9edt: Difference between revisions

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~~<h2>IMPORTED REVISION FROM WIKISPACES</h2>~~

~~This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>~~

~~: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-09-12 00:21:19 UTC</tt>.<br>~~

~~: The original revision id was <tt>591640954</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">The 9 equal division of 3, the tritave, divides it into 9 equal steps of size 211.328 cents each. It has a decent 7 and an excellent 13, but a 5 which is 39 cents flat; if octaves were added and it was a third, it would count as a neutral third. The corresponding 5/3 is 845 cents, which is a neutral sixth between 8/5 and 5/3, which is really more a 13/8, though this is allegedly a no-twos tuning. The 3.7.13 subgroup tempers out 351/343 and 2197/2187. 9edt is the third [[@The Riemann Zeta Function and Tuning#Removing%20primes|no-twos zeta peak edt]].

~~Following~~ [[~~@4edt~~]], ~~this~~ is the ~~next "Lambda"~~ (~~BP related~~) ~~equal division~~ of ~~the tritave~~; ~~in a certain sense analogous~~ to [[~~@7edo~~]] in ~~diatonic music~~.

== Theory ==

It has a decent seventh harmonic ([[7/1]]) which is 12.4¢ sharp, and an excellent [[13/1]] inherited from [[3edt]] which is only 2.6{{c}} flat. However, the [[5/1]] is 39{{c}} flat, thus 13 steps of 9edt (approximating the 5/1) can be described as a neutral seventeenth—or if tritave-reduced to 4 steps, a neutral sixth (approximating the 5/3). This neutral sixth has a size of 845{{c}}, which is between [[8/5]] and [[5/3]]; if this interval is also taken as an approximation to [[13/8]], it would temper out [[40/39]]—making 9edt an exotemperament in the 8.3.5.13 subgroup. Though, 9edt is more well behaved on the 3.7.13 [[subgroup]], of which it tempers out [[351/343]] and [[2197/2187]].

~~This scale~~ is ~~also related to~~ [[~~@17edo~~]] ~~by which it may be approximated by playing every third step~~ (~~the 17edo non~~-~~octave whole~~-~~tone~~ scale), the ~~discrepancy is only about four cents when it gets~~ to ~~3/1~~.

Following [[4edt]], this is the next edt that supports [[BPS]] temperament. For small edts, this property is virtually the same as supporting a [[4L 5s (3/1-equivalent)|3/1-equivalent "lambda" scale]], of which 9edt offers the "equalized" interpretation of {{nowrap|L {{=}} s}}, analogous to [[7edo]] in diatonic ([[5L 2s]]) music.

~~0: 1/1~~

9edt is the third [[the Riemann zeta function and tuning#Removing primes|no-twos zeta peak edt]].

~~1: 211.328 cents 9/8~~

~~2: 422.657 cents 9/7~~

=== Relation to edos ===

~~3: 633.985 cents 13/9~~

9edt is related to [[17edo]], by which it may be approximated by playing every third step (the 17edo non-octave whole-tone scale), the discrepancy is only about four cents when it gets to [[3/1]].

~~4: 845.313 cents 5/3~~

~~5: 1056.642 cents 9/5~~

=== Harmonics ===

~~6: 1267.970 cents~~

~~7: 1479.298 cents 7/3~~

~~8: 1690.627 cents 8/3~~

~~9: 3/1</pre></div>~~

{| class="wikitable"

~~<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"><html><head><title>9edt~~</title></head><body>The 9 equal division of 3,~~ the ~~tritave, divides it into 9 equal steps of size 211.328 cents each. It has a decent 7 and an excellent 13, but a 5 which is 39 cents flat; if octaves were added and it was a~~ third~~, it would count as a neutral third. The corresponding 5/3 is 845 cents, which is a neutral sixth between 8/5~~ and ~~5/3, which is really more a 13/8, though this is allegedly a no-twos~~ tuning~~. The 3.7.13 subgroup tempers out 351/343 and 2197/2187. 9edt is the third <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning~~#Removing~~%20primes" target="_blank">~~no-twos zeta peak edt~~</a>~~.~~<br />~~

! rowspan="2" | Steps

~~<br />~~

! colspan="2" | Size

~~Following <a class~~=~~"wiki_link" href~~=~~"/4edt" target~~=~~"_blank">4edt</a>, this is the next &quot;Lambda&quot; (BP related) equal division of the tritave; in a certain sense analogous~~ to ~~<a class~~=~~"wiki_link" href~~=~~"/7edo" target~~=~~"_blank">7edo</a> in diatonic music.<br />~~

