29edo: Difference between revisions
→Intervals and linear temperaments: Removed 7mus column |
updated ups/down notation, added color names to the comma list, general cleanup |
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<span style="display: block; text-align: right;">[[de:29edo]]</span> | <span style="display: block; text-align: right;">[[de:29edo]]</span>__FORCETOC__ | ||
=Theory= | |||
__FORCETOC__ | |||
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|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]]. | ||
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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. | ||
=Intervals | =Intervals= | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
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downminor 2nd | downminor 2nd | ||
| style="text-align:center;" | D | | style="text-align:center;" | ^D, vEb | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
|- | |- | ||
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| style="text-align:center;" | ^m2 | | style="text-align:center;" | ^m2 | ||
| style="text-align:center;" | upminor 2nd | | style="text-align:center;" | upminor 2nd | ||
| style="text-align:center;" | Eb | | style="text-align:center;" | ^Eb | ||
| style="text-align:center;" | [[Negri|Negri]]/[[Negril|Negril]] | | style="text-align:center;" | [[Negri|Negri]]/[[Negril|Negril]] | ||
|- | |- | ||
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| style="text-align:center;" | vM2 | | style="text-align:center;" | vM2 | ||
| style="text-align:center;" | downmajor 2nd | | style="text-align:center;" | downmajor 2nd | ||
| style="text-align:center;" | | | style="text-align:center;" | vE | ||
| style="text-align:center;" | [[Porcupine|Porcupine]]/[[Porky|Porky]]/[[Coendou|Coendou]] | | style="text-align:center;" | [[Porcupine|Porcupine]]/[[Porky|Porky]]/[[Coendou|Coendou]] | ||
|- | |- | ||
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downminor 3rd | downminor 3rd | ||
| style="text-align:center;" | E | | style="text-align:center;" | ^E, vF | ||
| style="text-align:center;" | [[Chromatic_pairs#Bridgetown|Bridgetown]]/[[Immunity|Immunity]] | | style="text-align:center;" | [[Chromatic_pairs#Bridgetown|Bridgetown]]/[[Immunity|Immunity]] | ||
|- | |- | ||
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| style="text-align:center;" | ^m3 | | style="text-align:center;" | ^m3 | ||
| style="text-align:center;" | upminor 3rd | | style="text-align:center;" | upminor 3rd | ||
| style="text-align:center;" | F | | style="text-align:center;" | ^F | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
|- | |- | ||
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| style="text-align:center;" | vM3 | | style="text-align:center;" | vM3 | ||
| style="text-align:center;" | downmajor 3rd | | style="text-align:center;" | downmajor 3rd | ||
| style="text-align:center;" | | | style="text-align:center;" | vF# | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
|- | |- | ||
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down 4th | down 4th | ||
| style="text-align:center;" | F# | | style="text-align:center;" | ^F#, vG | ||
| style="text-align:center;" | [[Ammonite|Ammonite]] | | style="text-align:center;" | [[Ammonite|Ammonite]] | ||
|- | |- | ||
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| style="text-align:center;" | ^4 | | style="text-align:center;" | ^4 | ||
| style="text-align:center;" | up 4th | | style="text-align:center;" | up 4th | ||
| style="text-align:center;" | G | | style="text-align:center;" | ^G | ||
| style="text-align:center;" | [[Wilsec|Wilsec]] | | style="text-align:center;" | [[Wilsec|Wilsec]] | ||
|- | |- | ||
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dim 5th | dim 5th | ||
| style="text-align:center;" | | | style="text-align:center;" | vG#, Ab | ||
| style="text-align:center;" | [[Tritonic|Tritonic]] | | style="text-align:center;" | [[Tritonic|Tritonic]] | ||
|- | |- | ||
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updim 5th | updim 5th | ||
| style="text-align:center;" | G#, Ab | | style="text-align:center;" | G#, ^Ab | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
|- | |- | ||
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| style="text-align:center;" | v5 | | style="text-align:center;" | v5 | ||
| style="text-align:center;" | down 5th | | style="text-align:center;" | down 5th | ||
| style="text-align:center;" | | | style="text-align:center;" | vA | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
|- | |- | ||
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downminor 6th | downminor 6th | ||
| style="text-align:center;" | A | | style="text-align:center;" | ^A, vBb | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
|- | |- | ||
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| style="text-align:center;" | ^m6 | | style="text-align:center;" | ^m6 | ||
| style="text-align:center;" | upminor 6th | | style="text-align:center;" | upminor 6th | ||
| style="text-align:center;" | Bb | | style="text-align:center;" | ^Bb | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
|- | |- | ||
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| style="text-align:center;" | vM6 | | style="text-align:center;" | vM6 | ||
| style="text-align:center;" | downmajor 6th | | style="text-align:center;" | downmajor 6th | ||
| style="text-align:center;" | | | style="text-align:center;" | vB | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
|- | |- | ||
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downminor 7th | downminor 7th | ||
| style="text-align:center;" | B | | style="text-align:center;" | ^B, vC | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
|- | |- | ||
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| style="text-align:center;" | ^m7 | | style="text-align:center;" | ^m7 | ||
| style="text-align:center;" | upminor 7th | | style="text-align:center;" | upminor 7th | ||
| style="text-align:center;" | C | | style="text-align:center;" | ^C | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
|- | |- | ||
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| style="text-align:center;" | vM7 | | style="text-align:center;" | vM7 | ||
| style="text-align:center;" | downmajor 7th | | style="text-align:center;" | downmajor 7th | ||
| style="text-align:center;" | | | style="text-align:center;" | vC# | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
|- | |- | ||
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down 8ve | down 8ve | ||
| style="text-align:center;" | C# | | style="text-align:center;" | ^C#, vD | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
|- | |- | ||
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| style="text-align:center;" | 6:7:9 | | style="text-align:center;" | 6:7:9 | ||
| style="text-align:center;" | 0-6-17 | | style="text-align:center;" | 0-6-17 | ||
| style="text-align:center;" | C | | style="text-align:center;" | C vEb G | ||
| style="text-align:center;" | | | style="text-align:center;" | Cvm | ||
| style="text-align:center;" | C downminor | | style="text-align:center;" | C downminor | ||
|- | |- | ||
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| style="text-align:center;" | 10:12:15 | | style="text-align:center;" | 10:12:15 | ||
| style="text-align:center;" | 0-8-17 | | style="text-align:center;" | 0-8-17 | ||
| style="text-align:center;" | C Eb | | style="text-align:center;" | C ^Eb G | ||
| style="text-align:center;" | C | | style="text-align:center;" | C^m | ||
| style="text-align:center;" | C upminor | | style="text-align:center;" | C upminor | ||
|- | |- | ||
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| style="text-align:center;" | 4:5:6 | | style="text-align:center;" | 4:5:6 | ||
| style="text-align:center;" | 0-9-17 | | style="text-align:center;" | 0-9-17 | ||
| style="text-align:center;" | C | | style="text-align:center;" | C vE G | ||
| style="text-align:center;" | | | style="text-align:center;" | Cv | ||
| style="text-align:center;" | C downmajor or C | | style="text-align:center;" | C downmajor or C down | ||
|- | |- | ||
| style="text-align:center;" | ru | | style="text-align:center;" | ru | ||
| style="text-align:center;" | 14:18:27 | | style="text-align:center;" | 14:18:27 | ||
| style="text-align:center;" | 0-11-17 | | style="text-align:center;" | 0-11-17 | ||
| style="text-align:center;" | C E | | style="text-align:center;" | C ^E G | ||
