Xenharmonic Wiki

17edo: Difference between revisions

← Older editNewer edit →
VisualWikitext
No edit summary
TallKite (talk | contribs)
autopatrolled, Sysops
4,223 edits
added M2, m2 and A1 to the template, moved the primes-error table up to the top, restored deleted ^v interval names
Line 8: Line 8:
| Prime factorization = 17
| Prime factorization = 17
| Subgroup = 2.3.7.11.13
| Subgroup = 2.3.7.11.13
| Step size = 70.588
| Step size = 70.588¢
| Fifth type = [[leapfrog]]/[[archy]] 10\17 705.88¢ (+3.927¢)
| Fifth type = [[leapfrog]]/[[archy]] 10\17 = 705.88¢
| Major 2nd = 3\17 = 212¢
| Minor 2nd = 1\17 = 71¢
| Augmented 1sn = 2\17 = 141¢
| Common uses = diatonic (often neo-medieval), Westernized maqam
| Common uses = diatonic (often neo-medieval), Westernized maqam
| Important MOS = diatonic ([[leapfrog]]/[[archy]]) 5L2s 1221222 (10\17, 1\1)<br/>[[maqamic]] 3L4s 3232322 (5\17, 1\1)<br/>[[maqamic]] 7L3s 2221221221 (5\17, 1\1)<br/>[[lovecraft]] 4L5s 313131311 (4\17, 1\1)
| Important MOS = diatonic ([[leapfrog]]/[[archy]]) 5L2s 1221222 (10\17, 1\1)<br/>[[maqamic]] 3L4s 3232322 (5\17, 1\1)<br/>[[maqamic]] 7L3s 2221221221 (5\17, 1\1)<br/>[[lovecraft]] 4L5s 313131311 (4\17, 1\1)
Line 18: Line 21:


17 tone equal temperament, or 17-EDO, divides the octave in 17 equal steps, each 70.588 [[cent]]s in size. It is the seventh [[prime numbers|prime]] [[EDO]], following [[13edo]] and coming before [[19edo]].
17 tone equal temperament, or 17-EDO, divides the octave in 17 equal steps, each 70.588 [[cent]]s in size. It is the seventh [[prime numbers|prime]] [[EDO]], following [[13edo]] and coming before [[19edo]].
== Introductory materials ==
* [[SeventeenTheory]], an introduction to 17-EDO theory, through the eyes of the [[SeventeenTonePianoProject]].
* [http://anaphoria.com/Secor17puzzle.pdf The 17-tone Puzzle] by George Secor, another introduction into 17-EDO theory.
* [[17edo Solfege]]
* [[17edo tetrachords]]
* [http://microtonalismo.com/proyecto-xvii Proyect 17-Perú] {{forbidden}}


== Theory ==
== Theory ==
17-EDO 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 the 3, 7, 11, and 13 are all 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]].
{| class="wikitable center-all"
! colspan="2" |
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
!prime 17
!prime 19
!prime 23
|-
! rowspan="2" | Error
! absolute ([[cent|¢]])
| 0
|  +3.93
|  -33.4
|  +19.4
|  +13.4
|  +6.5
|  -34.3
|  -15.2
|  +7.0
|-
![[Relative error|relative]] (%)
| 0
|  +6
|  -47
|  +27
|  +19
|  +9
|  -49
|  -21
|  +10
|-
! colspan="2" |[[nearest edomapping]]
|17
|10
|5
|14
|8
|12
|1
|4
|9
|-
! colspan="2" |[[fifthspan]]
| 0
|  +1
|  -8
|  -2
|  -6
|  +8
|  -5
|  -3
|  +6
|}17-EDO 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 the 3, 7, 11, and 13 are all 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]].


