les échelles nonoctaviantes qui ne se calculent pas à partir de la division du ton

 

ce qui ne signifie pas qu'elles ne sont pas multiples

4eme exploration : du connu vers l'inconnu (3)

EXPLORATION III

 

L'exploration des échelles octaviantes tonalisées nous a montré l'existence d'échelles nonoctaviantes au sein même du tempérament octaviant. Pouvait-on s'y attendre ?

Ces découvertes nous apprennent que la nonoctaviation existe aussi dans la division d'autres intervalles (inclus dans l'octaviation tempérée) comme la quarte, la quinte, la sixte mineure, la sixte majeure, la septième mineure et d'autres intervalles plus grands que l'octave. Dans les 53 échelles nonoctaviantes découvertes précédemment (à partir de la division du ton : voir abaque); 5 divisent la quarte, 4 divisent la quinte, 6 divisent la sixte mineure, 3 divisent la sixte majeure, 4 divisent la septième mineure, 3 divisent la septième majeure, 2 divisent une octave et seconde mineure, etc. En rapport avec tous ces intervalles tonaux divisés, nous allons dénombrer les autres échelles nonoctaviantes qui s'y dissimulent, explorer les fréquences de temps (les échelles de hauteurs sont des échelles de temps), pour découvrir d'autres échelles nonoctaviantes qui ne se calculent pas à partir de la division du ton.

 

]
. [Pierre Barbaud en nommant l'échelle tempéré de 12 1/2 tons Z12 l'ensemble de 12 entiers relatifs, nous ouvre une voie : les 8 échelles de base tonale 1/16e de ton, 1/15e de ton, 1/14e de ton, 1/13e de ton, 1/12e de ton, 1/11e de ton, 1/10e de ton et 1/9e de ton se nomment dans la continuité de Pierre Barbaud : Z96, Z90, Z84, Z78, Z72, Z66, Z60 et Z54. Ces appellations sont liées à leur nombre de divisions dans l'octave.]

. [Nous nous posons des limites arbitraires du Champ Scalaire pour les intervalles constituants les échelles nonoctaviantes à :

. entre 12,5 cents & 240 cents
. entre Z96 & Z5
. entre 1/16e de ton & 6/5e de ton
. entre 96√2 & 5√2 = 21/96 & 21/5
]

[

 

 

DIVISIONS des CYCLES

HORAIRES

1. Localisation de tous les degrés nonoctaviants
à partir des cycles multiples de Z12
cycles connus des échelles nonoctaviantes issus de la division du ton

 

.

 

