Arithmetic tuning: Difference between revisions

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=== Sequences ===

Other arithmetic tunings can be found by changing the step size. For example, if you vary the overtone series to have a step size of 3/4 instead of 1, then you get the tuning ~~~~<math>1, 1\frac 34, 2\frac 24, 3\frac14</math>, which is equivalent to ~~~~<math>\frac 44, \frac 74, \frac{10}{4}, \frac{13}{4}</math~~></span~~>, or in other words, a class iii [[isoharmonic_chords|isoharmonic]] tuning with starting position of 4. We call this the otonal sequence of 3 over 4, or OS3/4.

Other arithmetic tunings can be found by changing the step size. For example, if you vary the overtone series to have a step size of 3/4 instead of 1, then you get the tuning <math>1, 1\frac 34, 2\frac 24, 3\frac14</math>, which is equivalent to <math>\frac 44, \frac 74, \frac{10}{4}, \frac{13}{4}</math>, or in other words, a class iii [[isoharmonic_chords|isoharmonic]] tuning with starting position of 4. We call this the otonal sequence of 3 over 4, or OS3/4.

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~~></span~~> etc. we could have the AFSφ.

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 [[OS#Derivation|derivation of OS]].

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 === Sequences ===
-Other arithmetic tunings can be found by changing the step size. For example, if you vary the overtone series to have a step size of 3/4 instead of 1, then you get the tuning <span><math>1, 1\frac 34, 2\frac 24, 3\frac14</math><span>, which is equivalent to <span><math>\frac 44, \frac 74, \frac{10}{4}, \frac{13}{4}</math></span>, or in other words, a class iii [[isoharmonic_chords|isoharmonic]] tuning with starting position of 4. We call this the otonal sequence of 3 over 4, or OS3/4.
+Other arithmetic tunings can be found by changing the step size. For example, if you vary the overtone series to have a step size of 3/4 instead of 1, then you get the tuning <math>1, 1\frac 34, 2\frac 24, 3\frac14</math><span>, which is equivalent to <math>\frac 44, \frac 74, \frac{10}{4}, \frac{13}{4}</math>, or in other words, a class iii [[isoharmonic_chords|isoharmonic]] tuning with starting position of 4. We call this the otonal sequence of 3 over 4, or OS3/4.
-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.
+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 [[OS#Derivation|derivation of OS]].