edo, except for the 11th. Beyond that, the 27th, 31st, 35th, and 39th harmonics are likewise mapped identically.
 
[[34edo]], which doubles 17edo, provides a good correction to harmonics 5 and 17; while [[68edo]], which quadruples it, provides additionally the primes 7, 19, and 31.


== Intervals ==
== Intervals ==
{{See also|17edo solfege}}
{{See also| 17edo solfege }}
 
{| class="wikitable center-all right-2 left-3"
{| class="wikitable center-all right-2 left-3"
|-
|-
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! colspan="3" | [[Ups and downs notation]]<br>([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and ^d2)
! colspan="3" | [[Ups and downs notation]]<br>([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and ^d2)
! colspan="3" | [[SKULO interval names|SKULO notation]] {{nowrap|(U {{=}} 1)}}
! colspan="3" | [[SKULO interval names|SKULO notation]] {{nowrap|(U {{=}} 1)}}
! colspan="2" | [[3L&nbsp;4s]] notation
|-
|-
| 0
| 0
| 0.0
| 0.0
| 1/1
| [[1/1]]
| Unison
| Unison
| D
| D
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| unison
| unison
| P1
| P1
| D
| unison
| D
| D
|-
|-
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| U1, m2
| U1, m2
| UD, Eb
| UD, Eb
| augmented 1sn
| D#
|-
|-
| 2
| 2
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| N2
| N2
| UEb, uE
| UEb, uE
| minor 2nd
| Eb
|-
|-
| 3
| 3
Line 91: Line 85:
| major 2nd
| major 2nd
| M2
| M2
| E
| major 2nd
| E
| E
|-
|-
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| m3
| m3
| F
| F
| diminished 3rd
| Fb
|-
|-
| 5
| 5
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| N3
| N3
| UF, uF#
| UF, uF#
| perfect 3rd
| F
|-
|-
| 6
| 6
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| major 3rd
| major 3rd
| M3
| M3
| F#
| augmented 3rd
| F#
| F#
|-
|-
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| perfect 4th
| perfect 4th
| P4
| P4
| G
| minor 4th
| G
| G
|-
|-
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| U4/N4
| U4/N4
| UG
| UG
| major 4th
| G#
|-
|-
| 9
| 9
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| u5/N5
| u5/N5
| uA
| uA
| minor 5th
| Ab
|-
|-
| 10
| 10
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| perfect 5th
| perfect 5th
| P5
| P5
| A
| major 5th
| A
| A
|-
|-
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| minor 6th
| minor 6th
| m6
| m6
| Bb
| diminished 6th
| Bb
| Bb
|-
|-
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| N6
| N6
| UBb, uB
| UBb, uB
| perfect 6th
| B
|-
|-
| 13
| 13
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| M6
| M6
| B
| B
| augmented 6th
| B#
|-
|-
| 14
| 14
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| m7
| m7
| C
| C
| minor 7th
| Cb
|-
|-
| 15
| 15
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| N7
| N7
| UC, uC#
| UC, uC#
| major 7th
| C
|-
|-
| 16
| 16
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| M7, u8
| M7, u8
| C#, uD
| C#, uD
| diminished 8ve
| Db
|-
|-
| 17
| 17
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| octave
| octave
| P8
| P8
| D
| octave
| D
| D
|}
|}
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[[File:17edo Sagittal.png|800px]]
[[File:17edo Sagittal.png|800px]]
=== 3L 4s (mosh) notation ===
The notation of Neutral[7]. The generator is the perfect 3rd. Notes are denoted as {{nowrap|sLsLsLs {{=}} DEFGABCD}}, and raising and lowering by a chroma {{nowrap|(L − s)}}, 1 edostep in this instance, is denoted by ♯ and ♭.
{| class="wikitable center-all right-2 left-4 left-5 mw-collapsible mw-collapsed"
|-
! #
! Cents
! Note
! Name
! Associated ratios
|-
| 0
| 0.0
| D
| Perfect 1sn
| 1/1
|-
| 1
| 70.6
| D#
| Augmented 1sn
| 33/32
|-
| 2
| 141.2
| Eb
| Minor 2nd
| 12/11
|-
| 3
| 211.8
| E
| Major 2nd
| 9/8
|-
| 4
| 282.4
| Fb
| Diminished 3rd
| 32/27
|-
| 5
| 352.9
| F
| Perfect 3rd
| 11/9, 27/22
|-
| 6
| 423.5
| F#
| Augmented 3rd
| 81/64
|-
| 7
| 494.1
| G
| Minor 4th
| 4/3
|-
| 8
| 564.7
| G#
| Major 4th
| 11/8
|-
| 9
| 635.3
| Ab
| Minor 5th
| 16/11
|-
| 10
| 705.9
| A
| Major 5th
| 3/2
|-
| 11
| 776.5
| Bb
| Diminished 6th
| 128/81
|-
| 12
| 847.1
| B
| Perfect 6th
| 18/11, 44/27
|-
| 13
| 917.6
| B#
| Augmented 6th
| 27/16
|-
| 14
| 988.2
| Cb
| Minor 7th
| 16/9
|-
| 15
| 1058.8
| C
| Major 7th
| 11/6
|-
| 16
| 1129.4
| Db
| Diminished 8ve
| 64/33
|-
| 17
| 1200.0
| D
| Perfect 8ve
| 2/1
|}