cycle 4te

nombre
de degrés dans le cycle
Rapports en cent

notes par octave

en ton remarques


téléchargements pour
écoute

3 3√1,3.. = 1,10064 500/3 = 166,66.. 1200/166,66.. = 7,2 10/12e de ton élément des 53 Scala tuning script .scl
4 4√1,3.. = 1,07457 500/4 = 125 cents 1200/125 = 9,6 5/8e de ton élément des 53 Scala tuning script .scl
5 5√1,3.. = 1,05922 500/5 = 100 cents 1200/100 = 12 1/2 ton octaviation Z12  
6 6√1,3.. = 1,04912 500/6 = 83,33.. 1200/83,33.. = 14,4 5/12e de ton élément des 53 Scala tuning script .scl
7 7√1,3.. = 1,04195 500/7 = 71,428... 1200/71,428... = 16,8 5/14e de ton élément des 53 Scala tuning script .scl
8 8√1,3.. = 1,03661 500/8 = 62,5 1200/62,5 = 19,2 5/16e de ton élément des 53 Scala tuning script .scl
9 9√1,3.. = 1,03248 500/9 = 55,55.. 1200/55,55.. = 21,6     Scala tuning script .scl
10 10√1,3.. = 1,02919 500/10 = 50 1200/50 = 24 1/4 de ton octaviation Z24  
11 11√1,3.. = 1,0265 500/11 = 45,45.. 1200/45,45.. = 26,4     Scala tuning script .scl
12 12√1,3.. = 1,02426 500/12 = 41,66.. 1200/41,66.. = 28,8     Scala tuning script .scl
13 13√1,3.. = 1,02238 500/13 = 38,4615... 1200/38,4615... = 31,2     Scala tuning script .scl
14 14√1,3.. = 1,02076 500/14 = 35,714... 1200/35,714... = 33,6     Scala tuning script .scl
15 15√1,3.. = 1,01936 500/15 = 33,33.. 1200/33,33.. = 36 1/6e de ton octaviation Z36  
16 16√1,3.. = 1,01814 500/16 = 31,25 1200/31,25 = 27,428...     Scala tuning script .scl
17 17√1,3.. = 1,01707 500/17 = 29,4117... 1200/29,4117... = 29,1428...     Scala tuning script .scl
18 18√1,3.. = 1,01611 500/18 = 27,77.. 1200/27,77.. = 30,857...     Scala tuning script .scl
19 19√1,3.. = 1,01526 500/19 = 26,3157... 1200/26,3157... = 32,57...     Scala tuning script .scl
20 20√1,3.. = 1,01449 500/20 = 25 1200/25 = 48 1/8e de ton octaviation Z48  
21 21√1,3.. = 1,01379 500/21 = 23,809... 1200/23,809... = 36     Scala tuning script .scl
22 22√1,3.. = 1,01316 500/22 = 22,72.. 1200/22,72.. = 37,714...     Scala tuning script .scl
23 23√1,3.. = 1,01259 500/23 = 21,739... 1200/21,739... = 39,4285...     Scala tuning script .scl
24 24√1,3.. = 1,01206 500/24 = 20,833.. 1200/20,833.. = 41,1428...     Scala tuning script .scl
25 25√1,3.. = 1,01157 500/25 = 20 1200/20 = 60 1/10e de ton octaviation Z60  
26 26√1,3.. = 1,01113 500/26 = 19,23... 1200/19,23... = 62,4     Scala tuning script .scl
27 27√1,3.. = 1,01071 500/27 = 18,518... 1200/18,518... = 64,8     Scala tuning script .scl
28 28√1,3.. = 1,01033 500/28 = 17,857... 1200/17,857... = 67,2     Scala tuning script .scl
29 29√1,3.. = 1,00997 500/29 = 17,2413... 1200/17,2413... = 69,6     Scala tuning script .scl
30 30√1,3.. = 1,00964 500/30 = 16,66.. 1200/16,66.. = 72 1/12e de ton octaviation Z72  
31 31√1,3.. = 1,00932 500/31 = 16,129... 1200/16,129... = 74,4
    Scala tuning script .scl
32 32√1,3.. = 1,00903 500/32 = 15,625 1200/15,625 = 76,8     Scala tuning script .scl
33 33√1,3.. = 1,00876 500/33 = 15,15.. 1200/15,15.. = 79,2     Scala tuning script .scl
34 34√1,3.. = 1,0085 500/34 = 14,7058... 1200/14,7058... = 81,6     Scala tuning script .scl
35 35√1,3.. = 1,00825 500/35 = 14,2857... 1200/14,2857... = 84 1/14e de ton octaviation Z84  
36 36√1,3.. = 1,00802 500/36 = 13,88.. 1200/13,88.. = 86,4     Scala tuning script .scl
37 37√1,3.. = 1,00781 500/37 = 13,513... 1200/13,513... = 88,8     Scala tuning script .scl
38 38√1,3.. = 1,0076 500/38 = 13,1579... 1200/13,1579... = 91,2     Scala tuning script .scl
39 39√1,3.. = 1,0074 500/39 = 12,82... 1200/12,82... = 93,6     Scala tuning script .scl
40 40√1,3.. = 1,00722 500/40 = 12,5 1200/12,5 = 96 1/16e de ton octaviation Z96  
... ... ... ... ...      

De 3 à 40 divisions de la quarte, il y a pour 8 échelles octaviantes, 29 échelles nonoctaviantes.

dOwnlOad these 29 nonoctave scales dividing the 4th (.scl files zip 12Ko)

.

 

cycle 5te

nombre
de degrés dans le cycle
Rapports en cent

notes par octave

en ton remarques


téléchargements pour
écoute

3 3√1,5 = 1,14471 700/3 = 233,33.. 1200/233,33.. = 5,14... 7/6e de ton   Scala tuning script .scl
4 4√1,5 = 1,10668 700/4 = 175 cents 1200/175 = 6,857... 7/8e de ton élément des 53 Scala tuning script .scl
5 5√1,5 = 1,08447 700/5 = 140 cents 1200/140 = 8,57... 7/10e de ton élément des 53 Scala tuning script .scl
6 6√1,5 = 1,06991 700/6 = 116,66.. 1200/116,66.. = 10,2857... 7/12e de ton élément des 53 Scala tuning script .scl
7 7√1,5 = 1,05963 700/7 = 100 1200/100 = 12 1/2 ton octaviation Z12  
8 8√1,5 = 1,05199 700/8 = 87,5 1200/87,5 = 13,714... 7/16e de ton élément des 53 Scala tuning script .scl
9 9√1,5 = 1,04608 700/9 = 77,77.. 1200/77,77.. = 15,4285...   α de Wendy Carlos Scala tuning script .scl
10 10√1,5 = 1,04138 700/10 = 70 1200/70 = 17,1428...     Scala tuning script .scl
11 11√1,5 = 1,03755 700/11 = 63,63.. 1200/63,63.. = 18,857...   β de Wendy Carlos. Octaviation au 19e degré à 9 cents près Scala tuning script .scl
12 12√1,5 = 1,03437 700/12 = 58,33.. 1200/58,33.. = 20,57...   la 5te a pris la place de l'octave sur le clavier piano Scala tuning script .scl
13 13√1,5 = 1,0274 700/13 = 53,846... 1200/53,846... = 22,285...     Scala tuning script .scl
14 14√1,5 = 1,02939 700/14 = 50 1200/50 = 24 1/4 de ton octaviation Z24  
15 15√1,5 = 1,0274 700/15 = 46,66.. 1200/46,66.. = 25,714     Scala tuning script .scl
16 16√1,5 = 1,02567 700/16 = 43,75 1200/43,75 = 27,428...     Scala tuning script .scl
17 17√1,5 = 1,02414 700/17 = 41,176... 1200/41,176... = 29,1428...     Scala tuning script .scl
18 18√1,5 = 1,02278 700/18 = 38,88.. 1200/38,88.. = 30,857...   Octaviation au 31e degré à 5 cents près. Scala tuning script .scl
19 19√1,5 = 1,02157 700/19 = 36,842... 1200/36,842... = 32,57...     Scala tuning script .scl
20 20√1,5 = 1,02048 700/20 = 35 1200/35 = 34,2857...   γ de Wendy Carlos Scala tuning script .scl
21 21√1,5 = 1,0195 700/21 = 33,33.. 1200/33,33.. = 36 1/6e de ton octaviation Z36  
22 22√1,5 = 1,0186 700/22 = 31,81.. 1200/31,81.. = 37,714...     Scala tuning script .scl
23 23√1,5 = 1,01779 700/23 = 30,4347... 1200/30,4347... = 39,4285...     Scala tuning script .scl
24 24√1,5 = 1,01704 700/24 = 29,166.. 1200/29,166.. = 41,1428...     Scala tuning script .scl
25 25√1,5 = 1,01635 700/25 = 28 1200/28 = 42,857...     Scala tuning script .scl
26 26√1,5 = 1,01572 700/26 = 26,923... 1200/26,923... = 44,5714...     Scala tuning script .scl
27 27√1,5 = 1,01513 700/27 = 25,926... 1200/25,926... = 46,2857...     Scala tuning script .scl
28 28√1,5 = 1,01459 700/28 = 25 1200/25 = 48 1/8e de ton octaviation Z48  
29 29√1,5 = 1,01408 700/29 = 24,1379... 1200/24,1379... = 49,7142...     Scala tuning script .scl
30 30√1,5 = 1,01361 700/30 = 23,33.. 1200/23,33.. = 51,4285...     Scala tuning script .scl
31 31√1,5 = 1,01317 700/31 = 22,5806... 1200/22,5806... = 53,1428...
    Scala tuning script .scl
32 32√1,5 = 1,01275 700/32 = 21,875 1200/21,875 = 54,857...     Scala tuning script .scl
33 33√1,5 = 1,01236 700/33 = 21,21.. 1200/21,21.. = 56,57...     Scala tuning script .scl
34 34√1,5 = 1,012 700/34 = 20,58... 1200/20,58... = 58,28...     Scala tuning script .scl
35 35√1,5 = 1,01162 700/35 = 20 1200/20 = 60 1/10 de ton octaviation Z60  
36 36√1,5 = 1,01133 700/36 = 19,44.. 1200/19,44.. = 61,714...     Scala tuning script .scl
37 37√1,5 = 1,01102 700/37 = 18,92... 1200/18,92... = 63,43...     Scala tuning script .scl
38 38√1,5 = 1,01073 700/38 = 18,42... 1200/18,42... = 65,14...     Scala tuning script .scl
39 39√1,5 = 1,01045 700/39 = 17,95... 1200/17,95... = 66,85...     Scala tuning script .scl
40 40√1,5 = 1,01019 700/40 = 17,5 1200/17,5 = 68,57...     Scala tuning script .scl
41 41√1,5 = 1,00994 700/41 = 17,073... 1200/17,073.. = 70,28...     Scala tuning script .scl
42 42√1,5 = 1,0097 700/42 = 16,66.. 1200/16,66.. = 72 1/12e de ton octaviation Z72  
43 43√1,5 = 1,00947 700/43 = 16,28... 1200/16,28... = 73,71...     Scala tuning script .scl
44 44√1,5 = 1,00926 700/44 = 15,90.. 1200/15,90.. = 75,428...     Scala tuning script .scl
45 45√1,5 = 1,00905 700/45 = 15,55.. 1200/15,55 = 77,14...     Scala tuning script .scl
46 46√1,5 = 1,00885 700/46 = 15,21... 1200/15,21... = 78,857...     Scala tuning script .scl
47 47√1,5 = 1,00866 700/47 = 14,89... 1200/14,89... = 80,57...     Scala tuning script .scl
48 48√1,5 = 1,00848 700/48 = 14,5833.. 1200/14,5833... = 82,28...     Scala tuning script .scl
49 49√1,5 = 1,00831 700/49 = 14,28... 1200/14,28... = 84 1/14e de ton octaviation Z84  
50 50√1,5 = 1,00814 700/50 = 14 1200/14 = 85,714...     Scala tuning script .scl
51 51√1,5 = 1,00798 700/51 = 13,725... 1200/13,725 = 87,428...     Scala tuning script .scl
52 52√1,5 = 1,00783 700/52 = 13,761... 1200/13,461... = 89,143...     Scala tuning script .scl
53 53√1,5 = 1,00768 700/53 = 13,207... 1200/13,207... = 90,857...     Scala tuning script .scl
54 54√1,5 = 1,00754 700/54 = 12,962... 1200/12,962... = 92,57...     Scala tuning script .scl
55 55√1,5 = 1,0074 700/55 = 12,72.. 1200/12,72.. = 94,285...     Scala tuning script .scl
56 56√1,5 = 1,00727 700/56 = 12,5 1200/12,5 = 96 1/16e de ton octaviation Z96  
... ... ... ... ... ... ...  

