25edo: 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:

: This revision was by author [[User:~~Osmiorisbendi~~|~~Osmiorisbendi~~]] and made on <tt>2011-07-26 ~~20:02~~:09 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-08-07 15:26:38 UTC</tt>.

: The original revision id was <tt>~~242984401~~</tt>.

: The original revision id was <tt>244722873</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

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25EDO divides the [[octave]] in 25 equal steps of exact size 48 [[cent]]s each. It is a good way to tune the [[Blackwood temperament]], which takes the very sharp fifths of [[5EDO]] as a given, tempers out 28/27 and 49/48, and attempts to optimize the tunings for 5 ([[5_4|5/4]]) and 7 ([[7_4|7/4]]?).

25EDO has fifths 18 cents sharp, but its major thirds are excellent and its 7/4 is acceptable. It therefore makes sense to use it as a 2.5.7 [[Just intonation subgroups|subgroup]] tuning. Looking just at 2, 5, and 7, it equates five [[8_7|8/7]]s with the octave, and so tempers out (8/7)^5 / 2 = 16807/16384. It also equates a [[128_125|128/125]] [[diesis]] and two [[septimal tritones]] of [[7_5|7/5]] with the octave, and hence tempers out 3136/3125. If we want to temper out both of these and also have decent fifths, the obvious solution is [[50EDO]].

25EDO has fifths 18 cents sharp, but its major thirds are excellent and its 7/4 is acceptable. It therefore makes sense to use it as a 2.5.7 [[Just intonation subgroups|subgroup]] tuning. Looking just at 2, 5, and 7, it equates five [[8_7|8/7]]s with the octave, and so tempers out (8/7)^5 / 2 = 16807/16384. It also equates a [[128_125|128/125]] [[diesis]] and two [[septimal tritones]] of [[7_5|7/5]] with the octave, and hence tempers out 3136/3125. If we want to temper out both of these and also have decent fifths, the obvious solution is [[50EDO]]. And alternative fifth, 14\25, which is 672 cents, provides an alternative very flat fifth which can be used for mavila temperament.

If 5/4 and 7/4 aren't good enough, it also does 17/16 and 19/16, just like 12EDO. In fact, on the [[k*N subgroups|2*25 subgroup]] 2.9.5.7.33.39.17.19 it provides the same tuning and tempers out the same commas as 50et, which makes for wide range of harmony.

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25EDO divides the <a class="wiki_link" href="/octave">octave</a> in 25 equal steps of exact size 48 <a class="wiki_link" href="/cent">cent</a>s each. It is a good way to tune the <a class="wiki_link" href="/Blackwood%20temperament">Blackwood temperament</a>, which takes the very sharp fifths of <a class="wiki_link" href="/5EDO">5EDO</a> as a given, tempers out 28/27 and 49/48, and attempts to optimize the tunings for 5 (<a class="wiki_link" href="/5_4">5/4</a>) and 7 (<a class="wiki_link" href="/7_4">7/4</a>?).

25EDO has fifths 18 cents sharp, but its major thirds are excellent and its 7/4 is acceptable. It therefore makes sense to use it as a 2.5.7 <a class="wiki_link" href="/Just%20intonation%20subgroups">subgroup</a> tuning. Looking just at 2, 5, and 7, it equates five <a class="wiki_link" href="/8_7">8/7</a>s with the octave, and so tempers out (8/7)^5 / 2 = 16807/16384. It also equates a <a class="wiki_link" href="/128_125">128/125</a> <a class="wiki_link" href="/diesis">diesis</a> and two <a class="wiki_link" href="/septimal%20tritones">septimal tritones</a> of <a class="wiki_link" href="/7_5">7/5</a> with the octave, and hence tempers out 3136/3125. If we want to temper out both of these and also have decent fifths, the obvious solution is <a class="wiki_link" href="/50EDO">50EDO</a>.

25EDO has fifths 18 cents sharp, but its major thirds are excellent and its 7/4 is acceptable. It therefore makes sense to use it as a 2.5.7 <a class="wiki_link" href="/Just%20intonation%20subgroups">subgroup</a> tuning. Looking just at 2, 5, and 7, it equates five <a class="wiki_link" href="/8_7">8/7</a>s with the octave, and so tempers out (8/7)^5 / 2 = 16807/16384. It also equates a <a class="wiki_link" href="/128_125">128/125</a> <a class="wiki_link" href="/diesis">diesis</a> and two <a class="wiki_link" href="/septimal%20tritones">septimal tritones</a> of <a class="wiki_link" href="/7_5">7/5</a> with the octave, and hence tempers out 3136/3125. If we want to temper out both of these and also have decent fifths, the obvious solution is <a class="wiki_link" href="/50EDO">50EDO</a>. And alternative fifth, 14\25, which is 672 cents, provides an alternative very flat fifth which can be used for mavila temperament.

If 5/4 and 7/4 aren't good enough, it also does 17/16 and 19/16, just like 12EDO. In fact, on the <a class="wiki_link" href="/k%2AN%20subgroups">2*25 subgroup</a> 2.9.5.7.33.39.17.19 it provides the same tuning and tempers out the same commas as 50et, which makes for wide range of harmony.