! rowspan="2" | Comparable intervals (¢)

~~<br />~~

|-

~~This scale~~ is ~~also~~ related to ~~<a class="wiki_link" href="/~~17edo~~" target="_blank">17edo</a>~~ by which it may be approximated by playing every third step (the 17edo non-octave whole-tone scale), the discrepancy is only about four cents when it gets to 3/1.~~<br />~~

! Cents

~~<br />~~

! [[Hekt]]s

0: 1/1~~<br />~~

|-

1: 211.328 ~~cents~~ 9/8~~<br />~~

! colspan="3" | 0

2: 422.657 ~~cents~~ 9/7~~<br />~~

| [[1/1]]

3: 633.985 ~~cents~~ 13/9~~<br />~~

|-

4: 845.313 ~~cents~~ 5/3~~<br~~ /~~>~~

| 1

5: 1056.642 ~~cents~~ 9/5~~<br~~ /~~>~~

| 211.328

6: 1267.970 ~~cents<br~~ /~~>~~

| 144.444

7: 1479.298 ~~cents~~ 7/3~~<br />~~

| [[9/8]] (204)

8: 1690.627 ~~cents~~ 8/3~~<br />~~

|-

9: 3/1~~<~~/~~body><~~/~~html><~~/~~pre><~~/~~div>~~

| 2

| 422.657

| 288.889

| [[9/7]] (435)

|-

| 3

| 633.985

| 433.333

| [[13/9]] (637)

|-

| 4

| 845.313

| 577.778

| [[13/8]] (841), [[5/3]] (884), [[8/5]] (814)

|-

| 5

| 1056.642

| 722.222

| [[9/5]] (1018), [[11/6]] (1049)

|-

| 6

| 1267.970

| 866.667

| [[27/13]] (1265)

|-

| 7

| 1479.298

| 1011.111

| [[7/3]] (1467)

|-

| 8

| 1690.627

| 1155.556

| [[8/3]] (1698)

|-

| 9

| 1901.955

| 1300

| [[3/1]]

|}

== Music ==

* [https://www.youtube.com/watch?v=sEQP1AtjPrA Far Away From Them / Spazzystackers] by [[Mandrake]]

[[Category:Macrotonal]]