De 3 à 56 divisions de la quinte, il y a pour 8 échelles octaviantes, 46 échelles nonoctaviantes.

dOwnlOad these 46 nonoctave scales dividing the 5th (.scl files zip 20Ko)

.

 

cycle 6te m

nombre
de degrés
dans le cycle

Rapports en cent

notes par octave

en ton remarques


téléchargements pour
écoute

3 3√27/17 = 1,16673 800/3 = 266,66.. 1200/266,66.. = 4,5   pentatonique nonoctaviante Scala tuning script .scl
4 4√27/17 = 1,12261 800/4 = 200 1200/200 = 6 1 ton octaviation Z6  
5 5√27/17 = 1,09694 800/5 = 160 1200/160 = 7,5 4/5e de ton élément des 53 Scala tuning script .scl
6 6√27/17 = 1,08015 800/6 = 133,33.. 1200/133,33.. = 9 2/3 de ton octaviation Z9  
7 7√27/17 = 1,06832 800/7 = 114,28... 1200/114,28... = 10,5 4/7e de ton élément des 53 Scala tuning script .scl
8 8√27/17 = 1,05953 800/8 = 100 1200/100 = 12 1/2 ton octaviation Z12  
9 9√27/17 = 1,05275 800/9 = 88,88.. 1200/88,88.. = 13,5 4/9e de ton élément des 53 Scala tuning script .scl
10 10√27/17 = 1,04735 800/10 = 80 1200/80 = 15 2/5e de ton octaviation Z15  
11 11√27/17 = 1,04295 800/11 = 72,72.. 1200/72,72.. = 16,5 4/11e de ton élément des 53 Scala tuning script .scl
12 12√27/17 = 1,0393 800/12 = 66,66.. 1200/66,66.. = 18 1/3 de ton octaviation Z18  
13 13√27/17 = 1,03623 800/13 = 61,538... 1200/61,538... = 19,5 4/13e de ton élément des 53 Scala tuning script .scl
14 14√27/17 = 1,0336 800/14 = 57,14... 1200/57,14... = 21 2/7e de ton octaviation Z21  
15 15√27/17 = 1,03132 800/15 = 53,33.. 1200/53,33.. = 22,5 4/15e de ton élément des 53 Scala tuning script .scl
16 16√27/17 = 1,02934 800/16 = 50 1200/50 = 24 1/4 de ton octaviation Z24  
17 17√27/17 = 1,02759 800/17 = 47,058... 1200/47,058... = 25,5     Scala tuning script .scl
18 18√27/17 = 1,02603 800/18 = 44,44.. 1200/44,44.. = 27 2/9e de ton octaviation Z27  
19 19√27/17 = 1,02465 800/19 = 42,105... 1200/42,105... = 28,5     Scala tuning script .scl
20 20√27/17 = 1,0234 800/20 = 40 1200/40 = 30 1/5e de ton octaviation Z30  
21 21√27/17 = 1,02227 800/21 = 38,095... 1200/38,095... = 31,5     Scala tuning script .scl
22 22√27/17 = 1,02125 800/22 = 36,36.. 1200/36,36.. = 33 2/11e de ton octaviation Z33  
23 23√27/17 = 1,02032 800/23 = 34,78... 1200/34,78... = 34,5     Scala tuning script .scl
24 24√27/17 = 1,01946 800/24 = 33,33.. 1200/ = 36 1/6e de ton octaviation Z36  
25 25√27/17 = 1,01868 800/25 = 32 1200/32 = 37,5     Scala tuning script .scl
26 26√27/17 = 1,01795 800/26 = 30,769... 1200/30,769... = 39 2/13e de ton octaviation Z39  
27 27√27/17 = 1,01728 800/27 = 29,62... 1200/29,62... = 40,5     Scala tuning script .scl
28 28√27/17 = 1,01666 800/28 = 28,57... 1200/28,57... = 42 1/7e de ton octaviation Z42  
29 29√27/17 = 1,01608 800/29 = 27,58... 1200/27,58... = 43,5     Scala tuning script .scl
30 30√27/17 = 1,01554 800/30 = 26,66.. 1200/26,66.. = 45 2/15e de ton octaviation Z45  
31 31√27/17 = 1,01504 800/31 = 25,806... 1200/25,806... = 46,5
