Mina: 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:hearneg|hearneg]] and made on <tt>2017-06-11 04:20:54 UTC</tt>.

: This revision was by author [[User:hearneg|hearneg]] and made on <tt>2017-06-11 04:28:49 UTC</tt>.

: The original revision id was <tt>~~614524401~~</tt>.

: The original revision id was <tt>614524443</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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The following table lists some intervals which may be represented exactly in minas and in degrees and minutes, with the sizes listed in both [[cent]]s and minas and expressed as degrees and minutes.

||~ interval ||~ size in

cents ||~ size in

minas ||~ size as degrees

and minutes ||

|| 1\2460 || 0.488 || 1 || 1' ||

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|| 2/1 || 1200 || 2460 || 41° ||

Another ~~helpful proponent~~ of the mina is the accuracy and breadth of it's approximation to just intervals. Accordingly it is hardly necessary to express intervals in non-integer values of mina, something that arguably cannot be said of cents. [[2460edo]] It is uniquely [[xenharmonic/consistent|consistent]] through to the [[xenharmonic/27-limit|27-limit]], which is not very remarkable in itself ([[xenharmonic/388edo|388edo]] is the first such system), but what is remarkable is the degree of accuracy to which it represents the 27-limit intervals. It is also a [[xenharmonic/The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]] and has a lower 19-limit [[xenharmonic/Tenney-Euclidean temperament measures#TE%20simple%20badness|relative error]] than any edo until [[xenharmonic/3395edo|3395]], and a lower 23-limit relative error than any until [[xenharmonic/8269edo|8269]]. Also it has a lower 23-limit [[xenharmonic/Tenney-Euclidean metrics#Logflat%20TE%20badness| TE loglfat badness]] than any smaller edo and less than any until [[xenharmonic/16808edo|16808]].

Another notable feature of the mina is the accuracy and breadth of it's approximation to just intervals. Accordingly it is hardly necessary to express intervals in non-integer values of mina, something that arguably cannot be said of cents. [[2460edo]] It is uniquely [[xenharmonic/consistent|consistent]] through to the [[xenharmonic/27-limit|27-limit]], which is not very remarkable in itself ([[xenharmonic/388edo|388edo]] is the first such system), but what is remarkable is the degree of accuracy to which it represents the 27-limit intervals. It is also a [[xenharmonic/The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]] and has a lower 19-limit [[xenharmonic/Tenney-Euclidean temperament measures#TE%20simple%20badness|relative error]] than any edo until [[xenharmonic/3395edo|3395]], and a lower 23-limit relative error than any until [[xenharmonic/8269edo|8269]]. Also it has a lower 23-limit [[xenharmonic/Tenney-Euclidean metrics#Logflat%20TE%20badness| TE loglfat badness]] than any smaller edo and less than any until [[xenharmonic/16808edo|16808]].

Below the intervals of the [[27-limit]] [[tonality diamond]] are tabulated, with the sizes listed in both [[cent]]s and minas and expressed as degrees and minutes (rounded to the nearest minute). The value in minas, rounded to the nearest integer, can be found by applying the [[23-limit]] [[patent val]] <2460 3899 5712 6906 8510 9103 10055 10450 11128| for 2460edo; this will not work for [[1200edo]] and cents.

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<th>interval

</th>

<th>size in

<th>size in

cents

</th>

<th>size in

<th>size in

minas

</th>

<th>size as degrees

<th>size as degrees

and minutes

</th>

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Another ~~helpful proponent~~ of the mina is the accuracy and breadth of it's approximation to just intervals. Accordingly it is hardly necessary to express intervals in non-integer values of mina, something that arguably cannot be said of cents. <a class="wiki_link" href="/2460edo">2460edo</a> It is uniquely <a class="wiki_link" href="http://xenharmonic.wikispaces.com/consistent">consistent</a> through to the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/27-limit">27-limit</a>, which is not very remarkable in itself (<a class="wiki_link" href="http://xenharmonic.wikispaces.com/388edo">388edo</a> is the first such system), but what is remarkable is the degree of accuracy to which it represents the 27-limit intervals. It is also a <a class="wiki_link" href="http://xenharmonic.wikispaces.com/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists">zeta peak edo</a> and has a lower 19-limit <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Tenney-Euclidean%20temperament%20measures#TE%20simple%20badness">relative error</a> than any edo until <a class="wiki_link" href="http://xenharmonic.wikispaces.com/3395edo">3395</a>, and a lower 23-limit relative error than any until <a class="wiki_link" href="http://xenharmonic.wikispaces.com/8269edo">8269</a>. Also it has a lower 23-limit <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Tenney-Euclidean%20metrics#Logflat%20TE%20badness"> TE loglfat badness</a> than any smaller edo and less than any until <a class="wiki_link" href="http://xenharmonic.wikispaces.com/16808edo">16808</a>.

