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| <span style="display: block; text-align: right;">[[de:29edo]]</span> | | <span style="display: block; text-align: right;">[[de:29edo]]</span> |
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| __FORCETOC__
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| == Theory == | | == Theory == |
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| 29edo divides the 2:1 [[Octave|octave]] into 29 equal steps of approximately 41.37931 [[cent|cents]]. It is the 10th [[prime_numbers|prime]] edo, following [[23edo|23edo]] and coming before [[31edo|31edo]]. | | 29edo divides the 2:1 [[octave]] into 29 equal steps of approximately 41.37931 [[cent|cents]]. It is the 10th [[prime_numbers|prime]] edo, following [[23edo]] and coming before [[31edo]]. |
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| 29 is the lowest edo which approximates the [[3/2|3:2]] just fifth more accurately than [[12edo|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 Superpythagorean instead of a Meantone system. | | 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 Superpythagorean instead of a Meantone system. |
| {| class="wikitable" | | {| class="wikitable" |
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| |12edo diatonic major scale and cadence, for comparison | | |12edo diatonic major scale and cadence, for comparison |
| |} | | |} |
| 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 [[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 [[5-limit|5-limit]], 49/48 in the [[7-limit|7-limit]], 55/54 in the [[11-limit|11-limit]], and 65/64 in the [[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 [[19edo|19edo]] for [[Marvel_temperaments|negri]], as well as an alternative to [[22edo|22edo]] or [[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). | | 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 [[consistent|consistently]] 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]]. For those who enjoy the bizarre character of [[Father|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). |
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| Another possible use for 29edo is as an equally tempered para-pythagorean scale. Using its fifth as a generator leads to a variant of [[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 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 [[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 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|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. |
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| 29edo could be thought of as 12edo's "twin", since the 5-limit error for both is almost exactly the same, but in the opposite direction. 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). | | 29edo could be thought of as 12edo's "twin", since the 5-limit error for both is almost exactly the same, but in the opposite direction. 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). |
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| == Intervals == | | == Intervals == |
| {| class="wikitable" | | {| class="wikitable center-all" |
| |- | | |- |
| ! | Degree | | ! Degree |
| ! | Cents | | ! Cents |
| ! | Approx. ratios of the [[15-limit|15-limit]] | | ! Approx. ratios of the [[13-limit]] |
| ! colspan="3" | [[Ups_and_Downs_Notation|Ups and downs]] notation | | ! colspan="3" | [[Ups and Downs Notation]] |
| ! | Generator for temperaments | | ! Generator for temperaments |
| |- | | |- |
| | style="text-align:center;" | 0
| | | 0 |
| | style="text-align:center;" | 0.000
| | | 0.000 |
| | style="text-align:center;" | 1/1
| | | 1/1 |
| | style="text-align:center;" | P1
| | | P1 |
| | style="text-align:center;" | unison