| style="text-align:center;" | C | | style="text-align:center;" | C^ | ||
| style="text-align:center;" | C upmajor or C | | style="text-align:center;" | C upmajor or C up | ||
|} | |} | ||
For a more complete list, see [[Ups_and_Downs_Notation#Chord names in other EDOs|Ups and Downs Notation - Chord names in other EDOs]]. | For a more complete list, see [[Ups_and_Downs_Notation#Chord names in other EDOs|Ups and Downs Notation - Chord names in other EDOs]]. | ||
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=Commas= | =Commas= | ||
29 EDO tempers out the following commas. (Note: This assumes the val < [http://tel.wikispaces.com/29_46_67_81_100_107 29 46 67 81 100 107] |, cent values rounded to 5 digits.) | 29 EDO tempers out the following [[commas]]. (Note: This assumes the val < [http://tel.wikispaces.com/29_46_67_81_100_107 29 46 67 81 100 107] |, cent values rounded to 5 digits.) | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
! | | ! | [[Ratio]] | ||
! | Monzo | ! | [[Monzo]] | ||
! | | ! | [[Cents]] | ||
![[Color notation/Temperament Names|Color Name]] | |||
! | Name 1 | ! | Name 1 | ||
! | Name 2 | ! | Name 2 | ||
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| |<nowiki> | -14 3 4 </nowiki>> | | |<nowiki> | -14 3 4 </nowiki>> | ||
| style="text-align:right;" | 51.120 | | style="text-align:right;" | 51.120 | ||
| style="text-align:center;" |Laquadyo | |||
| style="text-align:center;" | Negri Comma | | style="text-align:center;" | Negri Comma | ||
| style="text-align:center;" | Double Augmentation Diesis | | style="text-align:center;" | Double Augmentation Diesis | ||
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| |<nowiki> | 1 -5 3 </nowiki>> | | |<nowiki> | 1 -5 3 </nowiki>> | ||
| style="text-align:right;" | 49.166 | | style="text-align:right;" | 49.166 | ||
| style="text-align:center;" |Triyo | |||
| style="text-align:center;" | Maximal Diesis | | style="text-align:center;" | Maximal Diesis | ||
| style="text-align:center;" | Porcupine Comma | | style="text-align:center;" | Porcupine Comma | ||
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| |<nowiki> | -15 8 1 </nowiki>> | | |<nowiki> | -15 8 1 </nowiki>> | ||
| style="text-align:right;" | 1.9537 | | style="text-align:right;" | 1.9537 | ||
| style="text-align:center;" |Layo | |||
| style="text-align:center;" | Schisma | | style="text-align:center;" | Schisma | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
| Line 506: | Line 504: | ||
| |<nowiki> | -9 1 2 1 </nowiki>> | | |<nowiki> | -9 1 2 1 </nowiki>> | ||
| style="text-align:right;" | 43.408 | | style="text-align:right;" | 43.408 | ||
| style="text-align:center;" |Lazoyoyo | |||
| style="text-align:center;" | Avicennma | | style="text-align:center;" | Avicennma | ||
| style="text-align:center;" | Avicenna's Enharmonic Diesis | | style="text-align:center;" | Avicenna's Enharmonic Diesis | ||
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| |<nowiki> | -4 -1 0 2 </nowiki>> | | |<nowiki> | -4 -1 0 2 </nowiki>> | ||
| style="text-align:right;" | 35.697 | | style="text-align:right;" | 35.697 | ||
| style="text-align:center;" |Zozo | |||
| style="text-align:center;" | Slendro Diesis | | style="text-align:center;" | Slendro Diesis | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
| Line 518: | Line 518: | ||
| |<nowiki> | 1 -3 -2 3 </nowiki>> | | |<nowiki> | 1 -3 -2 3 </nowiki>> | ||
| style="text-align:right;" | 27.985 | | style="text-align:right;" | 27.985 | ||
| style="text-align:center;" |Trizo-agugu | |||
| style="text-align:center;" | Senga | | style="text-align:center;" | Senga | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
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| |<nowiki> | -9 3 -3 4 </nowiki>> | | |<nowiki> | -9 3 -3 4 </nowiki>> | ||
| style="text-align:right;" | 22.227 | | style="text-align:right;" | 22.227 | ||
| style="text-align:center;" |Laquadzo-atrigu | |||
| style="text-align:center;" | Squalentine | | style="text-align:center;" | Squalentine | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
| Line 530: | Line 532: | ||
| |<nowiki> | 0 -2 5 -3 </nowiki>> | | |<nowiki> | 0 -2 5 -3 </nowiki>> | ||
