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 ==

Two or more [[chord~~|chords~~]] or intervals 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.

Two or more [[chord]]s or intervals 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 {{nowrap|{{Frac|32|27}}{{dot}}{{Frac|81|80}} {{=}} {{Frac|6|5}}}} and {{nowrap|{{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 {{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 ==
-Two or more [[chord|chords]] or intervals 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.
+Two or more [[chord]]s or intervals 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 {{nowrap|{{Frac|32|27}}{{dot}}{{Frac|81|80}} {{=}} {{Frac|6|5}}}} and {{nowrap|{{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 {{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]]