| | | unison |
| | style="text-align:center;" | D
| | | D |
| | style="text-align:center;" |
| | | |
| |- | | |- |
| | style="text-align:center;" | 1
| | | 1 |
| | style="text-align:center;" | 41.379
| | | 41.379 |
| | style="text-align:center;" | 25/24~33/32~56/55~81/80
| | | 25/24, 33/32, 56/55, 81/80 |
| | style="text-align:center;" | ^1, vm2
| | | ^1, vm2 |
| | style="text-align:center;" | up unison,
| | | up unison,<br>downminor 2nd |
| | | | ^D, vEb |
| downminor 2nd | | | |
| | style="text-align:center;" | ^D, vEb
| |
| | style="text-align:center;" |
| |
| |- | | |- |
| | style="text-align:center;" | 2
| | | 2 |
| | style="text-align:center;" | 82.759
| | | 82.759 |
| | style="text-align:center;" | 21/20
| | | 21/20 |
| | style="text-align:center;" | m2
| | | m2 |
| | style="text-align:center;" | minor 2nd
| | | minor 2nd |
| | style="text-align:center;" | Eb
| | | Eb |
| | style="text-align:center;" | [[Nautilus|Nautilus]]
| | | [[Nautilus]] |
| |- | | |- |
| | style="text-align:center;" | 3
| | | 3 |
| | style="text-align:center;" | 124.138
| | | 124.138 |
| | style="text-align:center;" | 16/15, 15/14, 14/13, 13/12
| | | 16/15, 15/14, 14/13, 13/12 |
| | style="text-align:center;" | ^m2
| | | ^m2 |
| | style="text-align:center;" | upminor 2nd
| | | upminor 2nd |
| | style="text-align:center;" | ^Eb
| | | ^Eb |
| | style="text-align:center;" | [[Negri|Negri]]/[[Negril|Negril]]
| | | [[Negri]]/[[Negril]] |
| |- | | |- |
| | style="text-align:center;" | 4
| | | 4 |
| | style="text-align:center;" | 165.517
| | | 165.517 |
| | style="text-align:center;" | 12/11, 11/10
| | | 12/11, 11/10 |
| | style="text-align:center;" | vM2
| | | vM2 |
| | style="text-align:center;" | downmajor 2nd
| | | downmajor 2nd |
| | style="text-align:center;" | vE
| | | vE |
| | style="text-align:center;" | [[Porcupine|Porcupine]]/[[Porky|Porky]]/[[Coendou|Coendou]]
| | | [[Porcupine]]/[[Porky]]/[[Coendou]] |
| |- | | |- |
| | style="text-align:center;" | 5
| | | 5 |
| | style="text-align:center;" | 206.897
| | | 206.897 |
| | style="text-align:center;" | 9/8
| | | 9/8 |
| | style="text-align:center;" | M2
| | | M2 |
| | style="text-align:center;" | major 2nd
| | | major 2nd |
| | style="text-align:center;" | E
| | | E |
| | style="text-align:center;" |
| | | |
| |- | | |- |
| | style="text-align:center;" | 6
| | | 6 |
| | style="text-align:center;" | 248.276
| | | 248.276 |
| | style="text-align:center;" | 8/7, 7/6, 15/13
| | | 8/7, 7/6, 15/13 |
| | style="text-align:center;" | ^M2, vm3
| | | ^M2, vm3 |
| | style="text-align:center;" | upmajor 2nd,
| | | upmajor 2nd,<br>downminor 3rd |
| | | | ^E, vF |
| downminor 3rd | | | [[Chromatic_pairs#Bridgetown|Bridgetown]]/[[Immunity]] |
| | style="text-align:center;" | ^E, vF
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| | style="text-align:center;" | [[Chromatic_pairs#Bridgetown|Bridgetown]]/[[Immunity|Immunity]]
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| |- | | |- |
| | style="text-align:center;" | 7· | | | ·7 |
| | style="text-align:center;" | 289.655
| | | 289.655 |
| | style="text-align:center;" | 13/11
| | | 13/11 |
| | style="text-align:center;" | m3
| | | m3 |
| | style="text-align:center;" | minor 3rd
| | | minor 3rd |
| | style="text-align:center;" | F
| | | F |
| | style="text-align:center;" |
| | | |
| |- | | |- |
| | style="text-align:center;" | 8
| | | 8 |
| | style="text-align:center;" | 331.0345
| | | 331.0345 |
| | style="text-align:center;" | 6/5, 11/9
| | | 6/5, 11/9 |
| | style="text-align:center;" | ^m3
| | | ^m3 |
| | style="text-align:center;" | upminor 3rd
| | | upminor 3rd |
| | style="text-align:center;" | ^F
| | | ^F |
| | style="text-align:center;" |
| | | |
| |- | | |- |
| | style="text-align:center;" | 9
| | | 9 |
