US: Difference between revisions

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A '''US''', or '''utonal sequence''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning.

~~The full specification of a US is (n-)USp: (n pitches of a) utonal sequence adding by p.~~

== Specification ==

The n is optional. If not provided, the sequence is open-ended~~. By specifying n, your sequence will be equivalent to some [[UD|UD (utonal division)]]. E.g. 8-US3/4 = 8-UD7, because 8(3/4) = 6, so you will have traveled 6 away from the root of 1, and reached 7~~.

The full specification of a US is (n-)USp: (n pitches of a) utonal sequence adding by p. The n is optional. If not provided, the sequence is open-ended.

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

== Formula ==

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 a USp 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 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. UD ===

By specifying n, your sequence will be equivalent to some [[UD|UD (utonal division)]]. E.g. 8-US3/4 = 8-UD7, because 8(3/4) = 6, so you will have traveled 6 away from the root of 1, and reached 7.

=== vs. ALS ===

A US is the rational version of [[ALS|ALS, or arithmetic length sequence]].

== Examples ==

{| class="wikitable"

@@ Line 1: / Line 1: @@
 A '''US''', or '''utonal sequence''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning.
-The full specification of a US is (n-)USp: (n pitches of a) utonal sequence adding by p.
+== Specification ==
-The n is optional. If not provided, the sequence is open-ended. By specifying n, your sequence will be equivalent to some [[UD|UD (utonal division)]]. E.g. 8-US3/4 = 8-UD7, because 8(3/4) = 6, so you will have traveled 6 away from the root of 1, and reached 7.
+The full specification of a US is (n-)USp: (n pitches of a) utonal sequence adding by p. The n is optional. If not provided, the sequence is open-ended.
-A US is a specific (rational) type of [[ALS|ALS, or arithmetic length sequence]]. By varying the undertone series step size to some rational number (other than 1) you can produce a US, and 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.
+== Formula ==
-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 a USp is:
@@ Line 14: / 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 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. UD ===
+By specifying n, your sequence will be equivalent to some [[UD|UD (utonal division)]]. E.g. 8-US3/4 = 8-UD7, because 8(3/4) = 6, so you will have traveled 6 away from the root of 1, and reached 7.
+=== vs. ALS ===
+A US is the rational version of [[ALS|ALS, or arithmetic length sequence]].
+== Examples ==
 {| class="wikitable"