@@ Line 1: / Line 1: @@
-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+{{Infobox ET}}
-This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
+{{ED intro}}
-: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-09-12 00:21:19 UTC</tt>.<br>
-: The original revision id was <tt>591640954</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">The 9 equal division of 3, the tritave, divides it into 9 equal steps of size 211.328 cents each. It has a decent 7 and an excellent 13, but a 5 which is 39 cents flat; if octaves were added and it was a third, it would count as a neutral third. The corresponding 5/3 is 845 cents, which is a neutral sixth between 8/5 and 5/3, which is really more a 13/8, though this is allegedly a no-twos tuning. The 3.7.13 subgroup tempers out 351/343 and 2197/2187. 9edt is the third [[@The Riemann Zeta Function and Tuning#Removing%20primes|no-twos zeta peak edt]].
-Following [[@4edt]], this is the next "Lambda" (BP related) equal division of the tritave; in a certain sense analogous to [[@7edo]] in diatonic music.
+== Theory ==
+It has a decent seventh harmonic ([[7/1]]) which is 12.4¢ sharp, and an excellent [[13/1]] inherited from [[3edt]] which is only 2.6{{c}} flat. However, the [[5/1]] is 39{{c}} flat, thus 13 steps of 9edt (approximating the 5/1) can be described as a neutral seventeenth—or if tritave-reduced to 4 steps, a neutral sixth (approximating the 5/3). This neutral sixth has a size of 845{{c}}, which is between [[8/5]] and [[5/3]]; if this interval is also taken as an approximation to [[13/8]], it would temper out [[40/39]]—making 9edt an exotemperament in the 8.3.5.13 subgroup. Though, 9edt is more well behaved on the 3.7.13 [[subgroup]], of which it tempers out [[351/343]] and [[2197/2187]].
-This scale is also related to [[@17edo]] by which it may be approximated by playing every third step (the 17edo non-octave whole-tone scale), the discrepancy is only about four cents when it gets to 3/1.
+Following [[4edt]], this is the next edt that supports [[BPS]] temperament. For small edts, this property is virtually the same as supporting a [[4L 5s (3/1-equivalent)|3/1-equivalent "lambda" scale]], of which 9edt offers the "equalized" interpretation of {{nowrap|L {{=}} s}}, analogous to [[7edo]] in diatonic ([[5L&nbsp;2s]]) music.
-: 1/1
+edt is the third [[the Riemann zeta function and tuning#Removing primes|no-twos zeta peak edt]].
-: 211.328 cents 9/8
-: 422.657 cents 9/7
+=== Relation to edos ===
-: 633.985 cents 13/9
+edt is related to [[17edo]], by which it may be approximated by playing every third step (the 17edo non-octave whole-tone scale), the discrepancy is only about four cents when it gets to [[3/1]].
-: 845.313 cents 5/3
-: 1056.642 cents 9/5
+=== Harmonics ===
-: 1267.970 cents
+{{Harmonics in equal|9|3|1|}}
-: 1479.298 cents 7/3
+{{Harmonics in equal|9|3|1|intervals=prime}}
-: 1690.627 cents 8/3
-: 3/1</pre></div>
+{| class="wikitable"
-<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;9edt&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The 9 equal division of 3, the tritave, divides it into 9 equal steps of size 211.328 cents each. It has a decent 7 and an excellent 13, but a 5 which is 39 cents flat; if octaves were added and it was a third, it would count as a neutral third. The corresponding 5/3 is 845 cents, which is a neutral sixth between 8/5 and 5/3, which is really more a 13/8, though this is allegedly a no-twos tuning. The 3.7.13 subgroup tempers out 351/343 and 2197/2187. 9edt is the third &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Removing%20primes" target="_blank"&gt;no-twos zeta peak edt&lt;/a&gt;.&lt;br /&gt;
+! rowspan="2" | Steps
-&lt;br /&gt;
+! colspan="2" | Size
-Following &lt;a class="wiki_link" href="/4edt" target="_blank"&gt;4edt&lt;/a&gt;, this is the next &amp;quot;Lambda&amp;quot; (BP related) equal division of the tritave; in a certain sense analogous to &lt;a class="wiki_link" href="/7edo" target="_blank"&gt;7edo&lt;/a&gt; in diatonic music.&lt;br /&gt;
+! rowspan="2" | Comparable intervals (¢)
-&lt;br /&gt;
+|-
-This scale is also related to &lt;a class="wiki_link" href="/17edo" target="_blank"&gt;17edo&lt;/a&gt; by which it may be approximated by playing every third step (the 17edo non-octave whole-tone scale), the discrepancy is only about four cents when it gets to 3/1.&lt;br /&gt;
+! Cents
-&lt;br /&gt;
+! [[Hekt]]s
-: 1/1&lt;br /&gt;
+|-
-: 211.328 cents 9/8&lt;br /&gt;
+! colspan="3" | 0
-: 422.657 cents 9/7&lt;br /&gt;
+| [[1/1]]
-: 633.985 cents 13/9&lt;br /&gt;
+|-
-: 845.313 cents 5/3&lt;br /&gt;
+| 1
-: 1056.642 cents 9/5&lt;br /&gt;
+| 211.328
-: 1267.970 cents&lt;br /&gt;
+| 144.444
-: 1479.298 cents 7/3&lt;br /&gt;
+| [[9/8]] (204)
-: 1690.627 cents 8/3&lt;br /&gt;
+|-
-: 3/1&lt;/body&gt;&lt;/html&gt;</pre></div>
+| 2
+| 422.657
+| 288.889
+| [[9/7]] (435)
+|-
+| 3
+| 633.985
+| 433.333
+| [[13/9]] (637)
+|-
+| 4
+| 845.313
+| 577.778
+| [[13/8]] (841), [[5/3]] (884), [[8/5]] (814)
+|-
+| 5
+| 1056.642
+| 722.222
+| [[9/5]] (1018), [[11/6]] (1049)
+|-
+| 6
+| 1267.970
+| 866.667
+| [[27/13]] (1265)
+|-
+| 7
+| 1479.298
+| 1011.111
+| [[7/3]] (1467)
+|-
+| 8
+| 1690.627
+| 1155.556
+| [[8/3]] (1698)
+|-
+| 9
+| 1901.955
+| 1300
+| [[3/1]]
+|}
+== Music ==
+* [https://www.youtube.com/watch?v=sEQP1AtjPrA Far Away From Them / Spazzystackers] by [[Mandrake]]
+[[Category:Macrotonal]]