29edo: 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:~~jdfreivald~~|~~jdfreivald~~]] and made on <tt>2012-07-~~12 14~~:00:31 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-07-27 21:01:30 UTC</tt>.<br>

: The original revision id was <tt>~~352843704~~</tt>.<br>

: The original revision id was <tt>355130628</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>

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The 3 is the only harmonic, of the intelligibly low ones anyway, that 29edo approximates very closely, and it does so quite well. Nonetheless, and rather surprisingly, 29 is the smallest equal division which [[xenharmonic/consistent|consistent]]ly represents the 15 odd limit. It is able to do this since it has an accurate 3, and the 5, 7, 11 and 13, while not very accurate, are all tuned flatly. Hence it tempers out a succession of fairly large commas: 250/243 in the [[xenharmonic/5-limit|5-limit]], 49/48 in the [[xenharmonic/7-limit|7-limit]], 55/54 in the [[xenharmonic/11-limit|11-limit]], and 65/64 in the [[xenharmonic/13-limit|13-limit]]. If using these approximations is desired, 29edo actually shines, and it can be used for such things as an alternative to [[xenharmonic/19edo|19edo]] for [[xenharmonic/Marvel temperaments|negri]], as well as an alternative to [[xenharmonic/22edo|22edo]] or [[xenharmonic/15edo|15edo]] for porcupine. For those who enjoy the bizarre character of Father temperament, 29edo can also be used to support that temperament, if one imagines 11\29 is approximating both 5/4 and 4/3 (ignoring the better approximations at 10\29 and 12\29, respectively).

Another possible use for 29edo is as an equally tempered para-pythagorean scale. Using its fifth as a generator leads to a variant of [[xenharmonic/Schismatic family|garibaldi temperament]] which is not very accurate but which has relatively low 13-limit complexity.

Another possible use for 29edo is as an equally tempered para-pythagorean scale. Using its fifth as a generator leads to a variant of [[xenharmonic/Schismatic family|garibaldi temperament]] which is not very accurate but which has relatively low 13-limit complexity. However, it gives the POL2 generator for [[Chromatic pairs#Edson|edson temperaament]] with essentially perfect accuracy, only 0.034 cents sharp of it.

~~Moreover~~, it represents the 2.3.11/5.13/5 subgroup to very high accuracy, and 2.3.7/5.11/5.13/5 to a lesser but still good accuracy, and so can be used with this subgroup, which is liberally supplied with chords such as the 1-11/7-13/7 (7:11:13) chord, the [[xenharmonic/The Archipelago|barbados triad]] 1-13/10-3/2 (10:13:15), the minor barbados triad 1-15/13-3/2, the 1-14/11-3/2 (22:28:33) triad, the 1-13/11-3/2 triad (22:26:33), and the [[xenharmonic/petrmic triad|petrmic triad]], a 13-limit [[xenharmonic/Dyadic chord|essentially tempered dyadic chord]]. 29 tempers out 352/351, 676/675 and 4000/3993 from the 2.3.11/5.13/5 subgroup, and in addition 196/195 and 364/363 from the 2.3.7/5.11/5.13/5 subgroup, so we have various relationships from the tempering, such as the fact that the 1-13/11-3/2 chord and the 1-14/11-3/2 chord are inverses of each other, a major-minor pairing. A larger subgroup containing both of these subgroups is the [[xenharmonic/k*N subgroups|3*29 subgroup]] 2.3.125.175.275.325; on this subgroup 29 tunes the same as 87, and the commas of 29 on this subgroup are the same as the 13-limit commas of 87. Still another subgroup of interest is the [[xenharmonic/k*N subgroups|2*29 subgroup]] 2.3.25.35.55.65.85; on this subgroup 29 tunes the same as 58 and has the same 17-limit commas.

