Harmonotonic tuning: Difference between revisions

Line 43:

If the new step size is irrational, the tuning is no longer JI, so we use a different acronym to distinguish it: AFS, for arithmetic frequency sequence. For example, if we wanted to move by steps of φ, like this: <math>1, 1+φ, 1+2φ, 1+3φ...</math> etc. we could have the AFSφ.

OS and AFS are equivalent to taking an overtone series and adding (or subtracting) a constant amount of frequency. By doing this, the step sizes remain equal in frequency, but their relationship in pitch changes. For a detailed explanation of this, see the later section on the [[~~Monotonic tunings~~#Derivation ~~of OS~~|derivation of OS]].

OS and AFS are equivalent to taking an overtone series and adding (or subtracting) a constant amount of frequency. By doing this, the step sizes remain equal in frequency, but their relationship in pitch changes. For a detailed explanation of this, see the later section on the [[OS#Derivation|derivation of OS]].

The same principles that were just described for frequency are also possible for length: by varying the undertone series step size to some rational number you can produce a utonal sequence (US), and varying it to an irrational number you can produce an arithmetic length sequence (ALS). In other words, by shifting the undertone series by a constant amount of string length, the step sizes remain equal in terms of length, but their relationship in pitch changes.

Line 277:

[[File:Pitch.svg|700px|link=https://en.xen.wiki/images/0/09/Pitch.svg]]

[[File:Length.svg|700px|link=https://en.xen.wiki/images/d/dc/Length.svg]]

~~== Derivation of OS ==~~

The tuning OS3/4 is the sequence <math>\frac 44, \frac 74, \frac{10}{4}, \frac{13}{4}...</math> and so on. Any OS is equivalent to shifting the overtone series by a constant amount of frequency. In the case of OS3/4, it is a shift by <math>\frac 13</math>. Let's show how.

~~Begin with the overtone series:~~

~~<math>1, 2, 3, 4...</math>~~

~~Shift it by <math>\frac 13</math>:~~

~~<math>~~

~~1\frac 13, 2\frac 13, 3\frac 13, 4\frac 13... \\~~

~~</math>~~

~~Convert to improper fractions by first expanding the whole number:~~

~~<math>~~

~~\frac 33 + \frac 13, \frac 63 + \frac 13, \frac 93 + \frac 13, \frac {12}{3} + \frac 13... \\~~

~~</math>~~

~~...then consolidating numerators:~~

~~<math>~~

~~\frac 43, \frac 73, \frac{10}{3}, \frac{13}{3}...~~

~~</math>~~

~~Resize to start at <math>\frac 11</math> by multiplying every term by the reciprocal of the first term, <math>\frac 43</math>, which is <math>\frac 34</math>:~~

~~<math>~~

~~\frac 43 \cdot \frac 34, \frac 73 \cdot \frac 34, \frac{10}{3} \cdot \frac 34, \frac{13}{3} \cdot \frac 34...~~

~~</math>~~

~~Cancel out:~~

~~<math>~~

~~\frac{4}{\cancel{3}} \cdot \frac{\cancel{3}}{4}, \frac{7}{\cancel{3}} \cdot \frac{\cancel{3}}{4}, \frac{10}{\cancel{3}} \cdot \frac{\cancel{3}}{4}, \frac{13}{\cancel{3}} \cdot \frac{\cancel{3}}{4}...~~

~~</math>~~

~~And we've arrived:~~

~~<math>~~

~~\frac 44, \frac 74, \frac{10}{4}, \frac{13}{4}...~~

~~</math>~~

So we can see that <math>\frac 13</math> was the right amount to shift by because it is the delta from the starting position <math>1</math> to <math>\frac 43</math>, the latter of which is the reciprocal of the target step size <math>\frac 34</math> and therefore the value that we need the starting position to equal in order to be sent ''back'' to <math>1</math> when we resize all steps from 1 to the target step size by multiplying everything by it.