| | style="text-align:center;" | 372.414
| | | 372.414 |
| | style="text-align:center;" | 5/4, 16/13
| | | 5/4, 16/13 |
| | style="text-align:center;" | vM3
| | | vM3 |
| | style="text-align:center;" | downmajor 3rd
| | | downmajor 3rd |
| | style="text-align:center;" | vF#
| | | vF# |
| | style="text-align:center;" |
| | | |
| |- | | |- |
| | style="text-align:center;" | 10
| | | 10 |
| | style="text-align:center;" | 413.793
| | | 413.793 |
| | style="text-align:center;" | 14/11
| | | 14/11 |
| | style="text-align:center;" | M3
| | | M3 |
| | style="text-align:center;" | major 3rd
| | | major 3rd |
| | style="text-align:center;" | F#
| | | F# |
| | style="text-align:center;" | [[Roman|Roman]]
| | | [[Roman]] |
| |- | | |- |
| | style="text-align:center;" | 11
| | | 11 |
| | style="text-align:center;" | 455.172
| | | 455.172 |
| | style="text-align:center;" | 9/7, 13/10
| | | 9/7, 13/10 |
| | style="text-align:center;" | ^M3, v4
| | | ^M3, v4 |
| | style="text-align:center;" | upmajor 3rd
| | | upmajor 3rd<br>down 4th |
| | | | ^F#, vG |
| down 4th | | | [[Ammonite]] |
| | style="text-align:center;" | ^F#, vG
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| | style="text-align:center;" | [[Ammonite|Ammonite]]
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| |- | | |- |
| | style="text-align:center;" | 12· | | | ·12 |
| | style="text-align:center;" | 496.552
| | | 496.552 |
| | style="text-align:center;" | 4/3
| | | 4/3 |
| | style="text-align:center;" | P4
| | | P4 |
| | style="text-align:center;" | 4th
| | | 4th |
| | style="text-align:center;" | G
| | | G |
| | style="text-align:center;" | [[cassandra|Cassandra]] [[Chromatic_pairs#Edson|Edson]] [[Chromatic_pairs#Pepperoni|Pepperoni]]
| | | [[Cassandra]] [[Chromatic_pairs#Edson|Edson]] [[Chromatic_pairs#Pepperoni|Pepperoni]] |
| |- | | |- |
| | style="text-align:center;" | 13
| | | 13 |
| | style="text-align:center;" | 537.931
| | | 537.931 |
| | style="text-align:center;" | 11/8, 15/11
| | | 11/8, 15/11 |
| | style="text-align:center;" | ^4
| | | ^4 |
| | style="text-align:center;" | up 4th
| | | up 4th |
| | style="text-align:center;" | ^G
| | | ^G |
| | style="text-align:center;" | [[Wilsec|Wilsec]]
| | | [[Wilsec]] |
| |- | | |- |
| | style="text-align:center;" | 14
| | | 14 |
| | style="text-align:center;" | 579.31
| | | 579.31 |
| | style="text-align:center;" | 7/5, 18/13
| | | 7/5, 18/13 |
| | style="text-align:center;" | vA4, d5
| | | vA4, d5 |
| | style="text-align:center;" | downaug 4th,
| | | downaug 4th,<br>dim 5th |
| | | | vG#, Ab |
| dim 5th | | | [[Tritonic]] |
| | style="text-align:center;" | vG#, Ab
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| | style="text-align:center;" | [[Tritonic|Tritonic]]
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| |- | | |- |
| | style="text-align:center;" | 15
| | | 15 |
| | style="text-align:center;" | 620.69
| | | 620.69 |
| | style="text-align:center;" | 10/7, 13/9
| | | 10/7, 13/9 |
| | style="text-align:center;" | A4, ^d5
| | | A4, ^d5 |
| | style="text-align:center;" | aug 4th,
| | | aug 4th,<br>updim 5th |
| | | | G#, ^Ab |
| updim 5th | | | |
| | style="text-align:center;" | G#, ^Ab
| |
| | style="text-align:center;" |
| |
| |- | | |- |
| | style="text-align:center;" | 16
| | | 16 |
| | style="text-align:center;" | 662.069
| | | 662.069 |
| | style="text-align:center;" | 16/11, 22/15
| | | 16/11, 22/15 |
| | style="text-align:center;" | v5
| | | v5 |
| | style="text-align:center;" | down 5th
| | | down 5th |
| | style="text-align:center;" | vA
| | | vA |
| | style="text-align:center;" |
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| |- | | |- |
| | style="text-align:center;" | 17· | | | ·17 |
| | style="text-align:center;" | 703.448
| | | 703.448 |