As a no-fives system, it is best used with timbres in which harmonic multiples of 5 are attenuated or absent. Also, the standard major chord (4:5:6) cannot be used since it includes the fifth harmonic.
As a no-fives system, it is best used with timbres in which harmonic multiples of 5 are attenuated or absent. Also, the standard major chord (4:5:6) cannot be used since it includes the fifth harmonic.
Line 39: Line 91:
! Edo steps
! Edo steps
! Cents
! Cents
! colspan="2" | Names of Intervals
! colspan="2" | Names of Intervals, extended
! Note Name
pythagorean note names
! [[Ups and Downs Notation]]
! colspan="3" |[[Ups and Downs Notation]]
! Approximate Ratios*
! Approximate Ratios*
! Temperament(s) generated
! Temperament(s) generated
Line 48: Line 100:
| 0.00
| 0.00
| Unison
| Unison
| P1
| C
| C
|unison
|P1
| C
| C
| 1/1
| 1/1
Line 57: Line 110:
| 70.59
| 70.59
| Super Unison/Minor Second
| Super Unison/Minor Second
| m2
| Db <br> (B#)
| Db <br> (B#)
|minor 2nd
|m2
| ^C
| ^C
| [[25/24]], [[26/25]], [[33/32]], [[24/23]]
| [[25/24]], [[26/25]], [[33/32]], [[24/23]]
Line 66: Line 120:
| 141.18
| 141.18
| Augmented Unison/Neutral Second
| Augmented Unison/Neutral Second
| ~2
| C#
| C#
|mid 2nd
|~2
| vD
| vD
| [[13/12]], [[12/11]], [[14/13]], [[25/23]]
| [[13/12]], [[12/11]], [[14/13]], [[25/23]]
Line 75: Line 130:
| 211.76
| 211.76
| Major Second/Sub Third
| Major Second/Sub Third
| M2
| D
| D
|major 2nd
|M2
| D
| D
| [[9/8]], [[8/7]], [[28/25]], [[25/22]], [[26/23]]
| [[9/8]], [[8/7]], [[28/25]], [[25/22]], [[26/23]]
Line 84: Line 140:
| 282.35
| 282.35
| Minor Third/Super Second
| Minor Third/Super Second
| m3
| Eb
| Eb
|minor 3rd
|m3
| ^D
| ^D
| [[13/11]], [[7/6]]
| [[13/11]], [[7/6]]
Line 92: Line 149:
| 5
| 5
| 352.94
| 352.94
| Augmented Second/Neutral Third/ <br> Diminished Fourth
| Augmented Second/Neutral
| ~3
Third/Diminished Fourth
| D# <br> (Fb)
| D# <br> (Fb)
|mid 3rd
|~3
| vE
| vE
| [[11/9]], [[16/13]], [[28/23]]
| [[11/9]], [[16/13]], [[28/23]]
Line 102: Line 161:
| 423.53
| 423.53
| Major Third/Sub Fourth
| Major Third/Sub Fourth
| M3
| E
| E
|major 3rd
|M3
| E
| E
| [[32/25]], [[9/7]], [[14/11]], [[33/26]], [[23/18]]
| [[32/25]], [[9/7]], [[14/11]], [[33/26]], [[23/18]]
Line 111: Line 171:
| 494.12
| 494.12
| Perfect Fourth
| Perfect Fourth
| P4
| F
| F
|perfect 4th
|P4
| F
| F
| [[4/3]]
| [[4/3]]
Line 120: Line 181:
| 564.71
| 564.71
| Super Fourth/Diminshed Fifth
| Super Fourth/Diminshed Fifth
| ^4, ~4, <br> d5
| Gb <br> (E#)
| Gb <br> (E#)
|mid 4th,
diminished 5th
|~4,
d5
| ^F
| ^F
| [[11/8]], [[18/13]], [[32/23]]
| [[11/8]], [[18/13]], [[32/23]]
Line 129: Line 193:
| 635.29
| 635.29
| Augmented Fourth/Sub Fifth
| Augmented Fourth/Sub Fifth
| A4, <br> v5, ~5
| F#
| F#
|augmented 4th,
mid 5th
|A4, ~5
| vG
| vG
| [[16/11]], [[13/9]], [[23/16]]
| [[16/11]], [[13/9]], [[23/16]]
Line 138: Line 204:
| 705.88
| 705.88
| Perfect Fifth
| Perfect Fifth
| P5
| G
| G
|perfect 5th
|P5
| G
| G
| [[3/2]]
| [[3/2]]
Line 147: Line 214:
| 776.47
| 776.47
| Super Fifth/Minor Sixth
| Super Fifth/Minor Sixth
| m6
| Ab
| Ab
|minor 6th
|m6
| ^G
| ^G
| [[25/16]], [[14/9]], [[11/7]], [[52/33]], [[36/23]]
| [[25/16]], [[14/9]], [[11/7]], [[52/33]], [[36/23]]
Line 155: Line 223:
| 12
| 12
| 847.06
| 847.06
| Augmented Fifth/Neutral Sixth/ <br> Diminished Seventh
| Augmented Fifth/Neutral  
| ~6
Sixth/Diminished Seventh
| G#
| G#
|mid 6th
|~6
| vA
| vA
| [[13/8]], [[18/11]], [[23/14]]
| [[13/8]], [[18/11]], [[23/14]]
Line 165: Line 235:
| 917.65
| 917.65
| Major Sixth/Sub Seventh
| Major Sixth/Sub Seventh
| M6
| A
| A
|major 6th
|M6
| A
| A
| [[17/10]], [[22/13]],[[12/7]]
| [[17/10]], [[22/13]],[[12/7]]
Line 174: Line 245:
| 988.24
| 988.24
| Minor Seventh/Super Sixth
| Minor Seventh/Super Sixth
| m7
| Bb
| Bb
|minor 7th
|m7
| ^A
| ^A
| [[16/9]], [[7/4]], [[25/14]], [[44/25]], [[23/13]]
| [[16/9]], [[7/4]], [[25/14]], [[44/25]], [[23/13]]
Line 182: Line 254:
| 15
| 15
| 1058.82
| 1058.82
| Augmented Sixth/Neutral Seventh/ <br> Diminished Octave
| Augmented Sixth/Neutral
| ~7
Seventh/Diminished Octave
| A# <br> (Cb)
| A# <br> (Cb)
|mid 7th
|~7
| vB
| vB
| [[11/6]], [[24/13]], [[13/7]], [[46/25]]
| [[11/6]], [[24/13]], [[13/7]], [[46/25]]
Line 192: Line 266:
| 1129.41
| 1129.41
| Major Seventh/Sub Octave
| Major Seventh/Sub Octave
| M7
| B
| B
|major 7th
|M7
| B
| B
| [[25/13]], [[48/25]], [[64/33]], [[23/12]]
| [[25/13]], [[48/25]], [[64/33]], [[23/12]]
Line 201: Line 276:
| 1200.00
| 1200.00
| Perfect Octave
| Perfect Octave
| P8
| C
| C
|octave
|P8
| C
| C
| [[2/1]]
| [[2/1]]
Line 309: Line 385:


