29edo: Difference between revisions
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== Theory == | == Theory == | ||
29 is the lowest edo which approximates the [[3/2]] just fifth more accurately than [[12edo]]: 3/2 = 701.955… cents; 17 degrees of 29edo = 703.448… cents. Since the fifth is slightly sharp, 29edo is a [[Erv Wilson's Linear Notations|positive temperament]] – a Parapythagorean instead of a | 29 is the lowest edo which approximates the [[3/2]] just fifth more accurately than [[12edo]]: 3/2 = 701.955… cents; 17 degrees of 29edo = 703.448… cents. Since the fifth is slightly sharp, 29edo is a [[Erv Wilson's Linear Notations|positive temperament]] – a [[Parapyth|Parapythagorean]] tuning instead of a meantone system. | ||
{| class="wikitable" | {| class="wikitable" | ||
| [[File:29edoSuperpythDiatonic.mp3]] [[:File:29edoSuperpythDiatonic.mp3|[File info]]] | | [[File:29edoSuperpythDiatonic.mp3]] [[:File:29edoSuperpythDiatonic.mp3|[File info]]] | ||
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| 12edo diatonic major scale and cadence, for comparison | | 12edo diatonic major scale and cadence, for comparison | ||
|} | |} | ||
The | |||
The third 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. | |||
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 [[Chromatic pairs #Edson|edson temperament]] with essentially perfect accuracy, only 0.034 cents sharp of 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 [[Chromatic pairs #Edson|edson temperament]] with essentially perfect accuracy, only 0.034 cents sharp of it. | ||
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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]]. 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 [[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 [[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 [[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]]. 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 [[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 [[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. | ||
29edo could be thought of as | 29edo could be thought of as the "opposite" of [[12edo]] in the 5-limit, since 5-limit intervals in 12edo and 29edo are tuned with almost exactly the same absolute errors, but in opposite directions. There are other ways in which they are counterparts (12 tempers out 50:49 but not 49:48; 29 does the opposite). Each supports a particularly good tonal framework (meantone[7] and nautilus[14], respectively). | ||
A more coincidental similarity is that just as the 12-tone scale is also a 1/2-tone scale (the whole tone being divided into 2 semitones), the 29-tone temperament may also be called 2/9-tone. This is because it has two different sizes of whole tone (4 and 5 steps wide, respectively). So the step size of 29edo may be called a 2/9-tone, just as 24edo's step size is called a quarter tone. | A more coincidental similarity is that just as the 12-tone scale is also a 1/2-tone scale (the whole tone being divided into 2 semitones), the 29-tone temperament may also be called 2/9-tone. This is because it has two different sizes of whole tone (4 and 5 steps wide, respectively). So the step size of 29edo may be called a 2/9-tone, just as 24edo's step size is called a quarter tone. | ||
| Line 50: | Line 51: | ||
| unison | | unison | ||
| D | | D | ||
|P1 | | P1 | ||
|unison | | unison | ||
|D | | D | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 342: | Line 343: | ||
| P8 | | P8 | ||
| 8ve | | 8ve | ||
| | | D | ||
|} | |} | ||
| Line 461: | Line 462: | ||
| {{monzo| 46 -29 }} | | {{monzo| 46 -29 }} | ||
| [{{val| 29 46 }}] | | [{{val| 29 46 }}] | ||
| | | −0.47 | ||
| 0.47 | | 0.47 | ||
| 1.14 | | 1.14 | ||
| Line 493: | Line 494: | ||
| 6.54 | | 6.54 | ||
|- | |- | ||
|2.3.5.7.11.13.19 | | 2.3.5.7.11.13.19 | ||
|49/48, 55/54, 65/64, 77/76, 100/99, 105/104 | | 49/48, 55/54, 65/64, 77/76, 100/99, 105/104 | ||
|[⟨29 46 67 81 100 107 123]] | | [⟨29 46 67 81 100 107 123]] | ||
| +2.91 | | +2.91 | ||
|2.55 | | 2.55 | ||
|6.16 | | 6.16 | ||
|- | |- | ||
|2.3.5.7.11.13.19.23 | | 2.3.5.7.11.13.19.23 | ||
|49/48, 55/54, 65/64, 70/69, 77/76, 100/99, 105/104 | | 49/48, 55/54, 65/64, 70/69, 77/76, 100/99, 105/104 | ||
|[⟨29 46 67 81 100 107 123 131]] | | [⟨29 46 67 81 100 107 123 131]] | ||
| +2.76 | | +2.76 | ||
|2.42 | | 2.42 | ||
|5.85 | | 5.85 | ||
|} | |} | ||
29et (29g val) has a lower relative error than any previous equal temperament in the [[23-limit]]. The next equal temperament doing better in this subgroup is [[46edo|46]]. | 29et (29g val) has a lower relative error than any previous equal temperament in the [[23-limit]]. The next equal temperament doing better in this subgroup is [[46edo|46]]. | ||
| Line 761: | Line 763: | ||
| [[Edson]] | | [[Edson]] | ||
|} | |} | ||
=== The Tetradecatonic System === | === The Tetradecatonic System === | ||
| Line 804: | Line 787: | ||
Nicetone scale 5435453 and cadence in 29edo | Nicetone scale 5435453 and cadence in 29edo | ||
== Scales == | == Scales == | ||
{{Main|List of MOS scales in 29edo}} | |||
* [[Bridgetown9]] | * [[Bridgetown9]] | ||
* [[Bridgetown14]] | * [[Bridgetown14]] | ||
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* [[Blackdye]] / [[syntonic dipentatonic]]: 1 4 3 4 1 4 3 4 1 4 | * [[Blackdye]] / [[syntonic dipentatonic]]: 1 4 3 4 1 4 3 4 1 4 | ||
* [[Blackville]] / [[Marvel dipentatonic]]: 2 3 4 3 2 3 4 3 2 3 | * [[Blackville]] / [[Marvel dipentatonic]]: 2 3 4 3 2 3 4 3 2 3 | ||
=== Well temperaments === | === Well temperaments === | ||