| | style="text-align:center;" | 3/2
| | | 3/2 |
| | style="text-align:center;" | P5
| | | P5 |
| | style="text-align:center;" | 5th
| | | 5th |
| | style="text-align:center;" | A
| | | A |
| | style="text-align:center;" |
| | | |
| |- | | |- |
| | style="text-align:center;" | 18
| | | 18 |
| | style="text-align:center;" | 744.828
| | | 744.828 |
| | style="text-align:center;" | 14/9, 20/13
| | | 14/9, 20/13 |
| | style="text-align:center;" | ^5, vm6
| | | ^5, vm6 |
| | style="text-align:center;" | up 5th,
| | | up 5th,<br>downminor 6th |
| | | | ^A, vBb |
| downminor 6th | | | |
| | style="text-align:center;" | ^A, vBb
| |
| | style="text-align:center;" |
| |
| |- | | |- |
| | style="text-align:center;" | 19
| | | 19 |
| | style="text-align:center;" | 786.207
| | | 786.207 |
| | style="text-align:center;" | 11/7
| | | 11/7 |
| | style="text-align:center;" | m6
| | | m6 |
| | style="text-align:center;" | minor 6th
| | | minor 6th |
| | style="text-align:center;" | Bb
| | | Bb |
| | style="text-align:center;" |
| | | |
| |- | | |- |
| | style="text-align:center;" | 20
| | | 20 |
| | style="text-align:center;" | 827.586
| | | 827.586 |
| | style="text-align:center;" | 8/5, 13/8
| | | 8/5, 13/8 |
| | style="text-align:center;" | ^m6
| | | ^m6 |
| | style="text-align:center;" | upminor 6th
| | | upminor 6th |
| | style="text-align:center;" | ^Bb
| | | ^Bb |
| | style="text-align:center;" |
| | | |
| |- | | |- |
| | style="text-align:center;" | 21
| | | 21 |
| | style="text-align:center;" | 868.9655
| | | 868.9655 |
| | style="text-align:center;" | 5/3, 18/11
| | | 5/3, 18/11 |
| | style="text-align:center;" | vM6
| | | vM6 |
| | style="text-align:center;" | downmajor 6th
| | | downmajor 6th |
| | style="text-align:center;" | vB
| | | vB |
| | style="text-align:center;" |
| | | |
| |- | | |- |
| | style="text-align:center;" | 22· | | | ·22 |
| | style="text-align:center;" | 910.345
| | | 910.345 |
| | style="text-align:center;" | 22/13
| | | 22/13 |
| | style="text-align:center;" | M6
| | | M6 |
| | style="text-align:center;" | major 6th
| | | major 6th |
| | style="text-align:center;" | B
| | | B |
| | style="text-align:center;" |
| | | |
| |- | | |- |
| | style="text-align:center;" | 23
| | | 23 |
| | style="text-align:center;" | 951.724
| | | 951.724 |
| | style="text-align:center;" | 7/4, 12/7, 26/15
| | | 7/4, 12/7, 26/15 |
| | style="text-align:center;" | ^M6, vm7
| | | ^M6, vm7 |
| | style="text-align:center;" | upmajor 6th,
| | | upmajor 6th,<br>downminor 7th |
| | | | ^B, vC |
| downminor 7th | | | |
| | style="text-align:center;" | ^B, vC
| |
| | style="text-align:center;" |
| |
| |- | | |- |
| | style="text-align:center;" | 24
| | | 24 |
| | style="text-align:center;" | 993.103
| | | 993.103 |
| | style="text-align:center;" | 16/9
| | | 16/9 |
| | style="text-align:center;" | m7
| | | m7 |
| | style="text-align:center;" | minor 7th
| | | minor 7th |
| | style="text-align:center;" | C
| | | C |
| | style="text-align:center;" |
| | | |
| |- | | |- |
| | style="text-align:center;" | 25
| | | 25 |
| | style="text-align:center;" | 1034.483
| | | 1034.483 |
| | style="text-align:center;" | 11/6, 20/11
| | | 11/6, 20/11 |
| | style="text-align:center;" | ^m7
| | | ^m7 |
| | style="text-align:center;" | upminor 7th
| | | upminor 7th |
| | style="text-align:center;" | ^C
| | | ^C |
| | style="text-align:center;" |
| | | |
| |- | | |- |
| | style="text-align:center;" | 26
| | | 26 |
| | style="text-align:center;" | 1075.862
| | | 1075.862 |
| | style="text-align:center;" | 15/8, 28/15, 13/7, 24/13
| | | 15/8, 28/15, 13/7, 24/13 |
| | style="text-align:center;" | vM7
| | | vM7 |
| | style="text-align:center;" | downmajor 7th
| | | downmajor 7th |
| | style="text-align:center;" | vC#
| | | vC# |
