Millioktave

Physikalische Einheit
Einheitenname	Millioktave
Einheitenzeichen	$\mathrm {mO}$
Physikalische Größe	musikalisches Intervall
Formelzeichen	$\Delta$
Dimension	${\mathsf {{\frac {T^{-1}}{T^{-1}}}=1}}$
In SI-Einheiten	$1$
Benannt nach	altgriechisch ὄκτω óktō = „acht“
Abgeleitet von	Oktave
Siehe auch: Cent, Neper, Savart

Die Millioktave (mO) ist eine Hilfsmaßeinheit für die Größe musikalischer Intervalle. 1000 mO entsprechen einer Oktave bzw. 1200 Cent bzw. einem Intervall mit dem Frequenzverhältnis (Proportion $p$ ) von 2:1.

Gegenüber dem Centmaß hat sich die Millioktave nie durchsetzen können. Sie wird jedoch bis heute gelegentlich von Autoren verwendet, welche die naheliegende Assoziation von Cent-Angaben mit gleichstufigen Intervallen vermeiden wollen.

Definition

Diatonische Intervalle
Prime Sekunde Terz Quarte Quinte Sexte Septime Oktave None Dezime Undezime Duodezime Tredezime Halbton/Ganzton
Besondere Intervalle
Mikrointervall Komma Diësis Limma Apotome Ditonus Tritonus Wolfsquinte Naturseptime
Maßeinheiten
Cent Millioktave Oktave Savart

Die Millioktave mO ist definiert durch:

1000 mO = 1 Oktave

Ist p das Frequenzverhältnis des Intervalls i, so ist die Größe des Intervalls

1000\log _{2}(p)\mathrm {mO}

oder taschenrechnerfreundlich mit dem 10er Logarithmus:

i=1000\cdot {\frac {\log _{10}(p)}{\log _{10}(2)}}\mathrm {mO}

Beispiel: Die Quinte hat das Frequenzverhältnis 2:3, also ist

Quinte =

1000\log _{2}({\frac {2}{3}})

mO =585 mO

Beispiel:

Intervall	Frequenzverhältnis	in Millioktave	In Cent
1 Oktave	2:1	1000	1200
2 Oktaven	4:1	2000	2400
3 Oktaven	8:1	3000	3600
Quinte	3:2	585	702
Quarte	4:3	415	498
große Terz	5:4	322	386

Umrechnungen

1 mO = 1,2 Cent = log₁₀(2) Savart ≈ 0,301 Savart

Geschichte

Eingeführt wurde die Millioktave 1903 vom deutschen Physiker Arthur von Oettingen in seinem Aufsatz Das duale System der Harmonie.^[1] Bereits im Jahr 1871 hatte George Biddell Airy in On Sound and Atmospheric Vibrations with the Mathematical Elements of Music^[2] den Vorschlag von John Frederick William Herschel diskutiert, die Oktave in 1000 Teile zu teilen.

Siehe auch

Stimmung

Einzelnachweise

↑ Arthur von Oettingen: Das duale System der Harmonie. (Memento des Originals vom 24. August 2011 im Internet Archive) Info: Der Archivlink wurde automatisch eingesetzt und noch nicht geprüft. Bitte prüfe Original- und Archivlink gemäß Anleitung und entferne dann diesen Hinweis.@1@2 In: Annalen der Naturphilosophie 1 (1902), S. 62–75; 2 (1903/4), S. 375–403; 3 (1904), S. 241–269; 4 (1905), S. 116–152 und 301–338; 5 (1906), S. 449–503. „Die Millioctave ist der 83. Theil eines Halbtones und ein so kleines Intervall, daß es als Differenz zweier Töne nicht mehr unterschieden wird.“ S. 388 f.
↑ George Biddell Airy: On sound and atmospheric vibrations: with the mathematical elements of music. 2. Auflage. London 1871, S. 222. “We are permitted by Sir John Herschel to explain a system proposed by him which possesses that advantage. It consists in using such a modulus that the logarithm of 2 is 1000.”

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