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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox ET}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
'''17EDT''' is the [[EDT|equal division of the third harmonic]] into 17 parts of 111.880 cents each (corresponding to 10.726 [[EDO]]).
: This revision was by author [[User:Kosmorsky|Kosmorsky]] and made on <tt>2011-08-10 15:12:01 UTC</tt>.<br>
: The original revision id was <tt>245287451</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">=17 Tone Equal Divided Tritave=


== Properties ==
Following [[13edt]], 17edt is the first EDT that can reasonably be described as a [[3.5.7 subgroup]] temperament, though one that sacrifices much accuracy compared to 13edt, compensating for that by representing primes 11 and 17. By the coincidence of the 11-limit commas 17edt tempers out, 5/3 and [[11/9]] are off by practically the same amount in opposite directions (+10.7 cents and -11.8 cents), leading to an excellent approximation of [[55/27]] (only 1.1 cents flat), as are 11/9 and 9/7 (-11.8 cents and +12.4 cents), leading to an excellent approximation of [[11/7]] (only .6 cents flat). 17edt's step is also, notably, only 0.15 cents sharp of [[16/15]].


17edt is closely related to the Bohlen-Pierce scale, I might go as far as call it an 11-limit version of it. Both have 4L+5s nonatonic modes, but whereas the ratio of large to small steps in BP is a calm 2:1, in 17edt it is a hard 3:1. Thus, the approximation of 5/3 and 7/3 suffers slightly, in return for a very good approximation of 11/9, which is in fact the size of the large step! Nifty right?
In the no-twos subgroup, 17edt tempers out [[245/243]] and 16807/15625 in the 7-limit, 77/75 and [[1331/1323]] in the 11-limit, and 175/169 and 121/117 in the 13-limit. It [[support]]s the no-twos temperament tempering out 245/243 and 77/75, which in terms of tritave patent vals could be written b17&amp;b21.


If the major chord is defined by degrees 0,8,13 (fibonacci numbers, coincidence?), and a major chord is built atop the two chord tones, an hexatonic set of 0,4,8,9,13,16,17 results. Excluding #16 produces a LLsLL pentatonic scale. Continuing the trend by building major chords on the three new chord tones (4, 9, and 16) gives you a set of the notes 0,3,4,7,8,9,12,13,15,16,17. Excluding #15 produces a LsLssLsLs moment of symmetry, which is called "Moll I" or "Delta", if I'm not mistaken.</pre></div>
17EDT is the sixth [[The Riemann zeta function and tuning#Removing primes|zeta peak tritave division]].
<h4>Original HTML content:</h4>
 
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;17edt&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x17 Tone Equal Divided Tritave"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;17 Tone Equal Divided Tritave&lt;/h1&gt;
{{Harmonics in equal|17|3|1|intervals=prime|columns=15}}
&lt;br /&gt;
 
&lt;br /&gt;
== Discussion ==
17edt is closely related to the Bohlen-Pierce scale, I might go as far as call it an 11-limit version of it. Both have 4L+5s nonatonic modes, but whereas the ratio of large to small steps in BP is a calm 2:1, in 17edt it is a hard 3:1. Thus, the approximation of 5/3 and 7/3 suffers slightly, in return for a very good approximation of 11/9, which is in fact the size of the large step! Nifty right?&lt;br /&gt;
17edt is closely related to [[13edt]], the Bohlen-Pierce division, because they share the feature of tempering out 245/243, so that the interval of [[9/7]] stacked twice results in [[5/3]]. Therefore, like 13edt, 17edt's 9/7 generates an enneatonic Lambda ([[4L 5s (3/1-equivalent)|4L&nbsp;5s]]) scale. If 13edt can be considered an analogue of [[12edo]] as the basic tuning of this scale, 17edt is an analogue of [[17edo]] as the hard 3:1 tuning. While the approximation of 5/3 and 7/3 is less good than that of 13edt, this scale has a good approximation of 11/9 (given the context of the weak 5/3 and 7/3), which is in fact the size of the large step, as it is equated to [[25/21]] by virtue of tempering out 77/75.
&lt;br /&gt;
 