| | style="text-align:center;" |
| | | |
| |- | | |- |
| | style="text-align:center;" | 27
| | | 27 |
| | style="text-align:center;" | 1117.241
| | | 1117.241 |
| | style="text-align:center;" | 40/21
| | | 40/21 |
| | style="text-align:center;" | M7
| | | M7 |
| | style="text-align:center;" | major 7th
| | | major 7th |
| | style="text-align:center;" | C#
| | | C# |
| | style="text-align:center;" |
| | | |
| |- | | |- |
| | style="text-align:center;" | 28
| | | 28 |
| | style="text-align:center;" | 1158.621
| | | 1158.621 |
| | style="text-align:center;" | 48/25~64/33~55/28 ~160/81
| | | 48/25~64/33~55/28 ~160/81 |
| | style="text-align:center;" | ^M7, v8
| | | ^M7, v8 |
| | style="text-align:center;" | upmajor 7th,
| | | upmajor 7th,<br>down 8ve |
| | | | ^C#, vD |
| down 8ve | | | |
| | style="text-align:center;" | ^C#, vD
| |
| | style="text-align:center;" |
| |
| |- | | |- |
| | style="text-align:center;" | 29
| | | 29 |
| | style="text-align:center;" | 1200
| | | 1200 |
| | style="text-align:center;" | 2/1
| | | 2/1 |
| | style="text-align:center;" | P8
| | | P8 |
| | style="text-align:center;" | 8ve
| | | 8ve |
| | style="text-align:center;" | D
| | | D |
| | style="text-align:center;" |
| | | |
| |} | | |} |
| See also: [[29edo_solfege|29edo solfege]] | | ''See also: [[29edo solfege]]'' |
|
| |
|
| Combining ups and downs notation with [[Kite's_color_notation|color notation]], qualities can be loosely associated with colors: | | Combining ups and downs notation with [[Kite's_color_notation|color notation]], qualities can be loosely associated with colors: |
|
| |
|
| {| class="wikitable" | | {| class="wikitable center-all" |
| |- | | |- |
| ! | quality | | ! quality |
| ! | [[Kite's color notation|color]] | | ! [[Kite's color notation|color]] |
| ! | monzo format | | ! monzo format |
| ! | examples | | ! examples |
| |- | | |- |
| | style="text-align:center;" | downminor
| | | downminor |
| | style="text-align:center;" | zo
| | | zo |
| | style="text-align:center;" | {a, b, 0, 1}
| | | {a, b, 0, 1} |
| | style="text-align:center;" | 7/6, 7/4
| | | 7/6, 7/4 |
| |- | | |- |
| | style="text-align:center;" | minor
| | | minor |
| | style="text-align:center;" | fourthward wa
| | | fourthward wa |
| | style="text-align:center;" | {a, b}, b < -1
| | | {a, b}, b < -1 |
| | style="text-align:center;" | 32/27, 16/9
| | | 32/27, 16/9 |
| |- | | |- |
| | style="text-align:center;" | upminor
| | | upminor |
| | style="text-align:center;" | gu
| | | gu |
| | style="text-align:center;" | {a, b, -1}
| | | {a, b, -1} |
| | style="text-align:center;" | 6/5, 9/5
| | | 6/5, 9/5 |
| |- | | |- |
| | style="text-align:center;" | "
| | | " |
| | style="text-align:center;" | ilo
| | | ilo |
| | style="text-align:center;" | {a, b, 0, 0, 1}
| | | {a, b, 0, 0, 1} |
| | style="text-align:center;" | 11/9, 11/6
| | | 11/9, 11/6 |
| |- | | |- |
| | style="text-align:center;" | downmajor
| | | downmajor |
| | style="text-align:center;" | lu
| | | lu |
| | style="text-align:center;" | {a, b, 0, 0, -1}
| | | {a, b, 0, 0, -1} |
| | style="text-align:center;" | 12/11, 18/11
| | | 12/11, 18/11 |
| |- | | |- |
| | style="text-align:center;" | "
| | | " |
| | style="text-align:center;" | yo
| | | yo |
| | style="text-align:center;" | {a, b, 1}
| | | {a, b, 1} |
| | style="text-align:center;" | 5/4, 5/3
| | | 5/4, 5/3 |
| |- | | |- |
| | style="text-align:center;" | major
| | | major |
| | style="text-align:center;" | fifthward wa
| | | fifthward wa |
| | style="text-align:center;" | {a, b}, b > 1
| | | {a, b}, b > 1 |
| | style="text-align:center;" | 9/8, 27/16
| | | 9/8, 27/16 |
| |- | | |- |
| | style="text-align:center;" | upmajor
| | | upmajor |
| | style="text-align:center;" | ru
| | | ru |
| | style="text-align:center;" | {a, b, 0, -1}