=== Selected just intervals by error ===
=== Selected just intervals by error ===
{| class="wikitable center-all"
! colspan="2" |
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
!prime 17
!prime 19
!prime 23
|-
! rowspan="2" | Error
! absolute ([[cent|¢]])
| 0
| +3.9
| -33.4
| +19.4
| +13.4
| +6.5
| -34.3
| -15.2
| +7.0
|-
! [[Relative error|relative]] (%)
| 0
| +5.6
| -47.3
| +27.5
| +19.0
| +9.2
| -48.7
| -21.5
| +9.9
|-
! colspan="2" | [[fifthspan]]
| 0
| +1
| -8
| -2
| -6
| +8
| -5
| -3
| +6
|}
==== 15-odd-limit mappings ====
==== 15-odd-limit mappings ====
The following table shows how [[15-odd-limit intervals]] are represented in 17edo (ordered by absolute error). Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.  
The following table shows how [[15-odd-limit intervals]] are represented in 17edo (ordered by absolute error). Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.  
Line 674: Line 703:
* [[User:CritDeathX/Sam's 17-note Well Temperament|Sam's 17-note Well Temperament]]
* [[User:CritDeathX/Sam's 17-note Well Temperament|Sam's 17-note Well Temperament]]
* [[User:FloraC/Flora's 17-note well temperament|Flora's 17-note well temperament]]
* [[User:FloraC/Flora's 17-note well temperament|Flora's 17-note well temperament]]
== Introductory materials ==
* [[SeventeenTheory]], an introduction to 17-EDO theory, through the eyes of the [[SeventeenTonePianoProject]].
* [http://anaphoria.com/Secor17puzzle.pdf The 17-tone Puzzle] by George Secor, another introduction into 17-EDO theory.
* [[17edo Solfege]]
* [[17edo tetrachords]]
* [http://microtonalismo.com/proyecto-xvii Proyect 17-Perú] {{forbidden}}


== Music ==
== Music ==
Retrieved from "https://en.xen.wiki/w/17edo"