V. Annäherungen der durch Potenzen von 2 und 3 beschriebenen Intervalle durch das Pythagoräische Komma
Wir betrachten 5 besondere Verhältnisse von Potenzen der 2 und der 3:
| Zahlenverhältnis | Centmaß | ||||
| O = 2 | o = [O] | = | 1200,0000000000 ¢ | ||
| T = 32/23 | t = [T] | = | 203,9100017308 ¢ | ||
| S = 28/35 | s = [S] | = | 90,2249956731 ¢ | ||
| P = 312/219 | p = [P] | = | 23,4600103846 ¢ | ||
| R = 353/284 | r = [R] | = | 3,6150458655 ¢ |
| Aus | T·S-2 = P | [T] + 2·[S-1] = [P] | |
| S-1·P3 = Q | [S-1] + 3·[P] = [Q] | ||
| P·Q = R | [P] + [Q] = [R] |
folgt durch Rechnung
| S = P4/R | [S] = s = 4p – r | ||
| T = P9/R2 | [T] = t = 9p – 2r |
Das Semitonium ist also circa so groß wie 4, der Tonus ca. wie 9 Pythagoräische Kommata.
Der Unterschied ist r (3,6150 ¢) bzw. 2r (7,2301 ¢).
| und mit | O = T5·S2 | [O] = 5·[T] + 2·[S] | |
| folgt | O = P53/R12 | [O] = o = 53p – 12r |
Die Oktav ist um 12r (43,3806 ¢) kleiner als 53p, es gilt also nicht genau
| P = O1/53 | p = 1200/53 (das wäre 22.6415 Cent) | ||
| sondern | |||
| P = O1/51,15087... | p = 1200/51,15087....(23,4600 Cent) |
Folgende Intervalle können nun in Erweiterungen des modalen Systems durch Potenzen der 2 und der 3 gebildet und durch das Pythagoräische Komma angenähert werden:
| Semitonium | S = | 28/35 | = P4/R | [S] = | 4p | - | r | ||
| Apotome | A = | 37/211 | = P5/R | [A] = | 5p | - | r | ||
| kleiner Ganzton | S2 = | 216/310 | = P8/R2 | [S2] = | 8p | - | 2r | ||
| großer Ganzton | T = | 32/23 | = P9/R2 | [T] = | 9p | - | 2r | ||
| kleine Terz | k31 = T.S = | 25/33 | = P13/R3 | [k31] = | 13p | - | 3r | ||
| kleine Terz | k32 = T.A = | 39/214 | = P14/R3 | [k32] = | 14p | - | 3r | ||
| große Terz | g31 = T.S2 = | 213/38 | = P17/R4 | [g31] = | 17p | - | 4r | ||
| große Terz | g32 = T2 = | 34/26 | = P18/R4 | [g32] = | 18p | - | 4r | ||
| Quart | Qua = | 22/3 | = P22/R5 | [Qua] = | 22p | - | 5r | ||
| verminderte Quint | Tr1= O/T3 = Qua·S = | 210/36 | = P26/R6 | [Tr1] = | 26p | - | 6r | ||
| Tritonus | Tr2 = T3 = Qua·A = | 36/29 | = P27/R6 | [Tr2] = | 27p | - | 6r | ||
| Quint | Qui = | 3/2 | = P31/R7 | [Qui] = | 31p | - | 7r | ||
| kleine Sext | k61 = Qui·S = | 27/34 | = P35/R8 | [k61] = | 35p | - | 8r | ||
| kleine Sext | k62 = Qui·A = | 38/212 | = P36/R8 | [k62] = | 36p | - | 8r | ||
| große Sext | g61 = Qui·S2 = | 215/39 | = P39/R9 | [g61] = | 39p | - | 9r | ||
| große Sext | g62 = Qui·T = | 33/24 | = P40/R9 | [g62] = | 40p | - | 9r | ||
| kleine Septim | k71 = O/T = | 24/32 | = P44/R10 | [k71] = | 44p | - | 10r | ||
| kleine Septim | k72 = O/S2 = | 310/215 | = P45/R10 | [k72] = | 45p | - | 10r | ||
| große Septim | g71 = O/A = | 212/37 | = P48/R11 | [g71] = | 48p | - | 11r | ||
| große Septim | g72 = O/S = | 35/27 | = P49/R11 | [g72] = | 49p | - | 11r | ||
| Oktav | O = | 2 | = P53/R12 | [O] = | 53p | - | 12r |
Mit B = P3/R = 227/317 erhalten wir also folgende Intervalle zwischen obigen Intervallen:
│ S │P│ B │P│ S │P│ B │P│ S │ S │P│ S │ S │P│ B │P│ S │P│ B │P│ S │
Durch aufsteigende Quinten (· 3/2) bzw. absteigende Quarten (· 3/22) kann man folgende Ordnung („Quintenzirkel“) erhalten: