29edo: Difference between revisions

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=== Prime harmonics ===

[[3/1|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 [[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 [[5-limit]], [[49/48]] in the [[7-limit]], [[55/54]] in the [[11-limit]], and [[65/64]] in the [[13-limit]]. If using these approximations is desired, 29edo actually shines, and it can be used for such things as an alternative to [[19edo]] for [[negri]], as well as an alternative to [[22edo]] or [[15edo]] for [[porcupine]]. 29edo is also an [[oneirotonic]] tuning with generator 11\29, which generates [[ammonite]] temperament. ~~(''See [[regular temperament]] for more about what all this means and how to use it.'')~~

[[3/1|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 [[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 [[5-limit]], [[49/48]] in the [[7-limit]], [[55/54]] in the [[11-limit]], and [[65/64]] in the [[13-limit]]. If using these approximations is desired, 29edo actually shines, and it can be used for such things as an alternative to [[19edo]] for [[negri]], as well as an alternative to [[22edo]] or [[15edo]] for [[porcupine]]. 29edo is also an [[oneirotonic]] tuning with generator 11\29, which generates [[ammonite]] temperament.

=== Stacking fifths ===

Another possible use for 29edo is as an equally tempered para-pythagorean scale. Using its fifth as a generator leads to a variant of [[garibaldi temperament]] which is not very accurate but which has relatively low 13-limit complexity. However, it gives the POL2 generator for [[Subgroup temperaments #Edson (2.3.7/5.11/5.13/5 subgroup)|edson temperament]] with essentially perfect accuracy, only 0.034 cents sharp of it. ~~(''See [[regular temperament]] for more about what all this means and how to use it.'')~~

Another possible use for 29edo is as an equally tempered para-pythagorean scale. Using its fifth as a generator leads to a variant of [[garibaldi temperament]] which is not very accurate but which has relatively low 13-limit complexity. However, it gives the POL2 generator for [[Subgroup temperaments #Edson (2.3.7/5.11/5.13/5 subgroup)|edson temperament]] with essentially perfect accuracy, only 0.034 cents sharp of it.

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 [[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 [[petrmic triad]], a 13-limit [[Dyadic chord|essentially tempered dyadic chord]].

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== Regular temperament properties ==

~~''See [[regular temperament]] for more about what all this means and how to use it.''~~

{| class="wikitable center-4 center-5 center-6"

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@@ Line 23: / Line 23: @@
 === Prime harmonics ===
-[[3/1|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 [[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 [[5-limit]], [[49/48]] in the [[7-limit]], [[55/54]] in the [[11-limit]], and [[65/64]] in the [[13-limit]]. If using these approximations is desired, 29edo actually shines, and it can be used for such things as an alternative to [[19edo]] for [[negri]], as well as an alternative to [[22edo]] or [[15edo]] for [[porcupine]]. 29edo is also an [[oneirotonic]] tuning with generator 11\29, which generates [[ammonite]] temperament. (''See [[regular temperament]] for more about what all this means and how to use it.'')
+[[3/1|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 [[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 [[5-limit]], [[49/48]] in the [[7-limit]], [[55/54]] in the [[11-limit]], and [[65/64]] in the [[13-limit]]. If using these approximations is desired, 29edo actually shines, and it can be used for such things as an alternative to [[19edo]] for [[negri]], as well as an alternative to [[22edo]] or [[15edo]] for [[porcupine]]. 29edo is also an [[oneirotonic]] tuning with generator 11\29, which generates [[ammonite]] temperament.
 {{Harmonics in equal|29|columns=11}}
 === Stacking fifths ===
-Another possible use for 29edo is as an equally tempered para-pythagorean scale. Using its fifth as a generator leads to a variant of [[garibaldi temperament]] which is not very accurate but which has relatively low 13-limit complexity. However, it gives the POL2 generator for [[Subgroup temperaments #Edson (2.3.7/5.11/5.13/5 subgroup)|edson temperament]] with essentially perfect accuracy, only 0.034 cents sharp of it. (''See [[regular temperament]] for more about what all this means and how to use it.'')
+Another possible use for 29edo is as an equally tempered para-pythagorean scale. Using its fifth as a generator leads to a variant of [[garibaldi temperament]] which is not very accurate but which has relatively low 13-limit complexity. However, it gives the POL2 generator for [[Subgroup temperaments #Edson (2.3.7/5.11/5.13/5 subgroup)|edson temperament]] with essentially perfect accuracy, only 0.034 cents sharp of it.
 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 [[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 [[petrmic triad]], a 13-limit [[Dyadic chord|essentially tempered dyadic chord]].
@@ Line 507: / Line 507: @@
 == Regular temperament properties ==
-''See [[regular temperament]] for more about what all this means and how to use it.''
 {| class="wikitable center-4 center-5 center-6"
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