Another notable feature of the mina is the accuracy and breadth of it's approximation to just intervals. Accordingly it is hardly necessary to express intervals in non-integer values of mina, something that arguably cannot be said of cents. <a class="wiki_link" href="/2460edo">2460edo</a> It is uniquely <a class="wiki_link" href="http://xenharmonic.wikispaces.com/consistent">consistent</a> through to the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/27-limit">27-limit</a>, which is not very remarkable in itself (<a class="wiki_link" href="http://xenharmonic.wikispaces.com/388edo">388edo</a> is the first such system), but what is remarkable is the degree of accuracy to which it represents the 27-limit intervals. It is also a <a class="wiki_link" href="http://xenharmonic.wikispaces.com/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists">zeta peak edo</a> and has a lower 19-limit <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Tenney-Euclidean%20temperament%20measures#TE%20simple%20badness">relative error</a> than any edo until <a class="wiki_link" href="http://xenharmonic.wikispaces.com/3395edo">3395</a>, and a lower 23-limit relative error than any until <a class="wiki_link" href="http://xenharmonic.wikispaces.com/8269edo">8269</a>. Also it has a lower 23-limit <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Tenney-Euclidean%20metrics#Logflat%20TE%20badness"> TE loglfat badness</a> than any smaller edo and less than any until <a class="wiki_link" href="http://xenharmonic.wikispaces.com/16808edo">16808</a>.

Below the intervals of the <a class="wiki_link" href="/27-limit">27-limit</a> <a class="wiki_link" href="/tonality%20diamond">tonality diamond</a> are tabulated, with the sizes listed in both <a class="wiki_link" href="/cent">cent</a>s and minas and expressed as degrees and minutes (rounded to the nearest minute). The value in minas, rounded to the nearest integer, can be found by applying the <a class="wiki_link" href="/23-limit">23-limit</a> <a class="wiki_link" href="/patent%20val">patent val</a> &lt;2460 3899 5712 6906 8510 9103 10055 10450 11128| for 2460edo; this will not work for <a class="wiki_link" href="/1200edo">1200edo</a> and cents.