If the major chord is defined by degrees 0,8,13 (fibonacci numbers, coincidence?), and a major chord is built atop the two chord tones, an hexatonic set of 0,4,8,9,13,16,17 results. Excluding #16 produces a LLsLL pentatonic scale. Continuing the trend by building major chords on the three new chord tones (4, 9, and 16) gives you a set of the notes 0,3,4,7,8,9,12,13,15,16,17. Excluding #15 produces a LsLssLsLs moment of symmetry, which is called &amp;quot;Moll I&amp;quot; or &amp;quot;Delta&amp;quot;, if I'm not mistaken.&lt;/body&gt;&lt;/html&gt;</pre></div>
17edt's tuning of BPS ([[Bohlen–Pierce–Stearns]], the temperament defined in 3.5.7 by tempering out 245/243) is also very notable in that its generator (9/7) spans 4 steps, meaning that it is divisible into 2; and the interval [[27/7]] spans 21 steps, meaning it is divisible into 3. These lead to weak extensions of BPS known as [[Dubhe]] (which splits the 9/7 generator into two intervals of [[17/15]], tempering out [[2025/2023]] as the additional comma in 3.5.7.17), and [[Mintra]] (an extension of [[Mintaka]] which splits 27/7 into three intervals of 11/7, tempering out [[1331/1323]] as the additional comma in 3.5.7.11), respectively. 17edt in fact supports basic tunings of Dubhe[9] (which is [[8L 1s (3/1-equivalent)|8L&nbsp;1s]]) and Mintaka[12] (which is [[5L 7s (3/1-equivalent)|5L&nbsp;7s]], i.e. a macro-chromatic scale). Therefore 17edt is important as the smallest nontrivial tuning to support each, and it is remarkable for providing such an efficient intersection of temperaments in the 3.5.7.11.17 subgroup, despite being an extremal tuning of most of these (specifically since Mintaka asks for an 11/7 tuned flat, rather than close to just; and the 9/7 is highly overtempered for BPS) and losing much accuracy compared to more optimal tunings—its behavior in this subgroup is reminiscent of that of [[15edo]] in the [[11-limit]].
 
== Intervals ==
The notation schemes below are based on the BPS ''Lambda'' enneatonic scale presented in the symmetric (sLsLsLsLs, Cassiopeian) mode in J, and the Mintaka macrodiatonic scale presented in the macro-Phrygian (sLLLsLL) mode in E.
 
Interval approximations shown in the 3.5.7.11.17 [[subgroup]] column are those within the 77-throdd [[limit]] and less than 1/3 of a step off (all of these being [[consistent]]).
 
Interval approximations shown in the 3.5''.14.16''.17 subgroup column are those within the 77-[[integer limit]] and less than 15 [[cents]] off, consistency is not taken into account.
 
