Tempering out: Difference between revisions

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In both cases, that implies that we are introducing some error into our tunings: where we would use 3, for instance, we use a number slightly larger or smaller than 3. You can introduce error into any prime, and when tempering out a single comma you can choose to leave any given prime pure. In practice, many people leave 2 pure to achieve pure octaves. There are also options to temper the octave, such as [[TOP tuning]].

== Tempering together ==

Two or more [[chord|chords]] are said to be ''tempered together'' if the commas that relate all of their corresponding steps are ''tempered out''. If two chords are tempered together, then the representation of each of those chords in the temperament is identical.

For example, in [[meantone]], which tempers out [[81/80]], the chords [[54:64:81]] (with steps 32/27, 81/64) and [[10:12:15]] (with steps 6/5, 5/4) are tempered together. Since {{Frac|32|27}}{{dot}}{{Frac|81|80}} = {{Frac|6|5}} and {{Frac|81|64}} = {{Frac|5|4}}{{dot}}{{Frac|81|80}}, equating 81/80 with 1/1 also equates 32/27 with 6/5, and 81/64 with 5/4.

== Example ==

The syntonic comma is 81/80. That is (3×3×3×3) / (5×2×2×2×2) or, in [[monzo]] form, {{monzo| -4 4 -1 }}.

The syntonic comma is 81/80. That is (3×3×3×3) / (5×2×2×2×2) or, in [[monzo]] form, {{monzo| -4 4 -1}}.

19edo tempers out 81/80. (Technically, we should say that 19edo tempers out 81/80 when you use the [[patent val]].) You can see this in several ways:

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 In both cases, that implies that we are introducing some error into our tunings: where we would use 3, for instance, we use a number slightly larger or smaller than 3. You can introduce error into any prime, and when tempering out a single comma you can choose to leave any given prime pure. In practice, many people leave 2 pure to achieve pure octaves. There are also options to temper the octave, such as [[TOP tuning]].
+== Tempering together ==
+Two or more [[chord|chords]] are said to be ''tempered together'' if the commas that relate all of their corresponding steps are ''tempered out''. If two chords are tempered together, then the representation of each of those chords in the temperament is identical.
+For example, in [[meantone]], which tempers out [[81/80]], the chords [[54:64:81]] (with steps 32/27, 81/64) and [[10:12:15]] (with steps 6/5, 5/4) are tempered together. Since {{Frac|32|27}}{{dot}}{{Frac|81|80}} = {{Frac|6|5}} and {{Frac|81|64}} = {{Frac|5|4}}{{dot}}{{Frac|81|80}}, equating 81/80 with 1/1 also equates 32/27 with 6/5, and 81/64 with 5/4.
 == Example ==
-The syntonic comma is 81/80. That is (3×3×3×3) / (5×2×2×2×2) or, in [[monzo]] form, {{monzo| -4 4 -1 }}.
+The syntonic comma is 81/80. That is (3×3×3×3) / (5×2×2×2×2) or, in [[monzo]] form, {{monzo| -4 4 -1}}.
 edo tempers out 81/80. (Technically, we should say that 19edo tempers out 81/80 when you use the [[patent val]].) You can see this in several ways: