Douglas Blumeyer's RTT How-To: Difference between revisions
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And if you’ve previously worked with EDOs, you may already be familiar with covectors. Covectors are a compact way to express EDOs in terms of the count of its steps it takes to reach its approximation of each prime harmonic, in order. For example, 12-EDO is {{val|12 19 28}}. The first term is the same as the name of the EDO, because the first prime harmonic is 2/1, or in other words: the octave. So this covector tells us that it takes 12 steps to reach the octave, 19 steps get you to about 3/1 (the [[tritave]]), and 28 steps get you to about 5/1 (the 5ave). | And if you’ve previously worked with EDOs, you may already be familiar with covectors. Covectors are a compact way to express EDOs in terms of the count of its steps it takes to reach its approximation of each prime harmonic, in order. For example, 12-EDO is {{val|12 19 28}}. The first term is the same as the name of the EDO, because the first prime harmonic is 2/1, or in other words: the octave. So this covector tells us that it takes 12 steps to reach the octave, 19 steps get you to about 3/1 (the [[tritave]]), and 28 steps get you to about 5/1 (the 5ave). | ||
If the musical structure that the mathematical structure called a vector represents is a ''' | If the musical structure that the mathematical structure called a vector represents is a JI '''interval''', the musical structure that the mathematical structure called a covector represents is called a '''map'''. | ||
Note the different direction of the brackets between covectors and vectors: covectors {{val|}} point left, vectors {{monzo|}} point right. | Note the different direction of the brackets between covectors and vectors: covectors {{val|}} point left, vectors {{monzo|}} point right. | ||
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For another example, can quickly find the fifth size for 12-EDO from its map, because 3/2 is {{monzo|-1 1}}, and so {{val|12 19 28}}{{monzo|-1 1}} = (12 × -1) + (19 × 1) = 7. Similarly, the major third — 5/4, or {{monzo|-2 0 1}} — is simply 28 - 12 - 12 = 4. | For another example, can quickly find the fifth size for 12-EDO from its map, because 3/2 is {{monzo|-1 1}}, and so {{val|12 19 28}}{{monzo|-1 1}} = (12 × -1) + (19 × 1) = 7. Similarly, the major third — 5/4, or {{monzo|-2 0 1}} — is simply 28 - 12 - 12 = 4. | ||
=== tempering out commas === | === tempering out commas === | ||