{| class="wikitable center-2 center-3 center-4 center-5 mw-collapsible"
|+ style="font-size: 105%;" | Intervals of 17edt
|-
! Degree
! Note ([[4L 5s (3/1-equivalent)#Notation|BPS Lambda notation]])
! Note (Macrodiatonic notation)
! Approximate 3.5.7.11.17 [[subgroup]] intervals
! Additional 3.5''.14.16''.17 subgroup intervals^
! Cents
! Hekts
|-
| 0
| J
| E
| [[1/1]]
| |
| 0
| 0
|-
| 1
| K
| F
| [[35/33]] (+10.0¢); [[27/25]] (−21.4¢); [[81/77]] (+24.2¢); [[49/45]] (−35.5¢)
| [[16/15]] (+0.2¢); [[17/16]] (+7.1¢); [[15/14]] (−7.6¢)
| 111.9
| 76.5
|-
| 2
| K♯
| G♭
| [[17/15]] (+7.1¢); [[63/55]] (−11.3¢)
| 16/14 ([[8/7]]) (-7.5¢)
| 223.8
| 152.9
|-
| 3
| L♭
| F♯
| [[11/9]] (−11.8¢); [[21/17]] (−30.2¢); [[25/21]] (+33.8¢)
| [[17/14]] (−0.5¢)
| 335.6
| 229.4
|-
| 4
| L
| G
| [[35/27]] (−1.8¢); [[9/7]] (+12.4¢)
|
| 447.5
| 305.9
|-
| 5
| M
| A♭
| [[15/11]] (+22.4¢); [[7/5]] (−23.1¢)
|
| 559.4
| 382.35
|-
| 6
| M♯
| G♯
| [[81/55]] (+1.1¢); [[25/17]] (+3.6¢); [[49/33]] (−13.1¢); [[51/35]] (+19.5¢)
|
| 671.3
| 458.8
|-
| 7
| N♭
| A
| [[11/7]] (+0.7¢); [[27/17]] (−17.8¢); [[17/11]] (+29.5¢)
|
| 783.2
| 535.3
|-
| 8
| N
| B♭
| [[5/3]] (+10.7¢)
| [[27/16]] (−10.8¢)
| 895.1
| 611.8
|-
| 9
| O
| A♯
| [[9/5]] (−10.7¢)
| [[25/14]] (+3.1¢); [[16/9]] (+10.8¢)
| 1006.9
| 688.2
|-
| 10
| O♯
| B
| [[21/11]] (−0.7¢); [[17/9]] (+17.8¢); [[33/17]] (−29.5¢)
|
| 1118.8
| 764.7
|-
| 11
| P♭
| C
| [[55/27]] (−1.1¢); [[51/25]] (−3.6¢); [[99/49]] (+13.1¢); [[35/17]] (−19.5¢)
|
| 1230.7
| 841.2
|-
| 12
| P
| D♭
| [[11/5]] (−22.4¢); [[15/7]] (+23.1¢)
|
| 1342.6
| 917.65
|-
| 13
| Q
| C♯
| [[81/35]] (+1.8¢); [[7/3]] (−12.4¢)
|
| 1454.4
| 994.1
|-
| 14
| Q♯
| D
| [[27/11]] (+11.8¢); [[17/7]] (+30.2¢); [[63/25]] (−33.8¢)
|
| 1566.3
| 1070.6
|-
| 15
| R♭
| E♭
| [[45/17]] (−7.1¢); [[55/21]] (+11.3¢)
|
| 1678.2
| 1147.1
|-
| 16
| R
| D♯
| [[99/35]] (−10.0¢); [[25/9]] (+21.4¢); [[77/27]] (−24.2¢); [[135/49]] (+35.5¢)
| [[45/16]] (−0.1¢); [[14/5]] (+7.6¢)
| 1790.1
| 1223.5
|-
| 17
| J
| E
| [[3/1]]
|
| 1901.955
| 1300.
|}
 
It is a strange coincidence how the schemes of 17EDO and 17EDT diatonicism are so similar and how their approximations of 9/7 are off by such similar amounts in opposite directions (17EDO −11.6 cents and 17EDT +12.4 cents).
 
== Regular temperament ==
One step of 17edt can also be thought of as a generator of the [[vavoom]] temperament, since it is very close to [[16/15]]; interpreting it thus, an unnoticeable comma {{monzo|-68 18 17}} is tempered out in the vavoom temperament.
 
; <font style="font-size:1.15em">Vavoom (118&amp;783)</font>
'''5-limit'''<br>
Comma: {{monzo|-68 18 17}}<br>
Mapping: [{{val|1 0 4}}, {{val|0 17 -18}}]<br>
POTE generator: ~16/15 = 111.876<br>
{{Optimal ET sequence|legend=1|11, 32, 43, 75, 118, 429, 547, 665, 783, 901, 1684}}<br>
Badness: 0.098376<br><br>
 
== Z function ==
Below is a plot of the [[The Riemann zeta function and tuning#Removing primes|no-twos Z function]] in the vicinity of 17EDT.
 
[[File:17edt.png|alt=17edt.png|17edt.png]]     
 
== Music ==
; [[Togenom]]
* "Circle is Square" from ''Xenharmonics, Vol. 5'' (2024) – [https://open.spotify.com/track/7aY59bxbyy17SZhBGTTJHF Spotify] | [https://togenom.bandcamp.com/track/circle-is-square Bandcamp] | [https://www.youtube.com/watch?v=WPR52uMkZK8 YouTube]
 
[[Category:macrotonal]]
[[Category:tritave]]
Retrieved from "https://en.xen.wiki/w/17edt"