Edson is a 2.3.7/5.11/5.13/5 subgroup temperament, and 29 it represents the 2.3.11/5.13/5 subgroup to very high accuracy, and the 2.3.7/5.11/5.13/5 to a lesser but still good accuracy, and so can be used with this subgroup, which is liberally supplied with chords such as the 1-11/7-13/7 (7:11:13) chord, the [[xenharmonic/The Archipelago|barbados triad]] 1-13/10-3/2 (10:13:15), the minor barbados triad 1-15/13-3/2, the 1-14/11-3/2 (22:28:33) triad, the 1-13/11-3/2 triad (22:26:33), and the [[xenharmonic/petrmic triad|petrmic triad]], a 13-limit [[xenharmonic/Dyadic chord|essentially tempered dyadic chord]]. 29 tempers out 352/351, 676/675 and 4000/3993 from the 2.3.11/5.13/5 subgroup, and in addition 196/195 and 364/363 from the 2.3.7/5.11/5.13/5 subgroup, so we have various relationships from the tempering, such as the fact that the 1-13/11-3/2 chord and the 1-14/11-3/2 chord are inverses of each other, a major-minor pairing. A larger subgroup containing both of these subgroups is the [[xenharmonic/k*N subgroups|3*29 subgroup]] 2.3.125.175.275.325; on this subgroup 29 tunes the same as 87, and the commas of 29 on this subgroup are the same as the 13-limit commas of 87. Still another subgroup of interest is the [[xenharmonic/k*N subgroups|2*29 subgroup]] 2.3.25.35.55.65.85; on this subgroup 29 tunes the same as 58 and has the same 17-limit commas.

=Intervals and linear temperaments=

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The 3 is the only harmonic, of the intelligibly low ones anyway, that 29edo approximates very closely, and it does so quite well. Nonetheless, and rather surprisingly, 29 is the smallest equal division which <a class="wiki_link" href="http://xenharmonic.wikispaces.com/consistent">consistent</a>ly represents the 15 odd limit. It is able to do this since it has an accurate 3, and the 5, 7, 11 and 13, while not very accurate, are all tuned flatly. Hence it tempers out a succession of fairly large commas: 250/243 in the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/5-limit">5-limit</a>, 49/48 in the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/7-limit">7-limit</a>, 55/54 in the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/11-limit">11-limit</a>, and 65/64 in the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/13-limit">13-limit</a>. If using these approximations is desired, 29edo actually shines, and it can be used for such things as an alternative to <a class="wiki_link" href="http://xenharmonic.wikispaces.com/19edo">19edo</a> for <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Marvel%20temperaments">negri</a>, as well as an alternative to <a class="wiki_link" href="http://xenharmonic.wikispaces.com/22edo">22edo</a> or <a class="wiki_link" href="http://xenharmonic.wikispaces.com/15edo">15edo</a> for porcupine. For those who enjoy the bizarre character of Father temperament, 29edo can also be used to support that temperament, if one imagines 11\29 is approximating both 5/4 and 4/3 (ignoring the better approximations at 10\29 and 12\29, respectively).<br />

<br />

Another possible use for 29edo is as an equally tempered para-pythagorean scale. Using its fifth as a generator leads to a variant of <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Schismatic%20family">garibaldi temperament</a> which is not very accurate but which has relatively low 13-limit complexity.<br />

Another possible use for 29edo is as an equally tempered para-pythagorean scale. Using its fifth as a generator leads to a variant of <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Schismatic%20family">garibaldi temperament</a> which is not very accurate but which has relatively low 13-limit complexity. However, it gives the POL2 generator for <a class="wiki_link" href="/Chromatic%20pairs#Edson">edson temperaament</a> with essentially perfect accuracy, only 0.034 cents sharp of it.<br />

<br />

~~Moreover~~, it represents the 2.3.11/5.13/5 subgroup to very high accuracy, and 2.3.7/5.11/5.13/5 to a lesser but still good accuracy, and so can be used with this subgroup, which is liberally supplied with chords such as the 1-11/7-13/7 (7:11:13) chord, the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/The%20Archipelago">barbados triad</a> 1-13/10-3/2 (10:13:15), the minor barbados triad 1-15/13-3/2, the 1-14/11-3/2 (22:28:33) triad, the 1-13/11-3/2 triad (22:26:33), and the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/petrmic%20triad">petrmic triad</a>, a 13-limit <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Dyadic%20chord">essentially tempered dyadic chord</a>. 29 tempers out 352/351, 676/675 and 4000/3993 from the 2.3.11/5.13/5 subgroup, and in addition 196/195 and 364/363 from the 2.3.7/5.11/5.13/5 subgroup, so we have various relationships from the tempering, such as the fact that the 1-13/11-3/2 chord and the 1-14/11-3/2 chord are inverses of each other, a major-minor pairing. A larger subgroup containing both of these subgroups is the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/k%2AN%20subgroups">3*29 subgroup</a> 2.3.125.175.275.325; on this subgroup 29 tunes the same as 87, and the commas of 29 on this subgroup are the same as the 13-limit commas of 87. Still another subgroup of interest is the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/k%2AN%20subgroups">2*29 subgroup</a> 2.3.25.35.55.65.85; on this subgroup 29 tunes the same as 58 and has the same 17-limit commas.<br />

