Xenharmonic Wiki

17edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|17}}
{{Wikipedia|17 equal temperament}}
{{Wikipedia|17 equal temperament}}
'''17 equal divisions of the octave''' ('''17edo'''), or '''17(-tone) equal temperament''' ('''17tet''', '''17et''') when viewed from a [[regular temperament]] perspective, is the tuning system derived from dividing the octave in 17 [[equal]] steps, each around 70.6 [[cent]]s in size.


== Theory ==
== Theory ==
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{{Harmonics in equal|17|intervals=odd}}
{{Harmonics in equal|17|intervals=odd}}


=== Miscellaneous properties ===
=== Subsets and supersets ===
17edo is the seventh [[prime edo]], following [[13edo]] and coming before [[19edo]].  
17edo is the seventh [[prime edo]], following [[13edo]] and coming before [[19edo]]. [[34edo]], which doubles it, provides a good correction to the harmonic 5.  


== Intervals ==
== Intervals ==
{{See also|17edo solfege}}
{{See also|17edo solfege}}
{| class="wikitable center-all right-2 left-8"
{| class="wikitable center-all right-2 left-3"
|-
|-
! Edosteps
! #
! Cents
! Cents
! colspan="2" | Extended [[circle-of-fifths notation]] *
! Approximate Ratios†
! colspan="2" | [[Circle-of-fifths notation|Circle-of-fifths Notation]] *
! colspan="3" | [[Ups and Downs Notation]]
! colspan="3" | [[Ups and Downs Notation]]
! colspan="2" | [[3L 4s]] notation
! colspan="2" | [[3L 4s]] Notation
! Approximate Ratios†
|-
|-
| 0
| 0
| 0.00
| 0.00
| 1/1
| Unison
| Unison
| D
| D
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| unison
| unison
| D
| D
| 1/1
|-
|-
| 1
| 1
| 70.59
| 70.59
| [[22/21]], [[25/24]], [[26/25]], [[28/27]], [[33/32]], [[24/23]]
| Minor 2nd<br>(Semiaugmented 1sn)
| Minor 2nd<br>(Semiaugmented 1sn)
| Eb<br>(D+)
| Eb<br>(D+)
| up unison,
| up unison, <br>minor 2nd
minor 2nd
| ^1, m2
| ^1, m2
| Eb
| Eb
| augmented 1sn
| augmented 1sn
| D#
| D#
| [[22/21]], [[25/24]], [[26/25]], [[28/27]], [[33/32]], [[24/23]]
|-
|-
| 2
| 2
| 141.18
| 141.18
| [[13/12]], [[12/11]], [[14/13]], [[25/23]]
| Augmented 1sn<br>(Neutral 2nd)
| Augmented 1sn<br>(Neutral 2nd)
| D#<br>(Ed)
| D#<br>(Ed)
| augmented 1sn,
| augmented 1sn, <br>mid 2nd
mid 2nd
| A1, ~2
| A1, ~2
| vE
| vE
| minor 2nd
| minor 2nd
| Eb
| Eb
| [[13/12]], [[12/11]], [[14/13]], [[25/23]]
|-
|-
| 3
| 3
| 211.76
| 211.76
| [[9/8]], [[8/7]], [[28/25]], [[25/22]], [[26/23]]
| Major 2nd
| Major 2nd
| E
| E
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| major 2nd
| major 2nd
| E
| E
| [[9/8]], [[8/7]], [[28/25]], [[25/22]], [[26/23]]
|-
|-
| 4
| 4
| 282.35
| 282.35
| [[13/11]], [[7/6]]
| Minor 3rd
| Minor 3rd
| F
| F
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| diminished 3rd
| diminished 3rd
| Fb
| Fb
| [[13/11]], [[7/6]]
|-
|-
| 5
| 5
| 352.94
| 352.94
| [[11/9]], [[16/13]], [[28/23]]
| Diminished 4th<br>(Neutral 3rd)
| Diminished 4th<br>(Neutral 3rd)
| Gb<br>(F+)
| Gb<br>(F+)
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| perfect 3rd
| perfect 3rd
| F
| F
| [[11/9]], [[16/13]], [[28/23]]
|-
|-
| 6
| 6
| 423.53
| 423.53
| [[32/25]], [[9/7]], [[14/11]], [[33/26]], [[23/18]]
| Major 3rd<br>(Semidiminished 4th)
| Major 3rd<br>(Semidiminished 4th)
| F#<br>(Gd)
| F#<br>(Gd)
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| augmented 3rd
| augmented 3rd
| F#
| F#
| [[32/25]], [[9/7]], [[14/11]], [[33/26]], [[23/18]]
|-
|-
| 7
| 7
| 494.12
| 494.12
| [[4/3]], [[21/16]]
| Perfect 4th
| Perfect 4th
| G
| G
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| minor 4th
| minor 4th
| G
