User:VectorGraphics/Monzo notation: Difference between revisions

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: ''See also: [[Val]], [[Keenan's explanation of vals]], [[Vals and tuning space]] (more mathematical)''

Monzos in just intonation are important because they enable us to see how any JI interval "maps" onto a val. This mapping is expressed by writing the val and the monzo together, such as [12 19 28][-4 4 -1]. The mapping is extremely easily to calculate: simply multiply together each component in the same position on both sides of the line, and add the results together. This is perhaps best demonstrated by example:

Monzos in just intonation are also important because they enable us to see how any JI interval "maps" onto a val. This mapping is expressed by writing the val and the monzo together, such as [12 19 28][-4 4 -1]. The mapping is extremely easily to calculate: simply multiply together each component in the same position on both sides of the line, and add the results together. This is perhaps best demonstrated by example:

<math>

Line 36:

</math>

In this case, the val [12 19 28] is the [[patent val]] for [[12-equal]], and [-4 4 1] is 81/80, or the [[syntonic comma]]. [12 19 28][-4 4 -1] tells us that 81/80 is mapped to 0 steps in 12-TET—in other words, it is tempered out—which tells us that 12-TET is a [[meantone]] temperament. It is noteworthy that almost the entirety of Western music composed in the [[Historical temperaments|Renaissance]] and from the sixteenth century onwards, particularly Western music composed for 12-tone circulating temperaments ([[12edo|12 equal]] and unequal [[Well temperament|well temperaments]]), is made possible by the tempering out of 81/80, and that almost all aspects of modern common practice Western music theory (chords and scales) in both classical and non-classical music genres are based exclusively on meantone.

In this case, the val [12 19 28] is the [[patent val]] for [[12-equal|12-]]TET, which essentially tells us how many steps of 12edo, if taken as a 5-limit system, represent each of the primes of the 5-limit (2, 3, and 5), and can be seen as a very simple [[Mapping|mapping matrix]].

[-4 4 1] is the monzo notation of 81/80, or the [[syntonic comma]] separating simple 5-limit intervals from nearby simple 3-limit intervals.

[12 19 28][-4 4 -1] tells us that 81/80 is mapped to 0 steps in 12-TET—in other words, it is tempered out—which tells us that 12-TET is a [[meantone]] temperament. It is noteworthy that almost the entirety of Western music composed in the [[Historical temperaments|Renaissance]] and from the sixteenth century onwards, particularly Western music composed for 12-tone circulating temperaments ([[12edo|12 equal]] and unequal [[Well temperament|well temperaments]]), is made possible by the tempering out of 81/80, and that almost all aspects of modern common practice Western music theory (chords and scales) in both classical and non-classical music genres are based exclusively on meantone.

In general:

@@ Line 28: / Line 28: @@
 : ''See also: [[Val]], [[Keenan's explanation of vals]], [[Vals and tuning space]] (more mathematical)''
-Monzos in just intonation are important because they enable us to see how any JI interval "maps" onto a val. This mapping is expressed by writing the val and the monzo together, such as [12 19 28][-4 4 -1]. The mapping is extremely easily to calculate: simply multiply together each component in the same position on both sides of the line, and add the results together. This is perhaps best demonstrated by example:
+Monzos in just intonation are also important because they enable us to see how any JI interval "maps" onto a val. This mapping is expressed by writing the val and the monzo together, such as [12 19 28][-4 4 -1]. The mapping is extremely easily to calculate: simply multiply together each component in the same position on both sides of the line, and add the results together. This is perhaps best demonstrated by example:
 <math>
@@ Line 36: / Line 36: @@
 </math>
-In this case, the val [12 19 28] is the [[patent val]] for [[12-equal]], and [-4 4 1] is 81/80, or the [[syntonic comma]]. [12 19 28][-4 4 -1] tells us that 81/80 is mapped to 0 steps in 12-TET&#x2014;in other words, it is tempered out&#x2014;which tells us that 12-TET is a [[meantone]] temperament. It is noteworthy that almost the entirety of Western music composed in the [[Historical temperaments|Renaissance]] and from the sixteenth century onwards, particularly Western music composed for 12-tone circulating temperaments ([[12edo|12 equal]] and unequal [[Well temperament|well temperaments]]), is made possible by the tempering out of 81/80, and that almost all aspects of modern common practice Western music theory (chords and scales) in both classical and non-classical music genres are based exclusively on meantone.
+In this case, the val [12 19 28] is the [[patent val]] for [[12-equal|12-]]TET, which essentially tells us how many steps of 12edo, if taken as a 5-limit system, represent each of the primes of the 5-limit (2, 3, and 5), and can be seen as a very simple [[Mapping|mapping matrix]].
+[-4 4 1] is the monzo notation of 81/80, or the [[syntonic comma]] separating simple 5-limit intervals from nearby simple 3-limit intervals.
+[12 19 28][-4 4 -1] tells us that 81/80 is mapped to 0 steps in 12-TET&#x2014;in other words, it is tempered out&#x2014;which tells us that 12-TET is a [[meantone]] temperament. It is noteworthy that almost the entirety of Western music composed in the [[Historical temperaments|Renaissance]] and from the sixteenth century onwards, particularly Western music composed for 12-tone circulating temperaments ([[12edo|12 equal]] and unequal [[Well temperament|well temperaments]]), is made possible by the tempering out of 81/80, and that almost all aspects of modern common practice Western music theory (chords and scales) in both classical and non-classical music genres are based exclusively on meantone.
 In general: