Tempering out: Difference between revisions

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'''Tempering out''' is what a [[regular temperament]] (including rank-1 temperaments ~~aka~~ [[equal temperament]]s) does to a small interval like a [[comma]]: it makes it disappear, or as some authors put it, ''vanish''.<ref>''Vanish'' is notably used throughout ''[[A Middle Path]]''.</ref>

{{interwiki

| en = Tempering out

| ja = テンパーアウト

| zh = 调和

}}

'''Tempering out''' is what a [[regular temperament]] (including rank-1 temperaments a.k.a. [[equal temperament]]s) does to a small interval like a [[comma]]: it makes it disappear, or as some authors put it, ''vanish''.<ref group="note">''Vanish'' is notably used throughout ''[[A Middle Path]]''. However, ''vanish'' is broader than ''temper out'', as any abstract interval can ''vanish''. For example, you could say "in [[5edo]], the small step of the [[5L 2s|diatonic scale]] vanishes" without specifying a just interval at all. In this sense, ''being tempered out'' is a specific way of ''vanishing''. </ref>

~~== Overview ==~~

For a tone measured as a [[ratio]] to disappear, it must become equal to 1/1, so that multiplying by the ratio does not change anything. For a tone measured in [[cent]]s to disappear, it must become 0 cents, so that adding it does not change anything.

For a tone measured as a ratio to "disappear", it must become equal to 1/1, so that multiplying by the ratio does not change anything. For a tone measured in ~~cents~~ to "disappear", it must become 0 cents, so that adding it does not change anything.

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]].

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 interval|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 ==

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== Example ==

The syntonic comma is 81/80. That is {{sfrac|34|2 ~~× 5~~4}} or, in [[monzo]] form, {{monzo| -4 4 -1}}.

The syntonic comma is 81/80. That is {{sfrac|34|24 × 5}} 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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== Notes ==

[[Category:Regular temperament theory]]

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-'''Tempering out''' is what a [[regular temperament]] (including rank-1 temperaments aka [[equal temperament]]s) does to a small interval like a [[comma]]: it makes it disappear, or as some authors put it, ''vanish''.<ref>''Vanish'' is notably used throughout ''[[A Middle Path]]''.</ref>
+{{interwiki
+| en = Tempering out
+| ja = テンパーアウト
+| zh = 调和
+}}
+'''Tempering out''' is what a [[regular temperament]] (including rank-1 temperaments a.k.a. [[equal temperament]]s) does to a small interval like a [[comma]]: it makes it disappear, or as some authors put it, ''vanish''.<ref group="note">''Vanish'' is notably used throughout ''[[A Middle Path]]''. However, ''vanish'' is broader than ''temper out'', as any abstract interval can ''vanish''. For example, you could say "in [[5edo]], the small step of the [[5L 2s|diatonic scale]] vanishes" without specifying a just interval at all. In this sense, ''being tempered out'' is a specific way of ''vanishing''. </ref>
-== Overview ==
+For a tone measured as a [[ratio]] to disappear, it must become equal to 1/1, so that multiplying by the ratio does not change anything. For a tone measured in [[cent]]s to disappear, it must become 0 cents, so that adding it does not change anything.
-For a tone measured as a ratio to "disappear", it must become equal to 1/1, so that multiplying by the ratio does not change anything. For a tone measured in cents to "disappear", it must become 0 cents, so that adding it does not change anything.
-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]].
+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 interval|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 ==
@@ Line 12: / Line 16: @@
 == Example ==
-The syntonic comma is 81/80. That is {{sfrac|3<sup>4</sup>|2 × 5<sup>4</sup>}} or, in [[monzo]] form, {{monzo| -4 4 -1}}.
+The syntonic comma is 81/80. That is {{sfrac|3<sup>4</sup>|2<sup>4</sup> × 5}} 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:
@@ Line 49: / Line 53: @@
 == Notes ==
-<references />
+<references group="note"/>
 [[Category:Regular temperament theory]]