    Scala tuning script .scl
32 32√27/17 = 1,01456 800/32 = 25 1200/25 = 48 1/8e de ton octaviation Z48  
33 33√27/17 = 1,01412 800/33 = 24,24.. 1200/24,24.. = 49,5     Scala tuning script .scl
34 34√27/17 = 1,0137 800/34 = 23,53... 1200/23,53... = 51 2/17 de ton octaviation Z51  
35 35√27/17 = 1,01331 800/35 = 22,85... 1200/22,85... = 52,5     Scala tuning script .scl
36 36√27/17 = 1,01293 800/36 = 22,22.. 1200/22,22.. = 54 1/9e de ton octaviation Z54  
37 37√27/17 = 1,01258 800/37 = 21,62... 1200/21,62... = 55,5     Scala tuning script .scl
38 38√27/17 = 1,01225 800/38 = 21,05... 1200/21,05... = 57 2/19e de ton octaviation Z57  
39 39√27/17 = 1,01193 800/39 = 20,51... 1200/20,51... = 58,5     Scala tuning script .scl
40 40√27/17 = 1,01163 800/40 = 20 1200/20 = 60 1/10e de ton octaviation Z60  
41 41√27/17 = 1,01135 800/41 = 19,51... 1200/19,51... = 61,5     Scala tuning script .scl
42 42√27/17 = 1,01108 800/42 = 19,04... 1200/19,04... = 63 octaviation Z63  
43 43√27/17 = 1,01082 800/43 = 18,604... 1200/18,604... = 64,5     Scala tuning script .scl
44 44√27/17 = 1,01057 800/44 = 18,18.. 1200/18,18.. = 66 1/11e de ton octaviation Z66  
45 45√27/17 = 1,01033 800/45 = 17,77.. 1200/17,77.. = 67,5     Scala tuning script .scl
46 46√27/17 = 1,01011 800/46 = 17,39... 1200/17,39... = 69 octaviation Z69  
47 47√27/17 = 1,00989 800/47 = 17,02... 1200/17,02... = 70,5     Scala tuning script .scl
48 48√27/17 = 1,00968 800/48 = 16,66.. 1200/16,66.. = 72 1/12e de ton octaviation Z72  
49 49√27/17 = 1,00949 800/49 = 16,32... 1200/16,32... = 73,5     Scala tuning script .scl
50 50√27/17 = 1,0093 800/50 = 16 1200/16 = 75 octaviation Z75  
51 51√27/17 = 1,00911 800/51 = 15,68... 1200/15,68... = 76,5     Scala tuning script .scl
52 52√27/17 = 1,00894 800/52 = 15,38... 1200/15,38... = 78 1/13e de ton octaviation Z78  
53 53√27/17 = 1,00877 800/53 = 15,09434 1200/15,09434 = 79,5     Scala tuning script .scl
54 54√27/17 = 1,0086 800/54 = 14,814... 1200/14,814... = 81 octaviation Z81  
55 55√27/17 = 1,00845 800/55 = 14,54.. 1200/14,54.. = 82,5     Scala tuning script .scl
56 56√27/17 = 1,0083 800/56 = 14,28... 1200/14,28... = 84 1/14e de ton octaviation Z84  
57 57√27/17 = 1,00815 800/57 = 14,03... 1200/14,03... = 85,5     Scala tuning script .scl
58 58√27/17 = 1,00801 800/58 = 13,79... 1200/13,79... = 87 octaviation Z87  
59 59√27/17 = 1,00787 800/59 = 13,56... 1200/13,56... = 88,5     Scala tuning script .scl
60 60√27/17 = 1,00774 800/60 = 13,33.. 1200/13,33.. = 90 1/15e de ton octaviation Z90  
61 61√27/17 = 1,00761 800/61 = 13,11... 1200/13,11... = 91,5     Scala tuning script .scl
62 62√27/17 = 1,00749 800/62 = 12,903... 1200/12,903... = 93 octaviation Z93  
63 63√27/17 = 1,00737 800/63 = 12,69... 1200/12,69... = 94,5     Scala tuning script .scl
64 64√27/17 = 1,00725 800/64 = 12,5 1200/12,5 = 96 1/16e de ton octaviation Z96  
... ... ... ... ... ... ...  

