Val: Difference between revisions
m this is the second time something from the summary that i decided not to include has appeared on the page. am i really that careless or is this an obscure bug? |
m →Definition: i don't see any referent to 26edo being used as the example prior to where we state that prime 2 is mapped to 26 steps, which is disorientating, hence i've restored its correct place in the title. i know it's inelegant, but elegance is not the chiefest concern here. 26edo was picked for a small tuning that shows an inconsistency in the 5-limit which one might nonetheless want to use the 5-limit approximations for, so is not that arbitrary under closer analysis anyways |
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A [[val]] is a list of numbers telling you the approximation of each [[prime harmonic]] used in an edo in terms of steps, where by ''prime harmonic'' we mean each frequency ratio ''p''/1 (where ''p'' is a {{w|prime number}}. This list of integers by convention corresponds to all primes up to some largest prime (the [[limit]]) so that we can tell what number represents the ''mapping'' of what prime by its place in the list (1st place is prime 2's mapping (a.k.a. the edo), 2nd place is prime 3's mapping, 3rd place is prime 5's mapping, 4th is prime 7's, etc.). The val is used to understand the edo's approximations to ratios involving those primes, like 2 × 5 / 3 / 3 = [[10/9]] for primes {2, 3, 5}. This list [[#Warts and generalized patent vals|does not have to be the closest approximation]] for each prime, but it usually is. Thus a val is essentially just a list of numbers that we are interpreting as having a certain meaning. | A [[val]] is a list of numbers telling you the approximation of each [[prime harmonic]] used in an edo in terms of steps, where by ''prime harmonic'' we mean each frequency ratio ''p''/1 (where ''p'' is a {{w|prime number}}. This list of integers by convention corresponds to all primes up to some largest prime (the [[limit]]) so that we can tell what number represents the ''mapping'' of what prime by its place in the list (1st place is prime 2's mapping (a.k.a. the edo), 2nd place is prime 3's mapping, 3rd place is prime 5's mapping, 4th is prime 7's, etc.). The val is used to understand the edo's approximations to ratios involving those primes, like 2 × 5 / 3 / 3 = [[10/9]] for primes {2, 3, 5}. This list [[#Warts and generalized patent vals|does not have to be the closest approximation]] for each prime, but it usually is. Thus a val is essentially just a list of numbers that we are interpreting as having a certain meaning. | ||
=== Example === | === Example: 26edo === | ||
* prime 2 is ''mapped'' to 26 steps (by definition, as we are equally dividing [[2/1]] into 26 steps, each representing an interval with frequency ratio 2<sup>1/26</sup>/1) | * prime 2 is ''mapped'' to 26 steps (by definition, as we are equally dividing [[2/1]] into 26 steps, each representing an interval with frequency ratio 2<sup>1/26</sup>/1) | ||
* prime 3 is ''mapped '' to log<sub>2</sub>(3) × 26 = 41.209… steps, which we round to 41 steps (because otherwise we are using frequency ratios not present in 26edo), meaning 2<sup>41/26</sup> = 2.983…/1 is the frequency ratio of 26edo that we use to approximate the [[3/1]] frequency ratio, so the correct statement is prime 3 is ''mapped to 41 steps'' (not a fractional amount) | * prime 3 is ''mapped '' to log<sub>2</sub>(3) × 26 = 41.209… steps, which we round to 41 steps (because otherwise we are using frequency ratios not present in 26edo), meaning 2<sup>41/26</sup> = 2.983…/1 is the frequency ratio of 26edo that we use to approximate the [[3/1]] frequency ratio, so the correct statement is prime 3 is ''mapped to 41 steps'' (not a fractional amount) | ||