| | | {a, b, 0, -1} |
| | style="text-align:center;" | 9/7, 12/7
| | | 9/7, 12/7 |
| |} | | |} |
| All 29edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Here are the zo, gu, yo and ru triads: | | All 29edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Here are the zo, gu, yo and ru triads: |
|
| |
|
| {| class="wikitable" | | {| class="wikitable center-all" |
| |- | | |- |
| ! | [[Kite's color notation|color of the 3rd]] | | ! [[Kite's color notation|color of the 3rd]] |
| ! | JI chord | | ! JI chord |
| ! | notes as edosteps | | ! notes as edosteps |
| ! | notes of C chord | | ! notes of C chord |
| ! | written name | | ! written name |
| ! | spoken name | | ! spoken name |
| |- | | |- |
| | style="text-align:center;" | zo
| | | zo |
| | style="text-align:center;" | 6:7:9
| | | 6:7:9 |
| | style="text-align:center;" | 0-6-17
| | | 0-6-17 |
| | style="text-align:center;" | C vEb G
| | | C vEb G |
| | style="text-align:center;" | Cvm
| | | Cvm |
| | style="text-align:center;" | C downminor
| | | C downminor |
| |- | | |- |
| | style="text-align:center;" | gu
| | | gu |
| | style="text-align:center;" | 10:12:15
| | | 10:12:15 |
| | style="text-align:center;" | 0-8-17
| | | 0-8-17 |
| | style="text-align:center;" | C ^Eb G
| | | C ^Eb G |
| | style="text-align:center;" | C^m
| | | C^m |
| | style="text-align:center;" | C upminor
| | | C upminor |
| |- | | |- |
| | style="text-align:center;" | yo
| | | yo |
| | style="text-align:center;" | 4:5:6
| | | 4:5:6 |
| | style="text-align:center;" | 0-9-17
| | | 0-9-17 |
| | style="text-align:center;" | C vE G
| | | C vE G |
| | style="text-align:center;" | Cv
| | | Cv |
| | style="text-align:center;" | C downmajor or C down
| | | C downmajor or C down |
| |- | | |- |
| | style="text-align:center;" | ru
| | | ru |
| | style="text-align:center;" | 14:18:27
| | | 14:18:27 |
| | style="text-align:center;" | 0-11-17
| | | 0-11-17 |
| | style="text-align:center;" | C ^E G
| | | C ^E G |
| | style="text-align:center;" | C^
| | | C^ |
| | style="text-align:center;" | C upmajor or C up
| | | C upmajor or C up |
| |} | | |} |
| For a more complete list, see [[Ups and Downs Notation#Chords and Chord Progressions|Ups and Downs Notation - Chords and Chord Progressions]]. | | For a more complete list, see [[Ups and Downs Notation #Chords and Chord Progressions]]. |
|
| |
|
| {| class="wikitable" | | {| class="wikitable" |
| |- | | |- |
| | | [[File:29edothumb.png|alt=29edothumb.png|29edothumb.png]]
| | | [[File:29edothumb.png|alt=29edothumb.png|29edothumb.png]] |
| |- | | |- |
| | | this example in Sagittal notation shows 29-edo as a fifth-tone system.
| | | this example in Sagittal notation shows 29-edo as a fifth-tone system. |
| |} | | |} |
|
| |
|
| === Selected just intervals by error === | | === Selected just intervals by error === |
| The following table shows how [[Just-24|some prominent just intervals]] are represented in 29edo (ordered by absolute error). | | The following table shows how [[15-odd-limit intervals]] are represented in 29edo. Prime harmonics are in '''bold'''. |
|
| |
|
| {| class="wikitable" | | {| class="wikitable center-all" |
| |- | | |- |
| | | '''Interval, complement'''
| | ! '''Interval, complement''' |
| | | '''Error (abs., in [[cent|cents]])'''
| | ! '''Error (abs,[[cent|¢]])''' |
| |- | | |- |
| | style="text-align:center;" | [[13/11|13/11]], [[22/13|22/13]]
| | | [[13/11]], [[22/13]] |
| | style="text-align:center;" | 0.445
| | | 0.445 |
| |- | | |- |
| | style="text-align:center;" | [[11/10|11/10]], [[20/11|20/11]]
| | | [[11/10]], [[20/11]] |
| | style="text-align:center;" | 0.513
| | | 0.513 |
| |- | | |- |
| | style="text-align:center;" | [[15/13|15/13]], [[26/15|26/15]]
| | | [[15/13]], [[26/15]] |
