User:Holger Stoltenberg/sandbox: Difference between revisions

Line 61:

A just major third relates the first (m=1) interval of a Mode 4 overtone scale to the tonic of Mode 4 (n=4) and we get

:::: <math>\displaystyle

::::<math>\displaystyle

r=\frac{4+1}{4}=\frac{5}{4}</math>

::::<math>\displaystyle

r_{cents}= ln(r)\cdot\frac{1200}{ln(2)} \approx 386 </math> ¢

==Intonation==

Intonation is an adjustment of pitch applied to notes - live at performance time.

In the context of this model, we use 12edo pitches as the reference scale for measuring intonation. To describe intonation precisely (and without reference to concert pitch or absolute frequency), we define

''Intonation is the signed interval between...''

'' r<sub>~~cents~~</sub> ~~= ln (r) * (1200/ln(2)) ≈ 386~~'' ¢

* ''a pitch, generated by a key with a given key descriptor (such as A3, B3, C4, C#4, ...), that rings exactly in tune with 12-tone equal temperament (12edo) and''

* ''a deviating pitch referenced by the same key descriptor''

Typically, intonation is a small interval between -50 ¢ and +50 ¢ although larger values are allowed. The common tonic of all modes of the overtone scale has an intonation of 0 ¢ by definition.

According to this definition'','' the upper pitch of the just major third (Mode 4) above the tonic has a signed intonation interval (distance to the next vertical 12edo line) of -14 ¢ .

To calculate the intonation

* compute the ''remainder'' of the interval’s value in cents by a modulo division (386 ¢ mod100) – the interim result is 86 ¢

* Test the interim result: If it is greater than 50 ¢ then subtract 100 ¢

* The test is true and the final result is -14 ¢ .

To determine the 12edo interval the intonation is applied to, get the original interval ''r<sub>cents </sub>'' and calculate

@@ Line 61: / Line 61: @@
 A just major third relates the first (m=1) interval of a Mode 4 overtone scale to the tonic of Mode 4 (n=4) and we get
-:::: <math>\displaystyle
+::::<math>\displaystyle
 r=\frac{4+1}{4}=\frac{5}{4}</math>
+::::<math>\displaystyle
+r_{cents}= ln(r)\cdot\frac{1200}{ln(2)} \approx 386 </math> ¢
+==Intonation==
+Intonation is an adjustment of pitch applied to notes - live at performance time.
+In the context of this model, we use 12edo pitches as the reference scale for measuring intonation. To describe intonation precisely (and without reference to concert pitch or absolute frequency), we define
+''Intonation is the signed interval between...''
-''        r<sub>cents</sub> = ln (r) * (1200/ln(2)) ≈ 386'' ¢
+* ''a pitch, generated by a key with a given key descriptor (such as A3, B3, C4, C#4, ...), that rings exactly in tune with 12-tone equal temperament (12edo) and''
+* ''a deviating pitch referenced by the same key descriptor''
+Typically, intonation is a small interval between -50 ¢ and +50 ¢ although larger values are allowed. The common tonic of all modes of the overtone scale has an intonation of 0 ¢ by definition.
+According to this definition'','' the upper pitch of the just major third (Mode 4) above the tonic has a signed intonation interval (distance to the next vertical 12edo line) of -14 ¢ .
+To calculate the intonation
+* compute the ''remainder'' of the interval’s value in cents by a modulo division (386 ¢ mod100) – the interim result is 86 ¢
+* Test the interim result:  If it is greater than 50 ¢ then subtract 100 ¢
+* The test is true and the final result is -14 ¢ .
+To determine the 12edo interval the intonation is applied to, get the original interval ''r<sub>cents </sub>'' and calculate