{27/17 = 1,5882353 sixte mineure septendecimale} & {(12√2)8 = 1,58736 sixte mineure temperée}
Sur 62 échelles de 3 à 64, 31 sont octaviantes et 31 sont nonoctaviantes.
De 3 à 64 divisions de la sixte mineure, il y a pour 31 échelles octaviantes, 31 échelles nonoctaviantes.

dOwnlOad these 31 nonoctave scales dividing the 6th minor (.scl files zip 16Ko)

.

 

cycle 6te M

nombre
de degrés dans le cycle
Rapports

n coupures de 1,6875
en cent

notes par octave

en ton remarques


téléchargements pour
écoute

4 4√27/16 = 1,13975 900/4 = 225 1200/225 = 5,33..     Scala tuning script .scl
5 5√27/16 = 1,11032 900/5 = 180 1200/180 = 6,66..   élément des 53  
6 6√27/16 = 900/6 = 150 1200/150 = 8 octaviante Z8  
7 7√27/16 = 1,07761 900/7 = 128,57... 1200/128,57... = 9,33..   élément des 53  
8 8√27/16 = 1,06759 900/8 = 112,5 1200/112,5 = 10,66..   élément des 53  
9 9√27/16 = 900/9 = 100 1200/100 = 12 1/2 de ton octaviante Z12  
10 10√27/16 = 1,05372 900/10 = 90 1200/90 = 13,33..     Scala tuning script .scl
11 11√27/16 = 1,04872 900/11 = 81,81.. 1200/81,81.. = 14,66..     Scala tuning script .scl
12 12√27/16 = 900/12 = 75 1200/75 = 16 3/8e de ton octaviante Z16  
13 13√27/16 = 1,04107 900/13 = 69,23... 1200/69,23... = 17,33..     Scala tuning script .scl
14 14√27/16 = 1,03808 900/14 = 64,28... 1200/64,28... = 18,66..     Scala tuning script .scl
15 15√27/16 = 900/15 = 60 1200/60 = 20 3/10e de ton octaviante Z20  
16 16√27/16 = 1,03324 900/16 = 56,25 1200/56,25 = 21,33..     Scala tuning script .scl
17 17√27/16 = 1,03126 900/17 = 52,94... 1200/52,94... = 22,66..     Scala tuning script .scl
18 18√27/16 = 900/18 = 1200/ = 1/4 de ton octaviante Z24  
19 19√27/16 = 1,02792 900/19 = 47,36... 1200/47,36... = 25,33..     Scala tuning script .scl
20 20√27/16 = 1,02651 900/20 = 45 1200/45 = 26,66..     Scala tuning script .scl
21 21√27/16 = 900/21 = 1200/ = 3/14e de ton octaviante Z28  
22 22√27/16 = 1,02407 900/22 = 40,90.. 1200/40,90.. = 29,33..     Scala tuning script .scl
23 23√27/16 = 1,02301 900/23 = 39,13... 1200/39,13... = 30,66..     Scala tuning script .scl
24 24√27/16 = 900/24 = 1200/ = 3/16e de ton octaviante Z32  
25 25√27/16 = 1,02115 900/25 = 36 1200/36 = 33,33..     Scala tuning script .scl
26 26√27/16 = 1,02033 900/26 = 34,61... 1200/34,61... = 34,66..     Scala tuning script .scl
27 27√27/16 = 900/27 = 1200/ = 1/6e de ton octaviante Z36  
28 28√27/16 = 1,01886 900/28 = 32,14... 1200/32,14... = 37,33..     Scala tuning script .scl
29 29√27/16 = 1,01821 900/29 = 31,03... 1200/31,03... = 38,66..     Scala tuning script .scl
30 30√27/16 = 900/30 = 1200/ = 3/20e de ton octaviante Z40  
31 31√27/16 = 1,01702 900/31 = 29,03... 1200/29,03... = 41,33..
    Scala tuning script .scl
32 32√27/16 = 1,01649 900/32 = 28,125 1200/28,125 = 42,66..     Scala tuning script .scl
33 33√27/16 = 900/33 = 1200/ = 3/22e de ton octaviante Z44  
34 34√27/16 = 1,01551 900/34 = 26,47... 1200/26,47... = 45,33..     Scala tuning script .scl
35 35√27/16 = 1,01506 900/35 = 25,71... 1200/25,71... = 46,66..     Scala tuning script .scl