== Approximation to JI ==
== Approximation to JI ==
=== 15-odd-limit interval mappings ===
=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|17}}
{{Q-odd-limit intervals|17}}
{{Q-odd-limit intervals|17.04|apx=val|header=none|tag=none|title=15-odd-limit intervals by 17c val mapping}}


=== Selected 13-limit intervals ===
=== Selected 13-limit intervals ===
[[File:17ed2-001.svg|alt=alt : Your browser has no SVG support.]]
[[File:17ed2-001.svg|alt=alt : Your browser has no SVG support.]]
=== Zeta peak index ===
{{ZPI
| zpi = 56
| steps = 17.0445886606675
| step size = 70.4035764012981
| tempered height = 5.056957
| pure height = 4.528893
| integral = 1.032175
| gap = 14.269437
| octave = 1196.86079882207
| consistent = 4
| distinct = 4
}}


== Tuning by ear ==
== Tuning by ear ==
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| [[Lee]] / [[liese]] (17c) / [[pycnic]] (17)<br>[[Progress]] (17c)
| [[Lee]] / [[liese]] (17c) / [[pycnic]] (17)<br>[[Progress]] (17c)
|}
|}
== Octave stretch or compression ==
17edo's approximations of harmonics 3, 7, 11, and 13 are all tempered sharp, so 17edo adapts well to slightly [[stretched and compressed tuning|compressing the octave]], if that is acceptable. [[27edt]] and [[44ed6]] are great demonstrations of this, where the octaves are flattened by about 2.5 and 1.5 cents, respectively.
; 17edo
* Step size: 70.588{{c}}, octave size: 1200.000{{c}}
Pure-octaves 17edo approximates the 2.3.11.13 subgroup best. Its approximation to 7 is less good, and it does not really approximate 5. It might make tuning for exploring new harmonies with the 7th, 11th and 13th harmonics not found in 12edo, but its very sharp 7th harmonic might arguably hamper that use case.
{{Harmonics in equal|17|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 17edo}}
{{Harmonics in equal|17|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 17edo (continued)}}
; [[44ed6]]
* Step size: 70.499{{c}}, octave size: 1198.483{{c}}
Compressing the octave of 17edo by around 1.5{{c}} results in much improved primes 3, 7, 11 and 13, but a slightly worse prime 2. The tuning 44ed6 does this. Its primary purpose is to greatly improve 17edo's approximation of harmonic 7, to make it into an exceptional no-5's [[13-limit]] tuning for its size.
{{Harmonics in equal|44|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 44ed6}}
{{Harmonics in equal|44|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 44ed6 (continued)}}
; [[27edt]]
* Step size: 70.443{{c}}, octave size: 1197.527{{c}}
Compressing the octave of 17edo by around 2.5{{c}} results in even more improved primes 3, 7, 11 and 13 than 44ed6, but a with more error on prime 2 than 44ed6 also. The tuning 27edt does this. Its primary purpose is to greatly improve 17edo's approximation of harmonic 7, to make it into an exceptional no-5's [[13-limit]] tuning for its size.
{{Harmonics in equal|27|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 27edt}}
{{Harmonics in equal|27|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 27edt (continued)}}
; [[ZPI|56zpi]] / [[WE|17et, 2.3.7.11.13-subgroup WE tuning]]
* Step size: 70.404{{c}}, octave size: 1296.861{{c}}
Compressing the octave of 17edo by around 3{{c}} results in even more improved primes 3, 7, 11 and 13 than 27edt, but a with more error on prime 2 than 27edt also. Tunings that do this include:
* 56zpi
* 17et, 2.3.7.11-subgroup [[TE]] and [[WE]]
* 17et, 2.3.7.11.13-subgroup TE and WE
Their primary purpose is to greatly improve 17edo's approximation of harmonic 7, to make it into an exceptional no-5's [[13-limit]] tuning for its size.
{{Harmonics in cet|70.403576|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 56zpi}}
{{Harmonics in cet|70.403576|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 56zpi (continued)}}


== Scales ==
== Scales ==
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<references group="note" />
<references group="note" />


[[Category:3-limit]]
[[Category:Teentuning]]
[[Category:Teentuning]]
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