Harmonotonic tuning: Difference between revisions

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A monotonic tuning is one whose step sizes are [https://en.wikipedia.org/wiki/Monotonic_function monotonic]: they do not both increase and decrease.

A '''monotonic tuning''' is one whose ''step sizes'' are [https://en.wikipedia.org/wiki/Monotonic_function monotonic]: they do not both increase and decrease.

* A diatonic tuning is '''not''' monotonic because it goes back and forth between whole and half steps.

* A diatonic tuning is ''not'' monotonic because it goes back and forth between whole and half steps.

* A segment of the overtone series '''is''' monotonic because its steps always decrease in size (within the interval of repetition).

* A segment of the overtone series ''is'' monotonic because its steps always decrease in size (within the interval of repetition).

* An EDO tuning '''is''' monotonic because the steps are all the same size.

* An EDO tuning ''is'' monotonic because the steps are all the same size.

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Here are the three different '''shapes''', according to their pitches sorted in ascending order:

# decreasing step size (e.g. overtone series)

# '''decreasing''' step size (e.g. overtone series)

# equal step size (e.g. EDO)

# '''equal''' step size (e.g. EDO)

# increasing step size (e.g. undertone series)

# '''increasing''' step size (e.g. undertone series)

And here are the three different '''types''':

# [[Monotonic tunings#Arithmetic tunings|arithmetic]] & rational

# '''[[Monotonic tunings#Arithmetic tunings|arithmetic]] & rational''' (e.g. overtone or undertone series)

# arithmetic & irrational

# '''arithmetic & irrational''' (e.g. EDO)

# non-arithmetic & irrational

# '''non-arithmetic & irrational'''

== Arithmetic tunings ==

An arithmetic tuning is one which has equal step sizes ''of any kind of quantity'', whether that be pitch, frequency, or length (of the resonating entity producing the sound). All arithmetic tunings are monotonic tunings.

An arithmetic tuning is one which has equal step sizes ''of any kind of quantity'', whether that be '''pitch''', '''frequency''', or '''length''' (of the resonating entity producing the sound).

All arithmetic tunings are monotonic tunings.

Basic examples of arithmetic tunings:

# the overtone series has equal steps of frequency (1, 2, 3, 4, etc.; adding 1 each step)

# the '''overtone''' series has equal steps of '''frequency''' (1, 2, 3, 4, etc.; adding 1 each step)

# any EDO has equal steps of pitch (12-EDO goes 0/12, 1/12, 2/12, 3/12, etc.; adding 1/12 each step)

# any '''EDO''' has equal steps of '''pitch''' (12-EDO goes 0/12, 1/12, 2/12, 3/12, etc.; adding 1/12 each step)

# the undertone series has equal steps of length (to play the first four steps of the undertone series you would pluck the whole length of a string, then 3/4 the string, then 2/4, then 1/4; adding -1/4 length each step)

# the '''undertone''' series has equal steps of '''length''' (to play the first four steps of the undertone series you would pluck the whole length of a string, then 3/4 the string, then 2/4, then 1/4; adding -1/4 length each step)

=== 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.

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 — 1, 1+φ, 1+2φ, 1+3φ, 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 Derivation of OS.

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-A monotonic tuning is one whose step sizes are [https://en.wikipedia.org/wiki/Monotonic_function monotonic]: they do not both increase and decrease.
+A '''monotonic tuning''' is one whose ''step sizes'' are [https://en.wikipedia.org/wiki/Monotonic_function monotonic]: they do not both increase and decrease.
-* A diatonic tuning is '''not''' monotonic because it goes back and forth between whole and half steps.
+* A diatonic tuning is ''not'' monotonic because it goes back and forth between whole and half steps.
-* A segment of the overtone series '''is''' monotonic because its steps always decrease in size (within the interval of repetition).
+* A segment of the overtone series ''is'' monotonic because its steps always decrease in size (within the interval of repetition).
-* An EDO tuning '''is''' monotonic because the steps are all the same size.
+* An EDO tuning ''is'' monotonic because the steps are all the same size.
 {|
@@ Line 16: / Line 16: @@
 Here are the three different '''shapes''', according to their pitches sorted in ascending order:
-# decreasing step size (e.g. overtone series)
+# '''decreasing''' step size (e.g. overtone series)
-# equal step size (e.g. EDO)
+# '''equal''' step size (e.g. EDO)
-# increasing step size (e.g. undertone series)
+# '''increasing''' step size (e.g. undertone series)
 And here are the three different '''types''':
-# [[Monotonic tunings#Arithmetic tunings|arithmetic]] & rational
+# '''[[Monotonic tunings#Arithmetic tunings|arithmetic]] & rational''' (e.g. overtone or undertone series)
-# arithmetic & irrational
+# '''arithmetic & irrational''' (e.g. EDO)
-# non-arithmetic & irrational
+# '''non-arithmetic & irrational'''
 == Arithmetic tunings ==
-An arithmetic tuning is one which has equal step sizes ''of any kind of quantity'', whether that be pitch, frequency, or length (of the resonating entity producing the sound). All arithmetic tunings are monotonic tunings.
+An arithmetic tuning is one which has equal step sizes ''of any kind of quantity'', whether that be '''pitch''', '''frequency''', or '''length''' (of the resonating entity producing the sound).
+All arithmetic tunings are monotonic tunings.
 Basic examples of arithmetic tunings:
-# the overtone series has equal steps of frequency (1, 2, 3, 4, etc.; adding 1 each step)
+# the '''overtone''' series has equal steps of '''frequency''' (1, 2, 3, 4, etc.; adding 1 each step)
-# any EDO has equal steps of pitch (12-EDO goes 0/12, 1/12, 2/12, 3/12, etc.; adding 1/12 each step)
+# any '''EDO''' has equal steps of '''pitch''' (12-EDO goes 0/12, 1/12, 2/12, 3/12, etc.; adding 1/12 each step)
-# the undertone series has equal steps of length (to play the first four steps of the undertone series you would pluck the whole length of a string, then 3/4 the string, then 2/4, then 1/4; adding -1/4 length each step)
+# the '''undertone''' series has equal steps of '''length''' (to play the first four steps of the undertone series you would pluck the whole length of a string, then 3/4 the string, then 2/4, then 1/4; adding -1/4 length each step)
 === 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.
-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 — 1, 1+φ, 1+2φ, 1+3φ, 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 — 1, 1+φ, 1+2φ, 1+3φ, 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 Derivation of OS.