@@ 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:hearneg|hearneg]] and made on <tt>2017-06-11 04:20:54 UTC</tt>.<br>
+: This revision was by author [[User:hearneg|hearneg]] and made on <tt>2017-06-11 04:28:49 UTC</tt>.<br>
-: The original revision id was <tt>614524401</tt>.<br>
+: The original revision id was <tt>614524443</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: @@
 The following table lists some intervals which may be represented exactly in minas and in degrees and minutes, with the sizes listed in both [[cent]]s and minas and expressed as degrees and minutes.
 ||~ interval ||~ size in
 cents ||~ size in
 minas ||~ size as degrees
 and minutes ||
 || 1\2460 || 0.488 || 1 || 1' ||
@@ Line 37: / Line 37: @@
 || 2/1 || 1200 || 2460 || 41° ||
-Another helpful proponent of the mina is the accuracy and breadth of it's approximation to just intervals. Accordingly it is hardly necessary to express intervals in non-integer values of mina, something that arguably cannot be said of cents. [[2460edo]] It is uniquely [[xenharmonic/consistent|consistent]] through to the [[xenharmonic/27-limit|27-limit]], which is not very remarkable in itself ([[xenharmonic/388edo|388edo]] is the first such system), but what is remarkable is the degree of accuracy to which it represents the 27-limit intervals. It is also a [[xenharmonic/The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]] and has a lower 19-limit [[xenharmonic/Tenney-Euclidean temperament measures#TE%20simple%20badness|relative error]] than any edo until [[xenharmonic/3395edo|3395]], and a lower 23-limit relative error than any until [[xenharmonic/8269edo|8269]]. Also it has a lower 23-limit [[xenharmonic/Tenney-Euclidean metrics#Logflat%20TE%20badness| TE loglfat badness]] than any smaller edo and less than any until [[xenharmonic/16808edo|16808]].
+Another notable feature of the mina is the accuracy and breadth of it's approximation to just intervals. Accordingly it is hardly necessary to express intervals in non-integer values of mina, something that arguably cannot be said of cents. [[2460edo]] It is uniquely [[xenharmonic/consistent|consistent]] through to the [[xenharmonic/27-limit|27-limit]], which is not very remarkable in itself ([[xenharmonic/388edo|388edo]] is the first such system), but what is remarkable is the degree of accuracy to which it represents the 27-limit intervals. It is also a [[xenharmonic/The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]] and has a lower 19-limit [[xenharmonic/Tenney-Euclidean temperament measures#TE%20simple%20badness|relative error]] than any edo until [[xenharmonic/3395edo|3395]], and a lower 23-limit relative error than any until [[xenharmonic/8269edo|8269]]. Also it has a lower 23-limit [[xenharmonic/Tenney-Euclidean metrics#Logflat%20TE%20badness| TE loglfat badness]] than any smaller edo and less than any until [[xenharmonic/16808edo|16808]].
 Below the intervals of the [[27-limit]] [[tonality diamond]] are tabulated, with the sizes listed in both [[cent]]s and minas and expressed as degrees and minutes (rounded to the nearest minute). The value in minas, rounded to the nearest integer, can be found by applying the [[23-limit]] [[patent val]] &lt;2460 3899 5712 6906 8510 9103 10055 10450 11128| for 2460edo; this will not work for [[1200edo]] and cents.
@@ Line 213: / Line 213: @@
          &lt;th&gt;interval&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;size in &lt;br /&gt;
+         &lt;th&gt;size in&lt;br /&gt;
 cents&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;size in &lt;br /&gt;
+         &lt;th&gt;size in&lt;br /&gt;
 minas&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;size as degrees &lt;br /&gt;
+         &lt;th&gt;size as degrees&lt;br /&gt;
 and minutes&lt;br /&gt;
 &lt;/th&gt;
@@ Line 436: / Line 436: @@
 &lt;br /&gt;
-Another helpful proponent of the mina is the accuracy and breadth of it's approximation to just intervals. Accordingly it is hardly necessary to express intervals in non-integer values of mina, something that arguably cannot be said of cents. &lt;a class="wiki_link" href="/2460edo"&gt;2460edo&lt;/a&gt; It is uniquely &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/consistent"&gt;consistent&lt;/a&gt; through to the &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/27-limit"&gt;27-limit&lt;/a&gt;, which is not very remarkable in itself (&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/388edo"&gt;388edo&lt;/a&gt; is the first such system), but what is remarkable is the degree of accuracy to which it represents the 27-limit intervals. It is also a &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&gt;zeta peak edo&lt;/a&gt; and has a lower 19-limit &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Tenney-Euclidean%20temperament%20measures#TE%20simple%20badness"&gt;relative error&lt;/a&gt; than any edo until &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/3395edo"&gt;3395&lt;/a&gt;, and a lower 23-limit relative error than any until &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/8269edo"&gt;8269&lt;/a&gt;. Also it has a lower 23-limit &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Tenney-Euclidean%20metrics#Logflat%20TE%20badness"&gt; TE loglfat badness&lt;/a&gt; than any smaller edo and less than any until &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/16808edo"&gt;16808&lt;/a&gt;.&lt;br /&gt;
+Another notable feature of the mina is the accuracy and breadth of it's approximation to just intervals. Accordingly it is hardly necessary to express intervals in non-integer values of mina, something that arguably cannot be said of cents. &lt;a class="wiki_link" href="/2460edo"&gt;2460edo&lt;/a&gt; It is uniquely &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/consistent"&gt;consistent&lt;/a&gt; through to the &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/27-limit"&gt;27-limit&lt;/a&gt;, which is not very remarkable in itself (&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/388edo"&gt;388edo&lt;/a&gt; is the first such system), but what is remarkable is the degree of accuracy to which it represents the 27-limit intervals. It is also a &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&gt;zeta peak edo&lt;/a&gt; and has a lower 19-limit &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Tenney-Euclidean%20temperament%20measures#TE%20simple%20badness"&gt;relative error&lt;/a&gt; than any edo until &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/3395edo"&gt;3395&lt;/a&gt;, and a lower 23-limit relative error than any until &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/8269edo"&gt;8269&lt;/a&gt;. Also it has a lower 23-limit &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Tenney-Euclidean%20metrics#Logflat%20TE%20badness"&gt; TE loglfat badness&lt;/a&gt; than any smaller edo and less than any until &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/16808edo"&gt;16808&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
 Below the intervals of the &lt;a class="wiki_link" href="/27-limit"&gt;27-limit&lt;/a&gt; &lt;a class="wiki_link" href="/tonality%20diamond"&gt;tonality diamond&lt;/a&gt; are tabulated, with the sizes listed in both &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;s and minas and expressed as degrees and minutes (rounded to the nearest minute). The value in minas, rounded to the nearest integer, can be found by applying the &lt;a class="wiki_link" href="/23-limit"&gt;23-limit&lt;/a&gt; &lt;a class="wiki_link" href="/patent%20val"&gt;patent val&lt;/a&gt; &amp;lt;2460 3899 5712 6906 8510 9103 10055 10450 11128| for 2460edo; this will not work for &lt;a class="wiki_link" href="/1200edo"&gt;1200edo&lt;/a&gt; and cents.&lt;br /&gt;