| | style="text-align:center;" | 0.535
| | | 0.535 |
| |- | | |- |
| | style="text-align:center;" | [[13/10|13/10]], [[20/13|20/13]]
| | | [[13/10]], [[20/13]] |
| | style="text-align:center;" | 0.958
| | | 0.958 |
| |- | | |- |
| | style="text-align:center;" | [[15/11|15/11]], [[22/15|22/15]]
| | | [[15/11]], [[22/15]] |
| | style="text-align:center;" | 0.980
| | | 0.980 |
| |- | | |- |
| | style="text-align:center;" | '''[[4/3|4/3]], [[3/2|3/2]]'''
| | | '''[[4/3]], [[3/2]]''' |
| | style="text-align:center;" | '''1.493'''
| | | '''1.493''' |
| |- | | |- |
| | style="text-align:center;" | [[9/8|9/8]], [[16/9|16/9]]
| | | [[9/8]], [[16/9]] |
| | style="text-align:center;" | 2.987
| | | 2.987 |
| |- | | |- |
| | style="text-align:center;" | [[7/5|7/5]], [[10/7|10/7]]
| | | [[7/5]], [[10/7]] |
| | style="text-align:center;" | 3.202
| | | 3.202 |
| |- | | |- |
| | style="text-align:center;" | [[14/11|14/11]], [[11/7|11/7]]
| | | [[14/11]], [[11/7]] |
| | style="text-align:center;" | 3.715
| | | 3.715 |
| |- | | |- |
| | style="text-align:center;" | [[14/13|14/13]], [[13/7|13/7]]
| | | [[14/13]], [[13/7]] |
| | style="text-align:center;" | 4.160
| | | 4.160 |
| |- | | |- |
| | style="text-align:center;" | [[15/14|15/14]], [[28/15|28/15]]
| | | [[15/14]], [[28/15]] |
| | style="text-align:center;" | 4.695
| | | 4.695 |
| |- | | |- |
| | style="text-align:center;" | [[16/15|16/15]], [[15/8|15/8]]
| | | [[16/15]], [[15/8]] |
| | style="text-align:center;" | 12.407
| | | 12.407 |
| |- | | |- |
| | style="text-align:center;" | '''[[16/13|16/13]], [[13/8|13/8]]'''
| | | '''[[16/13]], [[13/8]]''' |
| | style="text-align:center;" | '''12.941'''
| | | '''12.941''' |
| |- | | |- |
| | style="text-align:center;" | '''[[11/8|11/8]], [[16/11|16/11]]'''
| | | '''[[11/8]], [[16/11]]''' |
| | style="text-align:center;" | '''13.387'''
| | | '''13.387''' |
| |- | | |- |
| | style="text-align:center;" | '''[[5/4|5/4]], [[8/5|8/5]]'''
| | | '''[[5/4]], [[8/5]]''' |
| | style="text-align:center;" | '''13.900'''
| | | '''13.900''' |
| |- | | |- |
| | style="text-align:center;" | [[13/12|13/12]], [[24/13|24/13]]
| | | [[13/12]], [[24/13]] |
| | style="text-align:center;" | 14.435
| | | 14.435 |
| |- | | |- |
| | style="text-align:center;" | [[12/11|12/11]], [[11/6|11/6]]
| | | [[12/11]], [[11/6]] |
| | style="text-align:center;" | 14.880
| | | 14.880 |
| |- | | |- |
| | style="text-align:center;" | [[6/5|6/5]], [[5/3|5/3]]
| | | [[6/5]], [[5/3]] |
| | style="text-align:center;" | 15.393
| | | 15.393 |
| |- | | |- |
| | style="text-align:center;" | [[18/13|18/13]], [[13/9|13/9]]
| | | [[18/13]], [[13/9]] |
| | style="text-align:center;" | 15.928
| | | 15.928 |
| |- | | |- |
| | style="text-align:center;" | [[11/9|11/9]], [[18/11|18/11]]
| | | [[11/9]], [[18/11]] |
| | style="text-align:center;" | 16.373
| | | 16.373 |
| |- | | |- |
| | style="text-align:center;" | [[10/9|10/9]], [[9/5|9/5]]
| | | [[10/9]], [[9/5]] |
| | style="text-align:center;" | 16.886
| | | 16.886 |
| |- | | |- |
| | style="text-align:center;" | '''[[8/7|8/7]], [[7/4|7/4]]'''
| | | '''[[8/7]], [[7/4]]''' |
| | style="text-align:center;" | '''17.102'''
| | | '''17.102''' |
| |- | | |- |
| | style="text-align:center;" | [[7/6|7/6]], [[12/7|12/7]]
| | | [[7/6]], [[12/7]] |
| | style="text-align:center;" | 18.595
| | | 18.595 |
| |- | | |- |
| | style="text-align:center;" | [[9/7|9/7]], [[14/9|14/9]]
| | | [[9/7]], [[14/9]] |
| | style="text-align:center;" | 20.088
| | | 20.088 |
| |} | | |} |
|
| |
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| Line 475: |
Line 458: |
| {| class="wikitable" | | {| class="wikitable" |
| |- | | |- |
| ! | [[Ratio]] | | ! [[Ratio]] |
| ! | [[Monzo]] | | ! [[Monzo]] |
| ! | [[Cents]] | | ! [[Cents]] |
