Monzo: Difference between revisions
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This page gives a pragmatic introduction to '''monzos'''. For the formal mathematical definition of visit the page [[Monzos_and_Interval_Space|Monzos and Interval Space]]. | This page gives a pragmatic introduction to '''monzos'''. For the formal mathematical definition of visit the page [[Monzos_and_Interval_Space|Monzos and Interval Space]]. | ||
=Etymology= | == Etymology == | ||
Monzos are named in honor of [[Joe Monzo]]. | Monzos are named in honor of [[Joe Monzo]]. | ||
=Definition= | == Definition == | ||
A '''monzo''' is a way of notating a JI interval that allows us to express directly how any "composite" interval is represented in terms of those simpler prime intervals. They are typically written using the notation | a b c d e f ... | A '''monzo''' is a way of notating a JI interval that allows us to express directly how any "composite" interval is represented in terms of those simpler prime intervals. They are typically written using the notation {{Monzo|a b c d e f ... }}, where the columns represent how the primes 2, 3, 5, 7, 11, 13, etc, in that order, contribute to the interval's prime factorization, up to some [[Harmonic Limit|prime limit]]. | ||
Monzos can be thought of as counterparts to [[ | Monzos can be thought of as counterparts to [[vals]]. | ||
For a more mathematical discussion, see also [[ | For a more mathematical discussion, see also [[Monzos and Interval Space]]. | ||
=Examples= | == Examples == | ||
For example, the interval 15/8 can be thought of as having 5*3 in the numerator, and 2*2*2 in the denominator. This can be compactly represented by the expression 2^-3 * 3^1 * 5^1, 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 | ... | For example, the interval 15/8 can be thought of as having 5*3 in the numerator, and 2*2*2 in the denominator. This can be compactly represented by the expression 2^-3 * 3^1 * 5^1, 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:'' Because the pipe symbol and the greater sign have special meaning in wiki syntax and HTML, there is a helper template ([[Template:Monzo]]) that can be used like this <code><nowiki>{{Monzo|arguments}}</nowiki></code> to get the monzo brackets (<code>{{Monzo|arguments}}</code>) from it. | |||
Here are some common 5-limit monzos, for your reference: | Here are some common 5-limit monzos, for your reference: | ||
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{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
! | ! Ratio | ||
! | ! Monzo | ||
|- | |- | ||
| style="text-align:center;" | 3/2 | | style="text-align:center;" | [[3/2]] | ||
| | | {{Monzo| -1 1 0 }} | ||
|- | |- | ||
| style="text-align:center;" | 5/4 | | style="text-align:center;" | [[5/4]] | ||
| | | {{Monzo| -2 0 1 }} | ||
|- | |- | ||
| style="text-align:center;" | 9/8 | | style="text-align:center;" | [[9/8]] | ||
| | | {{Monzo| -3 2 0 }} | ||
|- | |- | ||
| style="text-align:center;" | 81/80 | | style="text-align:center;" | [[81/80]] | ||
| | | {{Monzo| -4 4 -1 }} | ||
|} | |} | ||
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! | Monzo | ! | Monzo | ||
|- | |- | ||
| style="text-align:center;" | 7/4 | | style="text-align:center;" | [[7/4]] | ||
| | | {{Monzo| -2 0 0 1 }} | ||
|- | |- | ||
| style="text-align:center;" | 7/6 | | style="text-align:center;" | [[7/6]] | ||
| | | {{Monzo| -1 -1 0 1 }} | ||
|- | |- | ||
| style="text-align:center;" | 7/5 | | style="text-align:center;" | [[7/5]] | ||
| | | {{Monzo| 0 0 -1 1 }} | ||
|} | |} | ||
=Relationship with vals= | == Relationship with vals == | ||
''See also: [[ | ''See also: [[Vals]], [[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 this case, the val < 12 19 28 | is the [[Patent_val|patent val]] for 12-equal, and | -4 4 -1 > is 81/80, or the syntonic comma. The fact that < 12 19 28 | -4 4 -1 > tells us that 81/80 is mapped to 0 steps in 12-equal - aka it's tempered out - which tells us that 12-equal is a meantone temperament. It is noteworthy that almost the entirety of western music, particularly western music composed for 12-equal or 12-tone well temperaments, is made possible by the above equation. | In this case, the val < 12 19 28 | is the [[Patent_val|patent val]] for 12-equal, and | -4 4 -1 > is 81/80, or the syntonic comma. The fact that < 12 19 28 | -4 4 -1 > tells us that 81/80 is mapped to 0 steps in 12-equal - aka it's tempered out - which tells us that 12-equal is a meantone temperament. It is noteworthy that almost the entirety of western music, particularly western music composed for 12-equal or 12-tone well temperaments, is made possible by the above equation. | ||
'''In general: < a b c | d e f > = ad + be + cf''' | '''In general: < a b c | d e f > = ad + be + cf''' | ||
[[Category:definition]] | |||
[[Category:intervals]] | [[Category:intervals]] | ||
[[Category: | [[Category:prime limit]] | ||
[[Category:theory]] | [[Category:theory]] | ||