36 36√27/16 = 900/36 = 1200/ = 1/8e de ton octaviante Z48  
37 37√27/16 = 1,01424 900/37 = 24,324... 1200/24,324... = 49,33..     Scala tuning script .scl
38 38√27/16 = 1,01386 900/38 = 23,68... 1200/23,68... = 50,66..     Scala tuning script .scl
39 39√27/16 = 900/39 = 1200/ = 3/26e de ton octaviante Z52  
40 40√27/16 = 1,01317 900/40 = 22,5 1200/22,5 = 53,33..     Scala tuning script .scl
41 41√27/16 = 1,01284 900/41 = 21,95... 1200/21,95... = 54,66..     Scala tuning script .scl
42 42√27/16 = 900/42 = 1200/ = 3/28e de ton octaviante Z56  
43 43√27/16 = 1,01224 900/43 = 20,93... 1200/20,93... = 57,33..     Scala tuning script .scl
44 44√27/16 = 1,01196 900/44 = 20,45... 1200/20,45... = 58,66..     Scala tuning script .scl
45 45√27/16 = 900/45 = 1200/ = 1/10e de ton octaviante Z60  
46 46√27/16 = 1,01144 900/46 = 19,56... 1200/19,56... = 61,33..     Scala tuning script .scl
47 47√27/16 = 1,0112 900/47 = 19,14... 1200/19,14... = 62,66..     Scala tuning script .scl
48 48√27/16 = 900/48 = 1200/ = octaviante Z64  
49 49√27/16 = 1,01074 900/49 = 18,36... 1200/18,36... = 65,33..     Scala tuning script .scl
50 50√27/16 = 1,01052 900/50 = 18 1200/18 = 66,66..     Scala tuning script .scl
51 51√27/16 = 900/51 = 1200/ = octaviante Z68  
52 52√27/16 = 1,01011 900/52 = 17,31... 1200/17,31... = 69,33..     Scala tuning script .scl
53 53√27/16 = 1,00992 900/53 = 16,98... 1200/16,98... = 70,66..     Scala tuning script .scl
54 54√27/16 = 900/54 = 1200/ = 1/12e de ton octaviante Z72  
55 55√27/16 = 1,00956 900/55 = 16,36.. 1200/16,36.. = 73,33..     Scala tuning script .scl
56 56√27/16 = 1,00939 900/56 = 16,07... 1200/16,07... = 74,66..     Scala tuning script .scl
57 57√27/16 = 900/57 = 1200/ = octaviante Z76  
58 58√27/16 = 1,00906 900/58 = 15,51... 1200/15,51... = 77,33..     Scala tuning script .scl
59 59√27/16 = 1,00891 900/59 = 15,25... 1200/15,25... = 78,66..     Scala tuning script .scl
60 60√27/16 = 900/60 = 1200/ = octaviante Z80  
61 61√27/16 = 1,00861 900/61 = 14,75... 1200/14,75... = 81,33..     Scala tuning script .scl
62 62√27/16 = 1,00848 900/62 = 14,51... 1200/14,51... = 82,66..     Scala tuning script .scl
63 63√27/16 = 900/63 = 1200/ = 1/14e de ton octaviante Z84  
64 64√27/16 = 1,00821 900/64 = 14,0625 1200/14,0625 = 85,33..     Scala tuning script .scl
65 65√27/16 = 1,00808 900/65 = 13,84... 1200/13,84... = 86,66..     Scala tuning script .scl
66 66√27/16 = 900/66 = 1200/ = octaviante Z88  
67 67√27/16 = 1,00784 900/67 = 13,43... 1200/13,43... = 89,33..     Scala tuning script .scl
68 68√27/16 = 1,00772 900/68 = 13,23... 1200/13,23... = 90,66..     Scala tuning script .scl
69 69√27/16 = 900/69 = 1200/ =   octaviante Z92  
70 70√27/16 = 1,0075 900/70 = 12,85... 1200/12,85... = 93,33..     Scala tuning script .scl
71 71√27/16 = 1,0074 900/71 = 12,67... 1200/12,67... = 94,66..     Scala tuning script .scl
72 72√27/16 = 900/72 = 1200/ = 1/16e de ton octaviation Z96  
73 ... 900/73 = 12,32... ... ... ... ... Scala tuning script .scl

