Expanding tonal space/planar extensions: Difference between revisions

Line 76:

::::<math>

r_{cents}= ln(r)\cdot\frac{1200}{ln(2)} \approx 1902</math> ¢

==== Pentave ====

In a Mode '''2<sup>-2'''</sup> (=0.25) overtone scale the first interval above the fundamental is the [[pentave]] (5/1), which substitutes the octave (2/1), the first interval in the Mode 1 overtone scale. This fractional mode of the overtone scale contains no octaves. It contains only odd harmonics with a spacing of 4 (e.g. 5, 9, 13, 17, 21, 25).

::::<math>

r=\frac{0.25+1}{0.25}=\frac{5}{1}</math>

::::<math>

r_{cents}= ln(r)\cdot\frac{1200}{ln(2)} \approx 2786</math> ¢

==== Double octave ====

In a Mode'''''(1/3)''''' overtone scale the first interval above the fundamental is the [[4/1|double octave]] (4/1), sometimes called the ''tetrave''. It substitutes the octave (2/1), the first interval in the Mode 1 overtone scale.

This fractional mode of the overtone scale contains every second octave. Octaves with 1200 ¢, 3600 ¢,… are not present in this mode. It contains harmonics with a spacing of 3 (e.g. 4, 7, 10, 13, 16).

::::<math>

r=\frac{\frac{1}{3}+1}{\frac{1}{3}}=\frac{4}{1}</math>

::::<math>

r_{cents}= ln(r)\cdot\frac{1200}{ln(2)} = 2400</math> ¢

To find the exponent x for the mode number in exponential form we solve <math>2^x\,=\frac{1}{3}</math> for x:

::::<math>x = \frac{ln(\frac{1}{3})}{ln(2)} \approx -1.585</math>

== Find out more about tonal space… ==

====[[Expanding tonal space|Part I: Expanding tonal space]]====

== See also... ==

[[OS|Otonal sequence]] (OS)

@@ Line 76: / Line 76: @@
 ::::<math>
 r_{cents}= ln(r)\cdot\frac{1200}{ln(2)} \approx 1902</math> ¢
+==== Pentave ====
+In a Mode '''2<sup>-2'''</sup> (=0.25) overtone scale the first interval above the fundamental is the [[pentave]] (5/1), which substitutes the octave (2/1), the first interval in the Mode 1 overtone scale. This fractional mode of the overtone scale contains no octaves. It contains only odd harmonics with a spacing of 4 (e.g. 5, 9, 13, 17, 21, 25).
+::::<math>
+r=\frac{0.25+1}{0.25}=\frac{5}{1}</math>
+::::<math>
+r_{cents}= ln(r)\cdot\frac{1200}{ln(2)} \approx 2786</math> ¢
+==== Double octave ====
+In a Mode'''''(1/3)''''' overtone scale the first interval above the fundamental is the [[4/1|double octave]] (4/1), sometimes called the ''tetrave''. It substitutes the octave (2/1), the first interval in the Mode 1 overtone scale.
+This fractional mode of the overtone scale contains every second octave. Octaves with 1200 ¢, 3600 ¢,… are not present in this mode. It contains harmonics with a spacing of 3 (e.g. 4, 7, 10, 13, 16).
+::::<math>
+r=\frac{\frac{1}{3}+1}{\frac{1}{3}}=\frac{4}{1}</math>
+::::<math>
+r_{cents}= ln(r)\cdot\frac{1200}{ln(2)} = 2400</math> ¢
+To find the exponent x for the mode number in exponential form we solve <math>2^x\,=\frac{1}{3}</math> for x:
+::::<math>x = \frac{ln(\frac{1}{3})}{ln(2)} \approx -1.585</math>
+== Find out more about tonal space… ==
+====[[Expanding tonal space|Part I: Expanding tonal space]]====
+== See also... ==
+[[OS|Otonal sequence]] (OS)