Arithmetic tuning: Difference between revisions

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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).

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). This ''arithmetic'' does not refer to measurements in the arithmetic scale, as in [[arithmetic mean]], but any arithmetic-like scale.

All arithmetic tunings are [[harmonotonic tunings]].

=== Types ===

== Types ==

Equal frequency steps:

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* [[US|US, utonal sequence]]

=== Basic examples ===

== Basic examples ==

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'' (12edo 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 ===

== 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><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.

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: <math>1, 1+φ, 1+2φ, 1+3φ...</math> etc. we could have the AFSφ.

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Note that because frequency is the inverse of length, if a frequency lower than the root pitch's frequency is asked for, the length will be greater than 1; at this point the physical analogy to a length of string breaks down somewhat, since it is not easy to imagine dynamically extending the length of a string to accommodate such pitches. However, it is not much of a stretch (pun intended) to tolerate lengths > 1, if the analogy is adapted to a switching from one string to another, and any string length imaginable is instantly available.

=== Divisions ===

== Divisions ==

So far we~~'ve~~ looked at arithmetic tunings produced by sequencing a single step repeatedly. But if an arithmetic tuning is defined by having equal step sizes of some kind of quantity (frequency, pitch, or length), then it also follows that they can be produced by taking a larger interval and equally dividing it according to that kind of quantity.

So far we have looked at arithmetic tunings produced by sequencing a single step repeatedly. But if an arithmetic tuning is defined by having equal step sizes of some kind of quantity (frequency, pitch, or length), then it also follows that they can be produced by taking a larger interval and equally dividing it according to that kind of quantity.

The most common example of this type of tuning is ~~12-EDO~~, standard tuning, which takes the interval of the octave, and equally divides its pitch into 12 parts. For long, we could call this ~~12-EPDO~~, for 12 equal '''pitch''' divisions of the octave (whenever pitch is the chosen kind of quality, we can assume it, and skip pointing it out; that's why ~~12-EDO~~ is the better name).

The most common example of this type of tuning is 12edo, standard tuning, which takes the interval of the octave, and equally divides its pitch into 12 parts. For long, we could call this 12epdo, for 12 equal ''pitch'' divisions of the octave (whenever pitch is the chosen kind of quality, we can assume it, and skip pointing it out; that's why 12edo is the better name).

But it is also possible to — instead of equally dividing the octave in 12 equal parts by pitch — divide it into 12 equal parts by '''frequency''', or '''length'''. In the former case, you will have ~~12-EFDO~~, and in the latter case, you will have ~~12-ELDO~~. However, that's not exactly ideal because, as with arithmetic sequences, different acronyms are used to distinguish rational (JI) tunings from irrational (non-JI) tunings, and so EFD and ELD are typically reserved for irrational tunings, such as ~~12-EFDφ~~. So it would be more appropriate to name these two tunings ~~12-ODO~~ and ~~12-UDO~~, for otonal divisions of the octave and utonal divisions of the octave, respectively.

But it is also possible to — instead of equally dividing the octave in 12 equal parts by pitch — divide it into 12 equal parts by ''frequency'', or ''length''. In the former case, you will have 12efdo, and in the latter case, you will have 12eldo. However, that is not exactly ideal because, as with arithmetic sequences, different acronyms are used to distinguish rational (JI) tunings from irrational (non-JI) tunings, and so EFD and ELD are typically reserved for irrational tunings, such as 12efdφ. So it would be more appropriate to name these two tunings 12odo and 12udo, for otonal divisions of the octave and utonal divisions of the octave, respectively.

=== Comparing arithmetic tunings ===

== Comparing arithmetic tunings ==

We can state a few helpful analogies:

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[[Category:Subharmonic series‏‎]]

[[Category:Equal-step tuning‏‎]]

~~[[Category:Equal divisions of the octave‏‎]]~~

@@ Line 1: / Line 1: @@
-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).
+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). This ''arithmetic'' does not refer to measurements in the arithmetic scale, as in [[arithmetic mean]], but any arithmetic-like scale.
 All arithmetic tunings are [[harmonotonic tunings]].
-=== Types ===
+== Types ==
 Equal frequency steps:
@@ Line 22: / Line 22: @@
 * [[US|US, utonal sequence]]
-=== Basic examples ===
+== Basic examples ==
 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'' (12edo 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 ===
+== 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><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.
+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: <math>1, 1+φ, 1+2φ, 1+3φ...</math> etc. we could have the AFSφ.
@@ Line 41: / Line 41: @@
 Note that because frequency is the inverse of length, if a frequency lower than the root pitch's frequency is asked for, the length will be greater than 1; at this point the physical analogy to a length of string breaks down somewhat, since it is not easy to imagine dynamically extending the length of a string to accommodate such pitches. However, it is not much of a stretch (pun intended) to tolerate lengths > 1, if the analogy is adapted to a switching from one string to another, and any string length imaginable is instantly available.
-=== Divisions ===
+== Divisions ==
-So far we've looked at arithmetic tunings produced by sequencing a single step repeatedly. But if an arithmetic tuning is defined by having equal step sizes of some kind of quantity (frequency, pitch, or length), then it also follows that they can be produced by taking a larger interval and equally dividing it according to that kind of quantity.
+So far we have looked at arithmetic tunings produced by sequencing a single step repeatedly. But if an arithmetic tuning is defined by having equal step sizes of some kind of quantity (frequency, pitch, or length), then it also follows that they can be produced by taking a larger interval and equally dividing it according to that kind of quantity.
-The most common example of this type of tuning is 12-EDO, standard tuning, which takes the interval of the octave, and equally divides its pitch into 12 parts. For long, we could call this 12-EPDO, for 12 equal '''pitch''' divisions of the octave (whenever pitch is the chosen kind of quality, we can assume it, and skip pointing it out; that's why 12-EDO is the better name).
+The most common example of this type of tuning is 12edo, standard tuning, which takes the interval of the octave, and equally divides its pitch into 12 parts. For long, we could call this 12epdo, for 12 equal ''pitch'' divisions of the octave (whenever pitch is the chosen kind of quality, we can assume it, and skip pointing it out; that's why 12edo is the better name).
-But it is also possible to — instead of equally dividing the octave in 12 equal parts by pitch — divide it into 12 equal parts by '''frequency''', or '''length'''. In the former case, you will have 12-EFDO, and in the latter case, you will have 12-ELDO. However, that's not exactly ideal because, as with arithmetic sequences, different acronyms are used to distinguish rational (JI) tunings from irrational (non-JI) tunings, and so EFD and ELD are typically reserved for irrational tunings, such as 12-EFDφ. So it would be more appropriate to name these two tunings 12-ODO and 12-UDO, for otonal divisions of the octave and utonal divisions of the octave, respectively.
+But it is also possible to — instead of equally dividing the octave in 12 equal parts by pitch — divide it into 12 equal parts by ''frequency'', or ''length''. In the former case, you will have 12efdo, and in the latter case, you will have 12eldo. However, that is not exactly ideal because, as with arithmetic sequences, different acronyms are used to distinguish rational (JI) tunings from irrational (non-JI) tunings, and so EFD and ELD are typically reserved for irrational tunings, such as 12efdφ. So it would be more appropriate to name these two tunings 12odo and 12udo, for otonal divisions of the octave and utonal divisions of the octave, respectively.
-=== Comparing arithmetic tunings ===
+== Comparing arithmetic tunings ==
 We can state a few helpful analogies:
@@ Line 83: / Line 83: @@
 [[Category:Subharmonic series‏‎]]
 [[Category:Equal-step tuning‏‎]]
-[[Category:Equal divisions of the octave‏‎]]