Edson is a 2.3.7/5.11/5.13/5 subgroup temperament, and 29 it represents the 2.3.11/5.13/5 subgroup to very high accuracy, and the 2.3.7/5.11/5.13/5 to a lesser but still good accuracy, and so can be used with this subgroup, which is liberally supplied with chords such as the 1-11/7-13/7 (7:11:13) chord, the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/The%20Archipelago">barbados triad</a> 1-13/10-3/2 (10:13:15), the minor barbados triad 1-15/13-3/2, the 1-14/11-3/2 (22:28:33) triad, the 1-13/11-3/2 triad (22:26:33), and the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/petrmic%20triad">petrmic triad</a>, a 13-limit <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Dyadic%20chord">essentially tempered dyadic chord</a>. 29 tempers out 352/351, 676/675 and 4000/3993 from the 2.3.11/5.13/5 subgroup, and in addition 196/195 and 364/363 from the 2.3.7/5.11/5.13/5 subgroup, so we have various relationships from the tempering, such as the fact that the 1-13/11-3/2 chord and the 1-14/11-3/2 chord are inverses of each other, a major-minor pairing. A larger subgroup containing both of these subgroups is the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/k%2AN%20subgroups">3*29 subgroup</a> 2.3.125.175.275.325; on this subgroup 29 tunes the same as 87, and the commas of 29 on this subgroup are the same as the 13-limit commas of 87. Still another subgroup of interest is the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/k%2AN%20subgroups">2*29 subgroup</a> 2.3.25.35.55.65.85; on this subgroup 29 tunes the same as 58 and has the same 17-limit commas.<br />

<br />

<h1 id="toc1"><a name="Intervals and linear temperaments"></a>Intervals and linear temperaments</h1>

@@ 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:jdfreivald|jdfreivald]] and made on <tt>2012-07-12 14:00:31 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-07-27 21:01:30 UTC</tt>.<br>
-: The original revision id was <tt>352843704</tt>.<br>
+: The original revision id was <tt>355130628</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 18: / Line 18: @@
 The 3 is the only harmonic, of the intelligibly low ones anyway, that 29edo approximates very closely, and it does so quite well. Nonetheless, and rather surprisingly, 29 is the smallest equal division which [[xenharmonic/consistent|consistent]]ly represents the 15 odd limit. It is able to do this since it has an accurate 3, and the 5, 7, 11 and 13, while not very accurate, are all tuned flatly. Hence it tempers out a succession of fairly large commas: 250/243 in the [[xenharmonic/5-limit|5-limit]], 49/48 in the [[xenharmonic/7-limit|7-limit]], 55/54 in the [[xenharmonic/11-limit|11-limit]], and 65/64 in the [[xenharmonic/13-limit|13-limit]]. If using these approximations is desired, 29edo actually shines, and it can be used for such things as an alternative to [[xenharmonic/19edo|19edo]] for [[xenharmonic/Marvel temperaments|negri]], as well as an alternative to [[xenharmonic/22edo|22edo]] or [[xenharmonic/15edo|15edo]] for porcupine. For those who enjoy the bizarre character of Father temperament, 29edo can also be used to support that temperament, if one imagines 11\29 is approximating both 5/4 and 4/3 (ignoring the better approximations at 10\29 and 12\29, respectively).
-Another possible use for 29edo is as an equally tempered para-pythagorean scale. Using its fifth as a generator leads to a variant of [[xenharmonic/Schismatic family|garibaldi temperament]] which is not very accurate but which has relatively low 13-limit complexity.
+Another possible use for 29edo is as an equally tempered para-pythagorean scale. Using its fifth as a generator leads to a variant of [[xenharmonic/Schismatic family|garibaldi temperament]] which is not very accurate but which has relatively low 13-limit complexity. However, it gives the POL2 generator for [[Chromatic pairs#Edson|edson temperaament]] with essentially perfect accuracy, only 0.034 cents sharp of it.
-Moreover, it represents the 2.3.11/5.13/5 subgroup to very high accuracy, and 2.3.7/5.11/5.13/5 to a lesser but still good accuracy, and so can be used with this subgroup, which is liberally supplied with chords such as the 1-11/7-13/7 (7:11:13) chord, the [[xenharmonic/The Archipelago|barbados triad]] 1-13/10-3/2 (10:13:15), the minor barbados triad 1-15/13-3/2, the 1-14/11-3/2 (22:28:33) triad, the 1-13/11-3/2 triad (22:26:33), and the [[xenharmonic/petrmic triad|petrmic triad]], a 13-limit [[xenharmonic/Dyadic chord|essentially tempered dyadic chord]]. 29 tempers out 352/351, 676/675 and 4000/3993 from the 2.3.11/5.13/5 subgroup, and in addition 196/195 and 364/363 from the 2.3.7/5.11/5.13/5 subgroup, so we have various relationships from the tempering, such as the fact that the 1-13/11-3/2 chord and the 1-14/11-3/2 chord are inverses of each other, a major-minor pairing. A larger subgroup containing both of these subgroups is the [[xenharmonic/k*N subgroups|3*29 subgroup]] 2.3.125.175.275.325; on this subgroup 29 tunes the same as 87, and the commas of 29 on this subgroup are the same as the 13-limit commas of 87. Still another subgroup of interest is the [[xenharmonic/k*N subgroups|2*29 subgroup]] 2.3.25.35.55.65.85; on this subgroup 29 tunes the same as 58 and has the same 17-limit commas.
+Edson is a 2.3.7/5.11/5.13/5 subgroup temperament, and 29 it represents the 2.3.11/5.13/5 subgroup to very high accuracy, and the 2.3.7/5.11/5.13/5 to a lesser but still good accuracy, and so can be used with this subgroup, which is liberally supplied with chords such as the 1-11/7-13/7 (7:11:13) chord, the [[xenharmonic/The Archipelago|barbados triad]] 1-13/10-3/2 (10:13:15), the minor barbados triad 1-15/13-3/2, the 1-14/11-3/2 (22:28:33) triad, the 1-13/11-3/2 triad (22:26:33), and the [[xenharmonic/petrmic triad|petrmic triad]], a 13-limit [[xenharmonic/Dyadic chord|essentially tempered dyadic chord]]. 29 tempers out 352/351, 676/675 and 4000/3993 from the 2.3.11/5.13/5 subgroup, and in addition 196/195 and 364/363 from the 2.3.7/5.11/5.13/5 subgroup, so we have various relationships from the tempering, such as the fact that the 1-13/11-3/2 chord and the 1-14/11-3/2 chord are inverses of each other, a major-minor pairing. A larger subgroup containing both of these subgroups is the [[xenharmonic/k*N subgroups|3*29 subgroup]] 2.3.125.175.275.325; on this subgroup 29 tunes the same as 87, and the commas of 29 on this subgroup are the same as the 13-limit commas of 87. Still another subgroup of interest is the [[xenharmonic/k*N subgroups|2*29 subgroup]] 2.3.25.35.55.65.85; on this subgroup 29 tunes the same as 58 and has the same 17-limit commas.
 =Intervals and linear temperaments=
@@ Line 114: / Line 114: @@