| style="text-align:right;" | 21.181 | | style="text-align:right;" | 21.181 | ||
| style="text-align:center;" |Triru-aquinyo | |||
| style="text-align:center;" | Gariboh | | style="text-align:center;" | Gariboh | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
| Line 536: | Line 539: | ||
| |<nowiki> | -4 1 -5 5 </nowiki>> | | |<nowiki> | -4 1 -5 5 </nowiki>> | ||
| style="text-align:right;" | 14.516 | | style="text-align:right;" | 14.516 | ||
| style="text-align:center;" |Quinzogu | |||
| style="text-align:center;" | Trimyna | | style="text-align:center;" | Trimyna | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
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| |<nowiki> | 5 -4 3 -2 </nowiki>> | | |<nowiki> | 5 -4 3 -2 </nowiki>> | ||
| style="text-align:right;" | 13.469 | | style="text-align:right;" | 13.469 | ||
| style="text-align:center;" |Rurutriyo | |||
| style="text-align:center;" | Octagar | | style="text-align:center;" | Octagar | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
| Line 548: | Line 553: | ||
| |<nowiki> | -5 2 2 -1 </nowiki>> | | |<nowiki> | -5 2 2 -1 </nowiki>> | ||
| style="text-align:right;" | 7.7115 | | style="text-align:right;" | 7.7115 | ||
| style="text-align:center;" |Ruyoyo | |||
| style="text-align:center;" | Septimal Kleisma | | style="text-align:center;" | Septimal Kleisma | ||
| style="text-align:center;" | Marvel Comma | | style="text-align:center;" | Marvel Comma | ||
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| |<nowiki> | 10 -6 1 -1 </nowiki>> | | |<nowiki> | 10 -6 1 -1 </nowiki>> | ||
| style="text-align:right;" | 5.7578 | | style="text-align:right;" | 5.7578 | ||
| style="text-align:center;" |Saruyo | |||
| style="text-align:center;" | Hemifamity | | style="text-align:center;" | Hemifamity | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
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| |<nowiki> | 25 -14 0 -1 </nowiki>> | | |<nowiki> | 25 -14 0 -1 </nowiki>> | ||
| style="text-align:right;" | 3.8041 | | style="text-align:right;" | 3.8041 | ||
| style="text-align:center;" |Sasaru | |||
| style="text-align:center;" | Garischisma | | style="text-align:center;" | Garischisma | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
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| |<nowiki> | 2 -2 2 0 -1 </nowiki>> | | |<nowiki> | 2 -2 2 0 -1 </nowiki>> | ||
| style="text-align:right;" | 17.399 | | style="text-align:right;" | 17.399 | ||
| style="text-align:center;" |Luyoyo | |||
| style="text-align:center;" | Ptolemisma | | style="text-align:center;" | Ptolemisma | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
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| |<nowiki> | -3 -1 -1 0 2 </nowiki>> | | |<nowiki> | -3 -1 -1 0 2 </nowiki>> | ||
| style="text-align:right;" | 14.367 | | style="text-align:right;" | 14.367 | ||
| style="text-align:center;" |Lologu | |||
| style="text-align:center;" | Biyatisma | | style="text-align:center;" | Biyatisma | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
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| |<nowiki> | 7 -4 0 1 -1 </nowiki>> | | |<nowiki> | 7 -4 0 1 -1 </nowiki>> | ||
| style="text-align:right;" | 9.6880 | | style="text-align:right;" | 9.6880 | ||
| style="text-align:center;" |Saluzo | |||
| style="text-align:center;" | Pentacircle | | style="text-align:center;" | Pentacircle | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
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| |<nowiki> | -3 2 -1 2 -1 </nowiki>> | | |<nowiki> | -3 2 -1 2 -1 </nowiki>> | ||
| style="text-align:right;" | 3.9302 | | style="text-align:right;" | 3.9302 | ||
| style="text-align:center;" |Luzozogu | |||
| style="text-align:center;" | Werckisma | | style="text-align:center;" | Werckisma | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
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| |<nowiki> | 5 -1 3 0 -3 </nowiki>> | | |<nowiki> | 5 -1 3 0 -3 </nowiki>> | ||
| style="text-align:right;" | 3.0323 | | style="text-align:right;" | 3.0323 | ||
| style="text-align:center;" |Trithuyo | |||
| style="text-align:center;" | Wizardharry | | style="text-align:center;" | Wizardharry | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