@@ Line 1: / Line 1: @@
 <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:Osmiorisbendi|Osmiorisbendi]] and made on <tt>2011-07-26 20:02:09 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-08-07 15:26:38 UTC</tt>.<br>
-: The original revision id was <tt>242984401</tt>.<br>
+: The original revision id was <tt>244722873</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>
@@ Line 11: / Line 11: @@
 EDO divides the [[octave]] in 25 equal steps of exact size 48 [[cent]]s each. It is a good way to tune the [[Blackwood temperament]], which takes the very sharp fifths of [[5EDO]] as a given, tempers out 28/27 and 49/48, and attempts to optimize the tunings for 5 ([[5_4|5/4]]) and 7 ([[7_4|7/4]]?).
-EDO has fifths 18 cents sharp, but its major thirds are excellent and its 7/4 is acceptable. It therefore makes sense to use it as a 2.5.7 [[Just intonation subgroups|subgroup]] tuning. Looking just at 2, 5, and 7, it equates five [[8_7|8/7]]s with the octave, and so tempers out (8/7)^5 / 2 = 16807/16384. It also equates a [[128_125|128/125]] [[diesis]] and two [[septimal tritones]] of [[7_5|7/5]] with the octave, and hence tempers out 3136/3125. If we want to temper out both of these and also have decent fifths, the obvious solution is [[50EDO]].
+EDO has fifths 18 cents sharp, but its major thirds are excellent and its 7/4 is acceptable. It therefore makes sense to use it as a 2.5.7 [[Just intonation subgroups|subgroup]] tuning. Looking just at 2, 5, and 7, it equates five [[8_7|8/7]]s with the octave, and so tempers out (8/7)^5 / 2 = 16807/16384. It also equates a [[128_125|128/125]] [[diesis]] and two [[septimal tritones]] of [[7_5|7/5]] with the octave, and hence tempers out 3136/3125. If we want to temper out both of these and also have decent fifths, the obvious solution is [[50EDO]]. And alternative fifth, 14\25, which is 672 cents, provides an alternative very flat fifth which can be used for mavila temperament.
 If 5/4 and 7/4 aren't good enough, it also does 17/16 and 19/16, just like 12EDO. In fact, on the [[k*N subgroups|2*25 subgroup]] 2.9.5.7.33.39.17.19 it provides the same tuning and tempers out the same commas as 50et, which makes for wide range of harmony.
@@ Line 77: / Line 77: @@
 EDO divides the &lt;a class="wiki_link" href="/octave"&gt;octave&lt;/a&gt; in 25 equal steps of exact size 48 &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;s each. It is a good way to tune the &lt;a class="wiki_link" href="/Blackwood%20temperament"&gt;Blackwood temperament&lt;/a&gt;, which takes the very sharp fifths of &lt;a class="wiki_link" href="/5EDO"&gt;5EDO&lt;/a&gt; as a given, tempers out 28/27 and 49/48, and attempts to optimize the tunings for 5 (&lt;a class="wiki_link" href="/5_4"&gt;5/4&lt;/a&gt;) and 7 (&lt;a class="wiki_link" href="/7_4"&gt;7/4&lt;/a&gt;?).&lt;br /&gt;
 &lt;br /&gt;
-EDO has fifths 18 cents sharp, but its major thirds are excellent and its 7/4 is acceptable. It therefore makes sense to use it as a 2.5.7 &lt;a class="wiki_link" href="/Just%20intonation%20subgroups"&gt;subgroup&lt;/a&gt; tuning. Looking just at 2, 5, and 7, it equates five &lt;a class="wiki_link" href="/8_7"&gt;8/7&lt;/a&gt;s with the octave, and so tempers out (8/7)^5 / 2 = 16807/16384. It also equates a &lt;a class="wiki_link" href="/128_125"&gt;128/125&lt;/a&gt; &lt;a class="wiki_link" href="/diesis"&gt;diesis&lt;/a&gt; and two &lt;a class="wiki_link" href="/septimal%20tritones"&gt;septimal tritones&lt;/a&gt; of &lt;a class="wiki_link" href="/7_5"&gt;7/5&lt;/a&gt; with the octave, and hence tempers out 3136/3125. If we want to temper out both of these and also have decent fifths, the obvious solution is &lt;a class="wiki_link" href="/50EDO"&gt;50EDO&lt;/a&gt;.&lt;br /&gt;
+EDO has fifths 18 cents sharp, but its major thirds are excellent and its 7/4 is acceptable. It therefore makes sense to use it as a 2.5.7 &lt;a class="wiki_link" href="/Just%20intonation%20subgroups"&gt;subgroup&lt;/a&gt; tuning. Looking just at 2, 5, and 7, it equates five &lt;a class="wiki_link" href="/8_7"&gt;8/7&lt;/a&gt;s with the octave, and so tempers out (8/7)^5 / 2 = 16807/16384. It also equates a &lt;a class="wiki_link" href="/128_125"&gt;128/125&lt;/a&gt; &lt;a class="wiki_link" href="/diesis"&gt;diesis&lt;/a&gt; and two &lt;a class="wiki_link" href="/septimal%20tritones"&gt;septimal tritones&lt;/a&gt; of &lt;a class="wiki_link" href="/7_5"&gt;7/5&lt;/a&gt; with the octave, and hence tempers out 3136/3125. If we want to temper out both of these and also have decent fifths, the obvious solution is &lt;a class="wiki_link" href="/50EDO"&gt;50EDO&lt;/a&gt;. And alternative fifth, 14\25, which is 672 cents, provides an alternative very flat fifth which can be used for mavila temperament.&lt;br /&gt;
 &lt;br /&gt;
 If 5/4 and 7/4 aren't good enough, it also does 17/16 and 19/16, just like 12EDO. In fact, on the &lt;a class="wiki_link" href="/k%2AN%20subgroups"&gt;2*25 subgroup&lt;/a&gt; 2.9.5.7.33.39.17.19 it provides the same tuning and tempers out the same commas as 50et, which makes for wide range of harmony.&lt;br /&gt;