Douglas Blumeyer's RTT How-To: Difference between revisions
Cmloegcmluin (talk | contribs) →maps: Steve Martin's suggestion to acknowledge 12-ET as an abbreviation of ⟨12 19 28] |
Cmloegcmluin (talk | contribs) m →vectors and covectors: revise wording per Steve Martin's suggestion not to potentially imply octaves must be pure |
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If you’ve previously worked with JI, you may already be familiar with vectors. Vectors are a compact way to express JI intervals in terms of their prime factorization, or in other words, their harmonic building blocks. In JI, and in most contexts in RTT, vectors simply list the count of each prime, in order. For example, 16/15 is {{monzo|4 -1 -1}} because it has four 2’s in its numerator, one 3 in its denominator, and also one 5 in its denominator. You can look at each term as an exponent: 2⁴ × 3⁻¹ × 5⁻¹ = 16/15. | If you’ve previously worked with JI, you may already be familiar with vectors. Vectors are a compact way to express JI intervals in terms of their prime factorization, or in other words, their harmonic building blocks. In JI, and in most contexts in RTT, vectors simply list the count of each prime, in order. For example, 16/15 is {{monzo|4 -1 -1}} because it has four 2’s in its numerator, one 3 in its denominator, and also one 5 in its denominator. You can look at each term as an exponent: 2⁴ × 3⁻¹ × 5⁻¹ = 16/15. | ||
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 | 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 2/1 (the [[octave]]), 19 steps to reach 3/1 (the [[tritave]]), and 28 steps to each 5/1 (the [[pentave]]). Any or all of those intervals may be approximate. | ||
If the musical structure that the mathematical structure called a vector represents is an '''interval''', the musical structure that the mathematical structure called a covector represents is called a '''map'''. | If the musical structure that the mathematical structure called a vector represents is an '''interval''', the musical structure that the mathematical structure called a covector represents is called a '''map'''. | ||