== A note on etymology ==

@@ Line 43: / Line 43: @@
 If the new step size is irrational, the tuning is no longer JI, so we use a different acronym to distinguish it: AFS, for arithmetic frequency sequence. For example, if we wanted to move by steps of φ, like this: <span><math>1, 1+φ, 1+2φ, 1+3φ...</math></span> etc. we could have the AFSφ.
-OS and AFS are equivalent to taking an overtone series and adding (or subtracting) a constant amount of frequency. By doing this, the step sizes remain equal in frequency, but their relationship in pitch changes. For a detailed explanation of this, see the later section on the [[Monotonic tunings#Derivation of OS|derivation of OS]].
+OS and AFS are equivalent to taking an overtone series and adding (or subtracting) a constant amount of frequency. By doing this, the step sizes remain equal in frequency, but their relationship in pitch changes. For a detailed explanation of this, see the later section on the [[OS#Derivation|derivation of OS]].
 The same principles that were just described for frequency are also possible for length: by varying the undertone series step size to some rational number you can produce a utonal sequence (US), and varying it to an irrational number you can produce an arithmetic length sequence (ALS). In other words, by shifting the undertone series by a constant amount of string length, the step sizes remain equal in terms of length, but their relationship in pitch changes.
@@ Line 277: / Line 277: @@
 [[File:Pitch.svg|700px|link=https://en.xen.wiki/images/0/09/Pitch.svg]]
 [[File:Length.svg|700px|link=https://en.xen.wiki/images/d/dc/Length.svg]]
-== Derivation of OS ==
-The tuning OS3/4 is the sequence <span><math>\frac 44, \frac 74, \frac{10}{4}, \frac{13}{4}...</math></span> and so on. Any OS is equivalent to shifting the overtone series by a constant amount of frequency. In the case of OS3/4, it is a shift by <span><math>\frac 13</math></span>. Let's show how.
-Begin with the overtone series:
-<math>1, 2, 3, 4...</math>
-Shift it by <span><math>\frac 13</math></span>:
-<math>
-\frac 13, 2\frac 13, 3\frac 13, 4\frac 13... \\
-</math>
-Convert to improper fractions by first expanding the whole number:
-<math>
-\frac 33 + \frac 13, \frac 63 + \frac 13, \frac 93 + \frac 13, \frac {12}{3} + \frac 13... \\
-</math>
-...then consolidating numerators:
-<math>
-\frac 43, \frac 73, \frac{10}{3}, \frac{13}{3}...
-</math>
-Resize to start at <span><math>\frac 11</math></span> by multiplying every term by the reciprocal of the first term, <span><math>\frac 43</math><span>, which is <span><math>\frac 34</math><span>:
-<math>
-\frac 43 \cdot \frac 34, \frac 73 \cdot \frac 34, \frac{10}{3} \cdot \frac 34, \frac{13}{3} \cdot \frac 34...
-</math>
-Cancel out:
-<math>
-\frac{4}{\cancel{3}} \cdot \frac{\cancel{3}}{4}, \frac{7}{\cancel{3}} \cdot \frac{\cancel{3}}{4}, \frac{10}{\cancel{3}} \cdot \frac{\cancel{3}}{4}, \frac{13}{\cancel{3}} \cdot \frac{\cancel{3}}{4}...
-</math>
-And we've arrived:
-<math>
-\frac 44, \frac 74, \frac{10}{4}, \frac{13}{4}...
-</math>
-So we can see that <span><math>\frac 13</math></span> was the right amount to shift by because it is the delta from the starting position <span><math>1</math></span> to <span><math>\frac 43</math></span>, the latter of which is the reciprocal of the target step size <span><math>\frac 34</math></span> and therefore the value that we need the starting position to equal in order to be sent ''back'' to <span><math>1</math></span> when we resize all steps from 1 to the target step size by multiplying everything by it.
 == A note on etymology ==