Frequency temperament: Difference between revisions

Line 1:

'''Arithmetic temperaments''' are the arithmetic counterpart to [[regular temperament]]s. Whereas regular temperaments are created by reducing integer powers of a [[generator]], an arithmetic temperament is created by reducing integer multiples of a generator. The n-th interval in an arithmetic temperament is given by ~~ng mod (p - 1)~~ + 1, where g is the generator ~~and p is the period~~.

'''Arithmetic temperaments''' are the arithmetic counterpart to [[regular temperament]]s. Whereas regular temperaments are created by reducing integer powers of a [[generator]], an arithmetic temperament is created by reducing integer multiples of a generator. The n-th interval in an arithmetic temperament prior to octave-reduction is given by n*g + 1, where g is the generator.

For example, this is the interval chain of an arithmetic temperament with a generator of 1.29 ~~(440 cents)~~ and period [[2/1]]:

For example, this is the interval chain of an arithmetic temperament with a generator of 0.29 and period [[2/1]]:

<pre>

1.29 ~~≈ 440¢~~

1+0.29 = 1.29 (440

2*1.29 ~~- 1~~ = 1.58 ~~≈ 791¢~~

1+2*0.29 = 1.58

3*1.29 ~~- 2~~ = 1.87 ~~≈ 1084¢~~

1+3*0.29 = 1.87

4*1.29 - ~~4 =~~ 1.~~16 ≈ 257¢~~

1+4*0.29 = 2.16 -> 1.08

5*1.29 - ~~5 =~~ 1.~~45 ≈ 643¢~~

1+5*0.29 = 2.45 -> 1.225

6*1.29 ~~- 6~~ = 1.74 ~~≈ 960¢~~

1+6*0.29 = 2.74 -> 1.37

~~7*1.29~~ - ~~8 =~~ 1.~~03 ≈ 51¢~~

...

</pre>

Arithmetic temperaments also temper out [[comma]]s, but these commas represent differences between intervals rather than ratios between them. For examp

== List of arithmetic temperaments ==

@@ Line 1: / Line 1: @@
-'''Arithmetic temperaments''' are the arithmetic counterpart to [[regular temperament]]s. Whereas regular temperaments are created by reducing integer powers of a [[generator]], an arithmetic temperament is created by reducing integer multiples of a generator. The n-th interval in an arithmetic temperament is given by ng mod (p - 1) + 1, where g is the generator and p is the period.
+'''Arithmetic temperaments''' are the arithmetic counterpart to [[regular temperament]]s. Whereas regular temperaments are created by reducing integer powers of a [[generator]], an arithmetic temperament is created by reducing integer multiples of a generator. The n-th interval in an arithmetic temperament prior to octave-reduction is given by n*g + 1, where g is the generator.
-For example, this is the interval chain of an arithmetic temperament with a generator of 1.29 (440 cents) and period [[2/1]]:
+For example, this is the interval chain of an arithmetic temperament with a generator of 0.29 and period [[2/1]]:
 <pre>
-.29 ≈ 440¢
++0.29 = 1.29 (440
-*1.29 - 1 = 1.58 ≈ 791¢
++2*0.29 = 1.58
-*1.29 - 2 = 1.87 ≈ 1084¢
++3*0.29 = 1.87
-*1.29 - 4 = 1.16 ≈ 257¢
++4*0.29 = 2.16 -> 1.08
-*1.29 - 5 = 1.45 ≈ 643¢
++5*0.29 = 2.45 -> 1.225
-*1.29 - 6 = 1.74 ≈ 960¢
++6*0.29 = 2.74 -> 1.37
-*1.29 - 8 = 1.03 ≈ 51¢
 ...
 </pre>
+Arithmetic temperaments also temper out [[comma]]s, but these commas represent differences between intervals rather than ratios between them. For examp
 == List of arithmetic temperaments ==