Mike's lecture on vector spaces and dual spaces: Difference between revisions
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If you have seen it, then to review, a '''monzo''' is a way to represent a JI interval that shows how it decomposes into a combination of simpler, "prime" intervals. It does so by directly representing an interval's prime factorization. A 5-limit monzo looks like <math>\ket{a \s b \s c}</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are the exponents for primes 2, 3, and 5, respectively. A 7-limit JI monzo looks like <math>\ket{a \s b \s c \s d}</math>, where <math>d</math> represents the additional exponent for 7. The 11-limit gets you another coefficient and so on. | If you have seen it, then to review, a '''monzo''' is a way to represent a JI interval that shows how it decomposes into a combination of simpler, "prime" intervals. It does so by directly representing an interval's prime factorization. A 5-limit monzo looks like <math>\ket{a \s b \s c}</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are the exponents for primes 2, 3, and 5, respectively. A 7-limit JI monzo looks like <math>\ket{a \s b \s c \s d}</math>, where <math>d</math> represents the additional exponent for 7. The 11-limit gets you another coefficient and so on. | ||
On the other hand, a '''val''' is a way to represent how JI intervals map to tempered steps along a chain of generators. A val does this by specifying the mapping for the primes, and in so doing ends up specifying the mapping for every JI interval as well: since every interval is a combination of primes, then we can find the mapping for any interval in some val by simply adding and subtracting the mapping for the primes in such a way that the original interval is recreated. A 5-limit val looks like , where , , and | On the other hand, a '''val''' is a way to represent how JI intervals map to tempered steps along a chain of generators. A val does this by specifying the mapping for the primes, and in so doing ends up specifying the mapping for every JI interval as well: since every interval is a combination of primes, then we can find the mapping for any interval in some val by simply adding and subtracting the mapping for the primes in such a way that the original interval is recreated. A 5-limit val looks like <math>\bra{a \s b \s c}</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are the number of steps along the chain that primes 2, 3, and 5 map to, respectively. A 7-limit val looks like , where represents the additional mapping for 7. Like with monzos, going to the 11-limit gets you another coefficient and so on.<math>\bra{x \s y \s z}</math><math>x</math><math>y</math><math>z</math><math>\bra{a \s b \s c \s d}</math><math>d</math>Again, if this is confusing, please go back to the pages on [[monzos|Monzos]] and [[Vals|Vals]] and read those first! | ||
Assuming you understand that, then we've reached our first new idea, which will help us gain a geometric intuition into what some of these abstract entities mean. That idea, which will enable us to rediscover vals and monzos in a much stronger mathematical and geometric context, is this: | Assuming you understand that, then we've reached our first new idea, which will help us gain a geometric intuition into what some of these abstract entities mean. That idea, which will enable us to rediscover vals and monzos in a much stronger mathematical and geometric context, is this: | ||