Monzo: Difference between revisions
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| ja = モンゾ | | ja = モンゾ | ||
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This page gives a pragmatic introduction to '''monzos'''. For the formal mathematical definition | This page gives a pragmatic introduction to '''monzos'''. For the formal mathematical definition, visit the page [[Monzos and Interval Space]]. | ||
== Definition == | == Definition == | ||
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For example, the interval 15/8 can be thought of as having <math>5⋅3</math> in the numerator, and <math>2⋅2⋅2</math> in the denominator. This can be compactly represented by the expression <math>2^{-3} \cdot 3^1 \cdot 5^1</math>, which is exactly equal to 15/8. We construct the monzo by taking the exponent from each prime, in order, and placing them within the {{monzo| ... }} brackets, hence yielding {{monzo|-3 1 1}}. | For example, the interval 15/8 can be thought of as having <math>5⋅3</math> in the numerator, and <math>2⋅2⋅2</math> in the denominator. This can be compactly represented by the expression <math>2^{-3} \cdot 3^1 \cdot 5^1</math>, which is exactly equal to 15/8. We construct the monzo by taking the exponent from each prime, in order, and placing them within the {{monzo| ... }} brackets, hence yielding {{monzo|-3 1 1}}. | ||
:'''Practical hint:''' the monzo template helps you getting correct brackets ([[Template:Monzo|read | :'''Practical hint:''' the monzo template helps you getting correct brackets ([[Template:Monzo|read more…]]). | ||
Here are some common 5-limit monzos, for your reference: | Here are some common 5-limit monzos, for your reference: | ||
{| class="wikitable" | {| class="wikitable center-1" | ||
|- | |- | ||
! Ratio | ! Ratio | ||
! Monzo | ! Monzo | ||
|- | |- | ||
| [[3/2]] | |||
| {{monzo| -1 1 0 }} | | {{monzo| -1 1 0 }} | ||
|- | |- | ||
| [[5/4]] | |||
| {{monzo| -2 0 1 }} | | {{monzo| -2 0 1 }} | ||
|- | |- | ||
| [[9/8]] | |||
| {{monzo| -3 2 0 }} | | {{monzo| -3 2 0 }} | ||
|- | |- | ||
| [[81/80]] | |||
| {{monzo| -4 4 -1 }} | | {{monzo| -4 4 -1 }} | ||
|} | |} | ||
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Here are a few 7-limit monzos: | Here are a few 7-limit monzos: | ||
{| class="wikitable" | {| class="wikitable center-1" | ||
|- | |- | ||
! Ratio | ! Ratio | ||
! Monzo | ! Monzo | ||
|- | |- | ||
| [[7/4]] | |||
| {{monzo| -2 0 0 1 }} | | {{monzo| -2 0 0 1 }} | ||
|- | |- | ||
| [[7/6]] | |||
| {{monzo| -1 -1 0 1 }} | | {{monzo| -1 -1 0 1 }} | ||
|- | |- | ||
| [[7/5]] | |||
| {{monzo| 0 0 -1 1 }} | | {{monzo| 0 0 -1 1 }} | ||
|} | |} | ||
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== Relationship with vals == | == Relationship with vals == | ||
''See also: [[ | ''See also: [[Val]], [[Keenan's explanation of vals]], [[Vals and Tuning Space]] (more mathematical)'' | ||
Monzos 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 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: | ||
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'''In general: < a b c | d e f > = ad + be + cf''' | '''In general: < a b c | d e f > = ad + be + cf''' | ||
[[Category: | [[Category:Regular temperament theory]] | ||
[[Category:Just intonation]] | |||
[[Category:Terms]] | [[Category:Terms]] | ||
[[Category:Notation]] | [[Category:Notation]] | ||