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| |<nowiki> | -3 4 -2 -2 2 </nowiki>> | | |<nowiki> | -3 4 -2 -2 2 </nowiki>> | ||
| style="text-align:right;" | 0.17665 | | style="text-align:right;" | 0.17665 | ||
| style="text-align:center;" |Bilorugu | |||
| style="text-align:center;" | Kalisma | | style="text-align:center;" | Kalisma | ||
| style="text-align:center;" | Gauss' Comma | | style="text-align:center;" | Gauss' Comma | ||
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| |<nowiki> | -1 -2 -1 1 0 1 </nowiki>> | | |<nowiki> | -1 -2 -1 1 0 1 </nowiki>> | ||
| style="text-align:right;" | 19.130 | | style="text-align:right;" | 19.130 | ||
| style="text-align:center;" |Thozogu | |||
| style="text-align:center;" | Superleap | | style="text-align:center;" | Superleap | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
|} | |} | ||
= | =Linear temperaments= | ||
[[List_of_29et_rank_two_temperaments_by_badness|List of 29et rank two temperaments by badness]] | |||
==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|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. | ||
| Line 622: | Line 639: | ||
If one can tolerate the tuning error (which is roughly equal to that of 12edo, albeit in the opposite direction for the 5- and 7-limits), this tetradecatonic scale is worth exploring. 29edo is often neglected since it falls so close to the much more popular and well-studied 31edo, but 29 does have its own advantages, and this is one of them. | If one can tolerate the tuning error (which is roughly equal to that of 12edo, albeit in the opposite direction for the 5- and 7-limits), this tetradecatonic scale is worth exploring. 29edo is often neglected since it falls so close to the much more popular and well-studied 31edo, but 29 does have its own advantages, and this is one of them. | ||
=Nicetone= | ==Nicetone== | ||
29edo is not a meantone system, but it could nonetheless be used as a basis for common-practice music if one considers the superfourth as a consonant, alternative type of fourth, and the 11:13:16 as an alternative type of consonant "doubly minor" triad. We can then use a diatonic scale such as 5435453 (which resembles Didymus' 5-limit JI diatonic scale, but with the syntonic comma being exaggerated in size). This scale has a very similar harmonic structure to a meantone diatonic scale, except that one of its minor triads is doubly-minor. | 29edo is not a meantone system, but it could nonetheless be used as a basis for common-practice music if one considers the superfourth as a consonant, alternative type of fourth, and the 11:13:16 as an alternative type of consonant "doubly minor" triad. We can then use a diatonic scale such as 5435453 (which resembles Didymus' 5-limit JI diatonic scale, but with the syntonic comma being exaggerated in size). This scale has a very similar harmonic structure to a meantone diatonic scale, except that one of its minor triads is doubly-minor. | ||
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[http://www.youtube.com/watch?v=uP2Z4Gy8lds Escala Tonal de 17 tonos - Charles Loli] | [http://www.youtube.com/watch?v=uP2Z4Gy8lds Escala Tonal de 17 tonos - Charles Loli] | ||
=Instruments= | |||
[http://www.microtonalismo.com/proyecto-xvii Guitar 29EDO from Peru - Charles Loli and Antonio Huamani] | |||
[[File:Loli-huamani-29edo-guitar-f_735065b21747.jpg|216x600px|none|thumb]] | |||
[http://www.microtonalismo.com/proyecto-xvii Bass 29EDO from Peru - Charles Loli and Antonio Huamani] | |||
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=Music= | =Music= | ||
[http://www.microtonalismo.com/el-teclado-29-edo Mp3 29EDO - Escala tonal de 17 notas]by [http://musicool.us/musicool/armonia.htm Charles Loli A.] | [http://www.microtonalismo.com/el-teclado-29-edo Mp3 29EDO - Escala tonal de 17 notas]by [http://musicool.us/musicool/armonia.htm Charles Loli A.] | ||
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[http://www.angelfire.com/mo/oljare/images/stranded.mid Stranded at Sea] by Mats Öljare | [http://www.angelfire.com/mo/oljare/images/stranded.mid Stranded at Sea] by Mats Öljare | ||
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