[72 divisions - 3 = 69]
{3/2 . 9/8 = 27/16 = 1,6875 sixte majeure pythagorienne}
& {(12√2)9 = 1,68175 sixte majeure tempérée à 900 cents}
De 4 à 72 divisions de la 6te Majeur, il y a 46+1 échelles nonoctaviantes pour 23 échelles octaviantes

dOwnlOad these 50 nonoctave scales dividing the 6th major (.scl files zip 26Ko)

 

 

Téléchargez les 157 échelles nonoctaviantes coupant 4te, 5te 6te mineur et majeur
Download these 157 nonoctave scales cutting the 4th the 5th the 6th minor & major
(.scl files zip 73Ko)

 

.

cycle 7e m

[80 divisions - 3 = 77]

...

 

cycle 7e M

[88 divisions - 4 = 84]

...

 

cycle 8ve+2de m

[104 divisions - 5 = 99]

...

 

 

 

Pour une division équidistante d'un intervalle-ambitus-cycle élément de Z12, il existe + d'échelles nonoctaviantes que d'échelles octaviantes :

div 4te : 29 échelles nonoctaviantes pour 8 échelles octaviantes
div 5te : 46 échelles nonoctaviantes pour 8 échelles octaviantes
div 6te m : 31 échelles nonoctaviantes pour 31 échelles octaviantes
div 6te M : 47 échelles nonoctaviantes pour 23 échelles octaviantes

...

 

 

A la recherche des échelles nonoctaviantes à partir des intervalles de la série harmonique

Les échelles : les suites en séries infinies pour les hôtes ou le savoir scalaire

Comment noter toutes les échelles pour les musiciens ?

L'harmonie nonoctaviante, révolution musicale

Théorie des Champs Scalaires

 

suite

re tour à la table des matières