VIII. Intervalle, die man durch Potenzen von 2 und 3 und durch den Faktor 7 beschreiben kann
Wenn man auch den 7. Teilton der Obertonreihe (F, 2F, 3F, 4F, 5F, 6F, 7F) berücksichtigt, ergeben sich noch weitere Intervalle:
| Zahlenverhältnis | Centmaß | ||||
| 8/7 | [8/7] = | 231,1740935 | |||
| 5 | |||||
| 7/6 | [7/6] = | 266,8709056 | |||
| 5 | |||||
| 9/7 | [9/7] = | 435,0840953 | |||
| 5 | |||||
| 21/16 | [21/16] = | 470,7809073 | |||
| 5 | |||||
| 32/21 | [32/21] = | 729,2190927 | |||
| 5 | |||||
| 14/9 | [14/9] = | 764,9159047 | |||
| 5 | |||||
| 12/7 | [12/7] = | 933,1290944 | |||
| 5 | |||||
| 7/4 | [7/4] = | 968,8259065 |
Nimmt man
| „7er-Komma“ Φ = 26/(7·32) = 64/63 | [Φ] = φ = 27,2640918 | ||||
| (Verhältnis zwischen Tonus 9/8 und dem Intervall 8/7) | |||||
| und | „7er-Schisma“ Y = Φ/P = 225/(7·314) | [Y] = y = 3,8048148 | |||
| (Verhältnis zwischen Pythagoräischem und 7er-Komma) | |||||
erhält man:
| 8/7 | = (P9Φ)/R2 | = (P10Y)/R2 | [8/7] = | 10p – 2r + y | |
| 7/6 | = P13/(ΦR3) | = P12/(R3Y) | [7/6] = | 12p – 3r – y | |
| 9/7 | = (P18Φ)/R4 | = (P19Y)/R4 | [9/7] = | 19p – 4r + y | |
| 21/16 | = P22/(ΦR5) | = P21/(R5Y) | [21/16] = | 21p – 5r – y | |
| 32/21 | = (P31Φ)/R7 | = (P32Y)/R7 | [32/21] = | 32p – 7r + y | |
| 14/9 | = P35/(ΦR8) | = P34/(R8Y) | [14/9] = | 34p – 8r – y | |
| 12/7 | = (P40Φ)/R9 | = (P41Y)/R9 | [12/7] = | 41p – 9r + y | |
| 7/4 | = P44/(ΦR10) | = P43/(R10Y) | [7/4] = | 43p – 10r – y |
Durch Potenzen der 2 und der 3 könnten die folgenden passenden Intervalle dargestellt werden:
| sehr großer Ganzton | A2 = | 314/222 = P10/R2 | [A2] = | 10p – 2r = | 10h + 14r/53 | |
| sehr kleine Terz | S3 = | 224/315 = P12/R3 | [S3] = | 12p – 3r = | 12h – 15r/53 | |
| sehr große Terz | T·A2 = | 316/225 = P19/R4 | [T·A2] = | 19p – 4r = | 19h + 16r/53 | |
| sehr große Terz | T·S3 = | 221/313 = P21/R5 | [T·S3] = | 21p – 5r = | 21h – 13r/53 | |
| große Quint | Qua·A2 = | 313/220 = P32/R7 | [Qua·A2] = | 32p – 7r = | 32h + 13r/53 | |
| sehr kleine Sext | Qua·S3 = | 226/316 = P34/R8 | [Qua·S3] = | 34p – 8r = | 34h – 16r/53 | |
| sehr große Sext | Qui·A2 = | 315/223 = P41/R9 | [Qui·A2] = | 41p – 9r = | 41h + 15r/53 | |
| sehr kleine Septim | Qui·S3 = | 223/314 = P43/R10 | [Qui·S3] = | 43p – 10r = | 43h – 14r/53 |
also erhält man:
| 8/7 | = (P10Y)/R2 | [8/7] = | 10p – 2r + y = | 10h + 14r/53 + y = 10h + 4,759733 | ||
| 7/6 | = P12/(R3Y) | [7/6] = | 12p – 3r – y = | 12h – 15r/53 – y = 12h – 4,827941 | ||
| 9/7 | = (P19Y)/R4 | [9/7] = | 19p – 4r + y = | 19h + 16r/53 + y = 19h + 4,896149 | ||
| 21/16 | = R21/(R5Y) | [21/16] = | 21p – 5r – y = | 21h – 13r/53 – y = 21h – 4,691524 | ||
| 32/21 | = (P32Y)/R7 | [32/21] = | 32p – 7r + y = | 32h + 13r/53 + y = 32h + 4,691524 | ||
| 14/9 | = P34/(R8Y) | [14/9] = | 34p – 8r – y = | 34h – 16r/53 – y = 34h – 4,896149 | ||
| 12/7 | = (P41Y)/R9 | [12/7] = | 41p – 9r + y = | 41h + 15r/53 + y = 41h + 4,827941 | ||
| 7/4 | = P43/(R10Y) | [7/4] = | 43p – 10r – y = | 43h – 14r/53 – y = 43h – 4,759733 |
Guido von Arezzo erwähnt im „Micrologus“ übrigens auch die Diësis, die Mitte eines Semitoniums, die man z.B. zwischen e und f finden kann, indem man von d das Saitenlängenverhältnis 6/7 misst, das entspricht einem Frequenzverhältnis von 7/6.