J o a c h i m M o h r Mathematik Musik Programmieren
Modulationen in Eulerschreibweise
Angegeben werden 12 Töne pro Tonart (Dur/Moll+2) in der der reinen Stimmung, zum Beispiel:c 'des d 'es ,e f ,fis g 'as ,a 'b ,h c 'des d 'es ,e f ,fis g 'as ,a 'b ,h c mit den 12 reinen Dur- und Molldreiklängen:
1. Stufe: c 'es g (moll) c e g (dur)
2. Stufe: 'des f 'as (dur)
3. Stufe: 'es g 'b (dur) ,e g ,h (moll)
4. Sufe: f ,a c (dur) f 'as c (moll)
5. Stufe: g ,h d (dur) g 'b d (moll)
6. Stufe: 'as c 'es (dur) ,a c ,e (moll)
7.Stufe: ,h d ,fis (moll)
Um besser vergleichen zu können: gleiche Töne untereinander
Die quintverwandten Tonarten:
des: ,c des 'eses es 'fes ,f ges ,g as 'heses ,b 'ces ,c des 'eses es 'fes ,f ges ,g as 'heses ,b 'ces ,c
as: ,c des ,d es 'fes ,f 'ges ,g as 'heses b 'ces ,c des ,d es 'fes ,f 'ges ,g as 'heses b 'ces ,c
↑ —— ————— ——— ↑
as: ,c des ,d es 'fes ,f 'ges ,g as 'heses b 'ces ,c des ,d es 'fes ,f 'ges ,g as 'heses b 'ces ,c
es: ,c 'des ,d es 'fes f 'ges ,g as ,a b 'ces ,c 'des ,d es 'fes f 'ges ,g as ,a b 'ces ,c
↑ —— ———— ——— ↑
es: ,c 'des ,d es 'fes f 'ges ,g as ,a b 'ces ,c 'des ,d es 'fes f 'ges ,g as ,a b 'ces ,c
b: c 'des ,d es ,e f 'ges ,g 'as ,a b 'ces c 'des ,d es ,e f 'ges ,g 'as ,a b 'ces c
↑ —— ———— ——— ↑
b: c 'des ,d es ,e f 'ges ,g 'as ,a b 'ces c 'des ,d es ,e f 'ges ,g 'as ,a b 'ces c
f: c 'des ,d 'es ,e f 'ges g 'as ,a b ,h c 'des ,d 'es ,e f 'ges g 'as ,a b ,h c
↑ —— ———— ——— ↑
f: c 'des ,d 'es ,e f 'ges g 'as ,a b ,h c 'des ,d 'es ,e f 'ges g 'as ,a b ,h c
c: c 'des d 'es ,e f ,fis g 'as ,a 'b ,h c 'des d 'es ,e f ,fis g 'as ,a 'b ,h c
↑ —— ———— —— ↑
c: c 'des d 'es ,e f ,fis g 'as ,a 'b ,h c 'des d 'es ,e f ,fis g 'as ,a 'b ,h c
g: c ,cis d 'es ,e f ,fis g 'as a 'b ,h c ,cis d 'es ,e 'f ,fis g 'as a 'b ,h c
↑ —— ———— —— ↑
g: c ,cis d 'es ,e 'f ,fis g 'as a 'b ,h c ,cis d 'es ,e 'f ,fis g 'as a 'b ,h c
d: 'c ,cis d 'es e 'f ,fis g ,gis a 'b ,h 'c ,cis d 'es e 'f ,fis g ,gis a 'b ,h 'c
↑ —— ———— —— ↑
d: 'c ,cis d 'es e 'f ,fis g ,gis a 'b ,h 'c ,cis d 'es e 'f ,fis g ,gis a 'b ,h 'c
a: 'c ,cis d ,dis e 'f ,fis 'g ,gis a 'b h 'c ,cis d ,dis e 'f ,fis 'g ,gis a 'b h 'c
↑ —— ———— —— ↑
a: 'c ,cis d ,dis e 'f ,fis 'g ,gis a 'b h 'c ,cis d ,dis e 'f ,fis 'g ,gis a 'b h 'c
e: 'c ,cis 'd ,dis e 'f fis 'g ,gis a ,ais h 'c ,cis 'd ,dis e 'f fis 'g ,gis a ,ais h 'c
↑ ——— ———— —— ↑
e: 'c ,cis 'd ,dis e 'f fis 'g ,gis a ,ais h 'c ,cis 'd ,dis e 'f fis 'g ,gis a ,ais h 'c
h: 'c cis 'd ,dis e ,eis fis 'g ,gis 'a ,ais h 'c cis 'd ,dis e ,eis fis 'g ,gis 'a ,ais h 'c
↑ ——— ———— —— ↑
h: 'c, cis 'd ,dis e ,eis fis 'g ,gis 'a ,ais h 'c cis 'd ,dis e ,eis fis 'g ,gis 'a ,ais h 'c
fis: ,his cis 'd ,dis 'e ,eis fis 'g gis 'a ,ais h ,his cis 'd ,dis 'e ,eis fis 'g gis 'a ,ais h ,his
↑ ——— ———— —— ↑
fis: ,his cis 'd ,dis 'e ,eis fis 'g gis 'a ,ais h ,his cis 'd dis 'e ,eis fis 'g gis 'a ,ais h ,his
cis: ,his cis 'd dis 'e ,eis fis ,fisis gis 'a ,ais 'h ,his cis 'd ,dis 'e ,eis fis ,fisis gis 'a ,ais 'h ,his
↑ ——— ———— —— ↑
Terzverwandte Tonarten:
,a: c ,,cis ,d ,,dis ,e f ,,fis g ,,gis ,a b ,h c ,,cis ,d ,,dis ,e f ,,fis g ,,gis ,a b ,h c
c: c 'des d 'es ,e f ,fis g 'as ,a 'b ,h c 'des d 'es ,e f ,fis g 'as ,a 'b ,h c
↑ —— ———— —— ———— ————— ————— ↑
Vergleich der Abweichungen der Tonhöhen
[,a a]=k=syntonisches Komma=21,51 Cent[des cis]=pythagoreisches Komma=23,46 Cent
[c ,his]=p-k=1,95 Cent=Schischma
[,cis 'des]=2k-p=19,55 Cent=Diaschisma
[,,cis 'des]=3k-p=41,06 Cent=kleine Diesis