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17edo: Difference between revisions

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{| class="wikitable center-all right-2 left-3"
{| class="wikitable center-all right-2 left-3"
|-
|-
! #
! #
! Cents
! Cents
! Approximate Ratios*
! Approximate Ratios<ref group="note">{{sg|limit=2.3.25.7.11.13.85.23&nbsp;subgroup}}</ref>
! colspan="2" | [[Circle-of-fifths notation|Circle-of-fifths Notation]]†
! colspan="2" | [[Circle-of-fifths notation|Circle-of-fifths Notation]]<sup>&dagger;</sup>
! colspan="3" | [[Ups and downs notation|Ups and Downs Notation]]
! colspan="3" | [[Ups and downs notation|Ups and Downs Notation]]
! colspan="3" | [[SKULO interval names|SKULO notation]] (U = 1)
! colspan="3" | [[SKULO interval names|SKULO notation]] {{nowrap|(U {{=}} 1)}}
! colspan="2" | [[3L 4s]] Notation
! colspan="2" | [[3L 4s]] Notation
|-
|-
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| 70.59
| 70.59
| [[24/23]], [[25/24]], [[26/25]], [[27/26]], [[28/27]]
| [[24/23]], [[25/24]], [[26/25]], [[27/26]], [[28/27]]
| Minor 2nd<br>(Semiaugmented 1sn)
| Minor 2nd<br />(Semiaugmented 1sn)
| Eb<br>(Dt)
| Eb<br />(Dt)
| up unison, <br>minor 2nd
| up unison, <br />minor 2nd
| ^1, m2
| ^1, m2
| Eb
| Eb
| uber unison, <br>minor 2nd
| uber unison, <br />minor 2nd
| U1, m2
| U1, m2
| UD, Eb
| UD, Eb
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| 141.18
| 141.18
| [[12/11]], [[13/12]], [[14/13]], [[25/23]]
| [[12/11]], [[13/12]], [[14/13]], [[25/23]]
| Augmented 1sn<br>(Neutral 2nd)
| Augmented 1sn<br />(Neutral 2nd)
| D#<br>(Ed)
| D#<br />(Ed)
| augmented 1sn, <br>mid 2nd
| augmented 1sn, <br />mid 2nd
| A1, ~2
| A1, ~2
| vE
| vE
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| 352.94
| 352.94
| [[11/9]], [[27/22]], [[16/13]], [[39/32]]
| [[11/9]], [[27/22]], [[16/13]], [[39/32]]
| Diminished 4th<br>(Neutral 3rd)
| Diminished 4th<br />(Neutral 3rd)
| Gb<br>(Ft)
| Gb<br />(Ft)
| mid 3rd
| mid 3rd
| ~3
| ~3
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| 423.53
| 423.53
| [[9/7]], [[14/11]], [[23/18]], [[32/25]], [[51/40]]
| [[9/7]], [[14/11]], [[23/18]], [[32/25]], [[51/40]]
| Major 3rd<br>(Semidiminished 4th)
| Major 3rd<br />(Semidiminished 4th)
| F#<br>(Gd)
| F#<br />(Gd)
| major 3rd
| major 3rd
| M3
| M3
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| 564.71
| 564.71
| [[11/8]], [[18/13]], [[25/18]], [[32/23]]
| [[11/8]], [[18/13]], [[25/18]], [[32/23]]
| Diminished 5th<br>(Semiaugmented 4th)
| Diminished 5th<br />(Semiaugmented 4th)
| Ab<br>(Gt)
| Ab<br />(Gt)
| mid 4th, <br>diminished 5th
| mid 4th, <br />diminished 5th
| ~4, d5
| ~4, d5
| ^G, Ab
| ^G, Ab
| uber 4th/<br>neutral 4th
| uber 4th/<br />neutral 4th
| U4/N4
| U4/N4
| UG
| UG
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| 635.29
| 635.29
| [[13/9]], [[16/11]], [[23/16]], [[36/25]]
| [[13/9]], [[16/11]], [[23/16]], [[36/25]]
| Augmented 4th<br>(Semidiminished 5th)
| Augmented 4th<br />(Semidiminished 5th)
| G#<br>(Ad)
| G#<br />(Ad)
| augmented 4th, <br>mid 5th
| augmented 4th, <br />mid 5th
| A4, ~5
| A4, ~5
| G#, vA
| G#, vA
| unter 5th/<br>neutral 5th
| unter 5th/<br />neutral 5th
| u5/N5
| u5/N5
| uA
| uA
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| 776.47
| 776.47
| [[11/7]], [[14/9]], [[25/16]], [[36/23]], [[80/51]]
| [[11/7]], [[14/9]], [[25/16]], [[36/23]], [[80/51]]
| Minor 6th<br>(Semiaugmented 5th)
| Minor 6th<br />(Semiaugmented 5th)
| Bb<br>(At)
| Bb<br />(At)
| minor 6th
| minor 6th
| m6
| m6
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| 847.06
| 847.06
| [[13/8]], [[18/11]], [[44/27]], [[64/39]]
| [[13/8]], [[18/11]], [[44/27]], [[64/39]]
| Augmented 5th<br>(Neutral 6th)
| Augmented 5th<br />(Neutral 6th)
| A#<br>(Bd)
| A#<br />(Bd)
| mid 6th
| mid 6th
| ~6
| ~6
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| 1058.82
| 1058.82
| [[11/6]], [[13/7]], [[24/13]], [[46/25]]
| [[11/6]], [[13/7]], [[24/13]], [[46/25]]
| Diminished 8ve<br>(Neutral 7th)
| Diminished 8ve<br />(Neutral 7th)
| Db<br>(Ct)
| Db<br />(Ct)
| mid 7th
| mid 7th
| ~7
| ~7
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| 1129.41
| 1129.41
| [[23/12]], [[25/13]], [[27/14]], [[48/25]], [[52/27]]
| [[23/12]], [[25/13]], [[27/14]], [[48/25]], [[52/27]]
| Major 7th<br>(Semidiminished 8ve)
| Major 7th<br />(Semidiminished 8ve)
| C#<br>(Dd)
| C#<br />(Dd)
| major 7th,<br>down 8ve
| major 7th,<br />down 8ve
| M7, v8
| M7, v8
| C#
| C#
| major 7th,<br>unter octave
| major 7th,<br />unter octave
| M7, u8
| M7, u8
| C#, uD
| C#, uD
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|}
|}