| ![[Color notation/Temperament Names|Color Name]] | | ! [[Color notation/Temperament Names|Color Name]] |
| ! | Name 1 | | ! Name 1 |
| ! | Name 2 | | ! Name 2 |
| |- | | |- |
| | style="text-align:center;" | 16875/16384 | | | style="text-align:center;" | 16875/16384 |
| Line 624: |
Line 607: |
|
| |
|
| == Linear temperaments == | | == Linear temperaments == |
| [[List_of_29et_rank_two_temperaments_by_badness|List of 29et rank two temperaments by badness]] | | * [[List of 29et rank two temperaments by badness]] |
|
| |
|
| === The Tetradecatonic System === | | === The Tetradecatonic System === |
| A variant of porcupine supported in 29edo is [[Nautilus|nautilus]], which splits the porcupine generator in half (tempering out 50:49 in the process), thus resulting in a different mapping for 7 than standard porcupine. Nautilus also extends to the 13-limit much more easily than does standard porcupine. | | A variant of porcupine supported in 29edo is [[nautilus]], which splits the porcupine generator in half (tempering out 50:49 in the process), thus resulting in a different mapping for 7 than standard porcupine. Nautilus also extends to the 13-limit much more easily than does standard porcupine. |
|
| |
|
| The MOS nautilus[14] contains both "even" tetrads (approximating 4:5:6:7 or its inverse) as well as "odd" tetrads (approximating the "Bohlen-Pierce-like" chord 9:11:13:15, or its inverse). Both types are recognizable and consonant, if somewhat heavily tempered. Moreover, one of the four types of tetrads may be built on '''each''' scale degree of nautilus[14], thus there are as many chords as there are notes, so nautilus[14] has a "circulating" quality to it with as much freedom of modulation as possible. To be exact, there are 4 "major-even", 4 "minor-even", 3 "major-odd", and 3 "minor-odd" chords. | | The MOS nautilus[14] contains both "even" tetrads (approximating 4:5:6:7 or its inverse) as well as "odd" tetrads (approximating the "Bohlen-Pierce-like" chord 9:11:13:15, or its inverse). Both types are recognizable and consonant, if somewhat heavily tempered. Moreover, one of the four types of tetrads may be built on '''each''' scale degree of nautilus[14], thus there are as many chords as there are notes, so nautilus[14] has a "circulating" quality to it with as much freedom of modulation as possible. To be exact, there are 4 "major-even", 4 "minor-even", 3 "major-odd", and 3 "minor-odd" chords. |
| Line 677: |
Line 660: |
| * [http://micro.soonlabel.com/gene_ward_smith/Others/Igs/NautilusReverie.mp3 Nautilus Reverie] by [[IgliashonJones|Igliashon Calvin Jones-Coolidge]] | | * [http://micro.soonlabel.com/gene_ward_smith/Others/Igs/NautilusReverie.mp3 Nautilus Reverie] by [[IgliashonJones|Igliashon Calvin Jones-Coolidge]] |
| * [http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Howling%20of%20the%20Holy.mp3 Howling of the Holy] by [[IgliashonJones|Igliashon Jones]] | | * [http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Howling%20of%20the%20Holy.mp3 Howling of the Holy] by [[IgliashonJones|Igliashon Jones]] |
| * [http://micro.soonlabel.com/tuning-survey/daily20111026-bridgetown-14.mp3 Route 14 in Bridgetown] by [[Chris_Vaisvil|Chris Vaisvil]] | | * [http://micro.soonlabel.com/tuning-survey/daily20111026-bridgetown-14.mp3 Route 14 in Bridgetown] by [[Chris Vaisvil]] |
| * [http://www.angelfire.com/mo/oljare/images/crowning.mid The Crowning Song] by Mats Öljare {{dead link}} | | * [http://www.angelfire.com/mo/oljare/images/crowning.mid The Crowning Song] by Mats Öljare {{dead link}} |
| * [http://www.angelfire.com/mo/oljare/images/ninedays.mid Nine Days Later] by Mats Öljare {{dead link}} | | * [http://www.angelfire.com/mo/oljare/images/ninedays.mid Nine Days Later] by Mats Öljare {{dead link}} |