| G
| [[4/3]], [[21/16]]
|-
|-
| 8
| 8
| 564.71
| 564.71
| [[11/8]], [[18/13]], [[32/23]]
| Diminshed 5th<br>(Semiaugmented 4th)
| Diminshed 5th<br>(Semiaugmented 4th)
| Ab<br>(G+)
| Ab<br>(G+)
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| major 4th
| major 4th
| G#
| G#
| [[11/8]], [[18/13]], [[32/23]]
|-
|-
| 9
| 9
| 635.29
| 635.29
| [[16/11]], [[13/9]], [[23/16]]
| Augmented 4th<br>(Semidiminished 5th)
| Augmented 4th<br>(Semidiminished 5th)
| G#<br>(Ad)
| G#<br>(Ad)
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| minor 5th
| minor 5th
| Ab
| Ab
| [[16/11]], [[13/9]], [[23/16]]
|-
|-
| 10
| 10
| 705.88
| 705.88
| [[3/2]], [[32/21]]
| Perfect 5th
| Perfect 5th
| A
| A
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| major 5th
| major 5th
| A
| A
| [[3/2]], [[32/21]]
|-
|-
| 11
| 11
| 776.47
| 776.47
| [[25/16]], [[14/9]], [[11/7]], [[52/33]], [[36/23]]
| Minor 6th<br>(Semiaugmented 5th)
| Minor 6th<br>(Semiaugmented 5th)
| Bb<br>(A+)
| Bb<br>(A+)
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| diminished 6th
| diminished 6th
| Bb
| Bb
| [[25/16]], [[14/9]], [[11/7]], [[52/33]], [[36/23]]
|-
|-
| 12
| 12
| 847.06
| 847.06
| [[13/8]], [[18/11]], [[23/14]]
| Augmented 5th<br>(Neutral 6th)
| Augmented 5th<br>(Neutral 6th)
| A#<br>(Bd)
| A#<br>(Bd)
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| perfect 6th
| perfect 6th
| B
| B
| [[13/8]], [[18/11]], [[23/14]]
|-
|-
| 13
| 13
| 917.65
| 917.65
| [[17/10]], [[22/13]], [[12/7]]
| Major 6th
| Major 6th
| B
| B
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| augmented 6th
| augmented 6th
| B#
| B#
| [[17/10]], [[22/13]], [[12/7]]
|-
|-
| 14
| 14
| 988.24
| 988.24
| [[16/9]], [[7/4]], [[25/14]], [[44/25]], [[23/13]]
| Minor 7th
| Minor 7th
| C
| C
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| minor 7th
| minor 7th
| Cb
| Cb
| [[16/9]], [[7/4]], [[25/14]], [[44/25]], [[23/13]]
|-
|-
| 15
| 15
| 1058.82
| 1058.82
| [[11/6]], [[24/13]], [[13/7]], [[46/25]]
| Diminished 8ve<br>(Neutral 7th)
| Diminished 8ve<br>(Neutral 7th)
| Db<br>(C+)
| Db<br>(C+)
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| major 7th
| major 7th
| C
| C
| [[11/6]], [[24/13]], [[13/7]], [[46/25]]
|-
|-
| 16
| 16
| 1129.41
| 1129.41
| [[21/11]], [[25/13]], [[48/25]], [[27/14]], [[64/33]], [[23/12]]
| Major 7th<br>(Semidiminished 8ve)
| Major 7th<br>(Semidiminished 8ve)
| C#<br>(Dd)
| C#<br>(Dd)
| major 7th,
| major 7th,<br>down 8ve
down 8ve
| M7, v8
| M7, v8
| C#
| C#
| diminished 8ve
| diminished 8ve
| Db
| Db
| [[21/11]], [[25/13]], [[48/25]], [[27/14]], [[64/33]], [[23/12]]
|-
|-
| 17
| 17
| 1200.00
| 1200.00
| [[2/1]]
| Octave
| Octave
| D
| D
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| octave
| octave
| D
| D
| [[2/1]]
|}
|}
<nowiki>*</nowiki> Half-sharps and half-flats (denoted "+" and "d", respectively) can be used to alter the note by a single step, since sharps and flats each span two edo steps. Using half-sharps and half-flats may be preferable for compatibility with the ups-and-downs notation in [[34edo]], in which an up or down respectively constitute a quarter-sharp or quarter-flat.
<nowiki>*</nowiki> Half-sharps and half-flats (denoted "+" and "d", respectively) can be used to alter the note by a single step, since sharps and flats each span two edo steps. Using half-sharps and half-flats may be preferable for compatibility with the ups-and-downs notation in [[34edo]], in which an up or down respectively constitute a quarter-sharp or quarter-flat.
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