<nowiki>*</nowiki> Ratios based on treating 17edo as a 2.3.25.7.11.13.85.23-subgroup temperament.
<sup>&dagger;</sup> Half-sharps and half-flats (denoted "t" and "d", respectively) can be used to alter the note by a single step, since sharps and flats each span two edosteps. 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.
 
† Half-sharps and half-flats (denoted "t" and "d", respectively) can be used to alter the note by a single step, since sharps and flats each span two edosteps. 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.


=== Interval quality and chord names in color notation ===
=== Interval quality and chord names in color notation ===
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! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br />8ve Stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" | Tuning Error
|-
|-
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| {{monzo| 27 -17 }}
| {{monzo| 27 -17 }}
| [{{val| 17 27 }}]
| [{{val| 17 27 }}]
| -1.24
| &minus;1.24
| 1.24
| 1.24
| 1.76
| 1.76
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| 64/63, 17496/16807
| 64/63, 17496/16807
| [{{val| 17 27 48 }}]
| [{{val| 17 27 48 }}]
| -3.13
| &minus;3.13
| 2.85
| 2.85
| 4.05
| 4.05
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| 64/63, 99/98, 243/242
| 64/63, 99/98, 243/242
| [{{val| 17 27 48 59 }}]
| [{{val| 17 27 48 59 }}]
| -3.31
| &minus;3.31
| 2.49
| 2.49
| 3.54
| 3.54
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| 64/63, 78/77, 99/98, 144/143
| 64/63, 78/77, 99/98, 144/143
| [{{val| 17 27 48 59 63 }}]
| [{{val| 17 27 48 59 63 }}]
| -3.00
| &minus;3.00
| 2.31
| 2.31
| 3.28
| 3.28
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{| class="commatable wikitable center-all left-3 right-4 left-6"
{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
|-
! [[Harmonic limit|Prime<br>Limit]]
! [[Harmonic limit|Prime<br />Limit]]
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Monzo]]
! [[Monzo]]
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| Diatisma
| Diatisma
|}
|}
<references/>
Note that despite their relatively large size, the 17-comma, the avicennma and the chromatic semitone are all tempered out by the 13-limit patent val, as stated.
Note that despite their relatively large size, the 17-comma, the avicennma and the chromatic semitone are all tempered out by the 13-limit patent val, as stated.


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{| class="wikitable center-all right-3 left-5"
{| class="wikitable center-all right-3 left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per 8ve
|-
! Periods<br />per 8ve
! Generator
! Generator
! Cents
! Cents
! Associated<br>Ratio
! Associated<br />Ratio
! Temperament
! Temperament
|-
|-
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| 352.94
| 352.94
| 11/9
| 11/9
| [[Suhajira]] / [[neutrominant]] (17c) / [[beatles]] (17c) / [[dicotic]] (17) <br>[[Hemif]] / [[mohamaq]] (17c) / [[salsa]] (17)
| [[Suhajira]] / [[neutrominant]] (17c) / [[beatles]] (17c) / [[dicotic]] (17) <br />[[Hemif]] / [[mohamaq]] (17c) / [[salsa]] (17)
|-
|-
| 1
| 1
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| 564.71
| 564.71
| 7/5
| 7/5
| [[Lee]] / [[liese]] (17c) / [[pycnic]] (17)<br>[[Progress]] (17c)
| [[Lee]] / [[liese]] (17c) / [[pycnic]] (17)<br />[[Progress]] (17c)
|}
|}
{{todo|complete table|comment=add warts for every temperament|inline=1}}
{{todo|complete table|comment=add warts for every temperament|inline=1}}
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* [[Lumatone mapping for 17edo]]
* [[Lumatone mapping for 17edo]]


[[Category:17edo| ]] <!-- main article -->
== Notes ==
<references group="note" />
 
[[Category:17edo| ]] <!-- Main article -->
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
[[Category:3-limit]]
[[Category:3-limit]]
[[Category:Teentuning]]
[[Category:Teentuning]]
Retrieved from "https://en.xen.wiki/w/17edo"