ALS: Difference between revisions

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An '''ALS''', or '''arithmetic length sequence''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning.

~~Is full specification is (n-)ALSp: (n pitches of an) arithmetic length sequence adding by p. It is equivalent to an undertone series shifted ± frequency.~~

== Specification ==

The n is optional. If not provided, the sequence is open-ended~~. By specifying n, your sequence will be equivalent to some [[ELD|ELD (equal length division)]]; specifically n-ALSp = n-ELD((p-1)/n)~~.

Is full specification is (n-)ALSp: (n pitches of an) arithmetic length sequence adding by p. It is equivalent to an undertone series shifted ± frequency. The n is optional. If not provided, the sequence is open-ended.

~~The analogous otonal equivalent of an ALS is an [[AFS|AFS (arithmetic frequency sequence)]].~~

== Formula ==

A [[US|US, or utonal sequence]], is a specific (rational) type of ALS. By varying the undertone series step size to some rational number (other than 1) you can produce a US, and by varying it to an irrational number you can produce an 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.

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.

The formula for length <span><math>k</math></span> of an ALSp is:

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L(k) = 1 + k⋅p

</math>

== Tip about tunings based on length ==

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.

== Relationship to other tunings ==

=== as shifted undertone series ===

By varying the undertone series step size to some rational number (other than 1) you can produce a US, and by varying it to an irrational number you can produce an 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.

=== vs. AFS ===

The analogous otonal equivalent of an ALS is an [[AFS|AFS (arithmetic frequency sequence)]].

=== vs. US ===

A [[US|US, or utonal sequence]], is the (rational) version of an ALS.

=== vs. ELD ===

By specifying n, your sequence will be equivalent to some [[ELD|ELD (equal length division)]]; specifically n-ALSp = n-ELD((p-1)/n).

== Examples ==

{| class="wikitable"

@@ Line 1: / Line 1: @@
 An '''ALS''', or '''arithmetic length sequence''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning.
-Is full specification is (n-)ALSp: (n pitches of an) arithmetic length sequence adding by p. It is equivalent to an undertone series shifted ± frequency.
+== Specification ==
-The n is optional. If not provided, the sequence is open-ended. By specifying n, your sequence will be equivalent to some [[ELD|ELD (equal length division)]]; specifically n-ALSp = n-ELD((p-1)/n).
+Is full specification is (n-)ALSp: (n pitches of an) arithmetic length sequence adding by p. It is equivalent to an undertone series shifted ± frequency. The n is optional. If not provided, the sequence is open-ended.
-The analogous otonal equivalent of an ALS is an [[AFS|AFS (arithmetic frequency sequence)]].
+== Formula ==
-A [[US|US, or utonal sequence]], is a specific (rational) type of ALS. By varying the undertone series step size to some rational number (other than 1) you can produce a US, and by varying it to an irrational number you can produce an 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.
-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.
 The formula for length <span><math>k</math></span> of an ALSp is:
@@ Line 16: / Line 12: @@
 L(k) = 1 + k⋅p
 </math>
+== Tip about tunings based on length ==
+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.
+== Relationship to other tunings ==
+=== as shifted undertone series ===
+By varying the undertone series step size to some rational number (other than 1) you can produce a US, and by varying it to an irrational number you can produce an 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.
+=== vs. AFS ===
+The analogous otonal equivalent of an ALS is an [[AFS|AFS (arithmetic frequency sequence)]].
+=== vs. US ===
+A [[US|US, or utonal sequence]], is the (rational) version of an ALS.
+=== vs. ELD ===
+By specifying n, your sequence will be equivalent to some [[ELD|ELD (equal length division)]]; specifically n-ALSp = n-ELD((p-1)/n).
+== Examples ==
 {| class="wikitable"