 The 3 is the only harmonic, of the intelligibly low ones anyway, that 29edo approximates very closely, and it does so quite well. Nonetheless, and rather surprisingly, 29 is the smallest equal division which &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/consistent"&gt;consistent&lt;/a&gt;ly represents the 15 odd limit. It is able to do this since it has an accurate 3, and the 5, 7, 11 and 13, while not very accurate, are all tuned flatly. Hence it tempers out a succession of fairly large commas: 250/243 in the &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/5-limit"&gt;5-limit&lt;/a&gt;, 49/48 in the &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/7-limit"&gt;7-limit&lt;/a&gt;, 55/54 in the &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/11-limit"&gt;11-limit&lt;/a&gt;, and 65/64 in the &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/13-limit"&gt;13-limit&lt;/a&gt;. If using these approximations is desired, 29edo actually shines, and it can be used for such things as an alternative to &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/19edo"&gt;19edo&lt;/a&gt; for &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Marvel%20temperaments"&gt;negri&lt;/a&gt;, as well as an alternative to &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/22edo"&gt;22edo&lt;/a&gt; or &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/15edo"&gt;15edo&lt;/a&gt; for porcupine. For those who enjoy the bizarre character of Father temperament, 29edo can also be used to support that temperament, if one imagines 11\29 is approximating both 5/4 and 4/3 (ignoring the better approximations at 10\29 and 12\29, respectively).&lt;br /&gt;
 &lt;br /&gt;
-Another possible use for 29edo is as an equally tempered para-pythagorean scale. Using its fifth as a generator leads to a variant of &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Schismatic%20family"&gt;garibaldi temperament&lt;/a&gt; which is not very accurate but which has relatively low 13-limit complexity.&lt;br /&gt;
+Another possible use for 29edo is as an equally tempered para-pythagorean scale. Using its fifth as a generator leads to a variant of &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Schismatic%20family"&gt;garibaldi temperament&lt;/a&gt; which is not very accurate but which has relatively low 13-limit complexity. However, it gives the POL2 generator for &lt;a class="wiki_link" href="/Chromatic%20pairs#Edson"&gt;edson temperaament&lt;/a&gt; with essentially perfect accuracy, only 0.034 cents sharp of it.&lt;br /&gt;
 &lt;br /&gt;
-Moreover, it represents the 2.3.11/5.13/5 subgroup to very high accuracy, and 2.3.7/5.11/5.13/5 to a lesser but still good accuracy, and so can be used with this subgroup, which is liberally supplied with chords such as the 1-11/7-13/7 (7:11:13) chord, the &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/The%20Archipelago"&gt;barbados triad&lt;/a&gt; 1-13/10-3/2 (10:13:15), the minor barbados triad 1-15/13-3/2, the 1-14/11-3/2 (22:28:33) triad, the 1-13/11-3/2 triad (22:26:33), and the &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/petrmic%20triad"&gt;petrmic triad&lt;/a&gt;, a 13-limit &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Dyadic%20chord"&gt;essentially tempered dyadic chord&lt;/a&gt;. 29 tempers out 352/351, 676/675 and 4000/3993 from the 2.3.11/5.13/5 subgroup, and in addition 196/195 and 364/363 from the 2.3.7/5.11/5.13/5 subgroup, so we have various relationships from the tempering, such as the fact that the 1-13/11-3/2 chord and the 1-14/11-3/2 chord are inverses of each other, a major-minor pairing. A larger subgroup containing both of these subgroups is the &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/k%2AN%20subgroups"&gt;3*29 subgroup&lt;/a&gt; 2.3.125.175.275.325; on this subgroup 29 tunes the same as 87, and the commas of 29 on this subgroup are the same as the 13-limit commas of 87. Still another subgroup of interest is the &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/k%2AN%20subgroups"&gt;2*29 subgroup&lt;/a&gt; 2.3.25.35.55.65.85; on this subgroup 29 tunes the same as 58 and has the same 17-limit commas.&lt;br /&gt;
+Edson is a 2.3.7/5.11/5.13/5 subgroup temperament, and 29 it represents the 2.3.11/5.13/5 subgroup to very high accuracy, and the 2.3.7/5.11/5.13/5 to a lesser but still good accuracy, and so can be used with this subgroup, which is liberally supplied with chords such as the 1-11/7-13/7 (7:11:13) chord, the &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/The%20Archipelago"&gt;barbados triad&lt;/a&gt; 1-13/10-3/2 (10:13:15), the minor barbados triad 1-15/13-3/2, the 1-14/11-3/2 (22:28:33) triad, the 1-13/11-3/2 triad (22:26:33), and the &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/petrmic%20triad"&gt;petrmic triad&lt;/a&gt;, a 13-limit &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Dyadic%20chord"&gt;essentially tempered dyadic chord&lt;/a&gt;. 29 tempers out 352/351, 676/675 and 4000/3993 from the 2.3.11/5.13/5 subgroup, and in addition 196/195 and 364/363 from the 2.3.7/5.11/5.13/5 subgroup, so we have various relationships from the tempering, such as the fact that the 1-13/11-3/2 chord and the 1-14/11-3/2 chord are inverses of each other, a major-minor pairing. A larger subgroup containing both of these subgroups is the &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/k%2AN%20subgroups"&gt;3*29 subgroup&lt;/a&gt; 2.3.125.175.275.325; on this subgroup 29 tunes the same as 87, and the commas of 29 on this subgroup are the same as the 13-limit commas of 87. Still another subgroup of interest is the &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/k%2AN%20subgroups"&gt;2*29 subgroup&lt;/a&gt; 2.3.25.35.55.65.85; on this subgroup 29 tunes the same as 58 and has the same 17-limit commas.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Intervals and linear temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Intervals and linear temperaments&lt;/h1&gt;