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Find the closest ratios for a scale

Extract from help for Fractal Tune Smithy

Cents or ratios Cents or n-et Decimal Herz

Scale:

1/1 in Hz (middle C = 261.62556 Hz, A = 440)

Max quotient Decimal places Tolerance cents

upper lower both closest

Primes (or composite factors)

Ratios

As cents

Cents diffs

Enter a single value to see successive ratio approximations to it.

Enter several values, e.g a scale, and the most accurate ratio found will be shown for each one. Scroll down this page to see the successive approximations for each scale degree

To convert the scale from cents to herz etc, change the selection for the scale notation. Values in herz are shown to two extra decimal places, and decimals are shown to four extra decimal places. When converting to n-et, the tolerance is used a tenth of the tolerance selected for the scale results, and all n-ets are checked up to 1200-et.

The n-et notation works like this: 7/17 means the 7th degree of seventeen equal temperament - i.e. 7*1200/17 cents. So it looks like a ratio but is a quick way of entering n-ets

With the cents or ratios and cents or n-et options, enter ratios or n-et notation with a '/': 5/4, and for values in cents notation, enter values without a '/': 250. I.e it treats it as a ratio or n-et notation if it looks like one.

The tolerance is the minimum distance you want the ratio to be from the cents value. If you set the max quotient high, and the tolerance low, then it will be slow, because this script goes through the quotients one at a time testing them all.

To halt the calculation, and try again, use your browser STOP button.

List all the primes you want to use as factors, or if you want to see all successive approximations, leave the primes field blank.

To set a maximum power for a prime, do it like this: "2^8 3^5" to set max powers of 2^8 and 3^5.

To exclude a prime, show it as a negative number. E.g. use -7 to search for all numbers except those divisible by 7. This can be combined with the positive primes, e.g. use 3 -9 to allow any multiple of 3, except for those that are a multiple of 9 (not sure why one would want to do it, but it comes for free!).

The entries in the factors field can also be composite. So for instance, if you enter 6 as a value, you will find ratios with denumerator or denumerator a multiple of 6 .

 

Successive ratio approximations for each degree of the scale

Ratios for degree 1

As cents

Cents diffs

Ratios for degree 2

As cents

Cents diffs

Ratios for degree 3

As cents

Cents diffs

Ratios for degree 4

As cents

Cents diffs

Ratios for degree 5

As cents

Cents diffs

Ratios for degree 6

As cents

Cents diffs

Ratios for degree 7

As cents

Cents diffs

Ratios for degree 8

As cents

Cents diffs

Ratios for degree 9

As cents

Cents diffs

Ratios for degree 10

As cents

Cents diffs

Ratios for degree 11

As cents

Cents diffs

Ratios for degree 12

As cents

Cents diffs

Ratios for degree 13

As cents

Cents diffs

Ratios for degree 14

As cents

Cents diffs

Ratios for degree 15

As cents

Cents diffs

Ratios for degree 16

As cents

Cents diffs

Ratios for degree 17

As cents

Cents diffs

Ratios for degree 18

As cents

Cents diffs

Ratios for degree 19

As cents

Cents diffs

Ratios for degree 20

As cents

Cents diffs

Ratios for degree 21

As cents

Cents diffs

Ratios for degree 22

As cents

Cents diffs

Ratios for degree 23

As cents

Cents diffs

Ratios for degree 24

As cents

Cents diffs

Ratios for degree 25

As cents

Cents diffs

Ratios for degree 26

As cents

Cents diffs

Ratios for degree 27

As cents

Cents diffs

Ratios for degree 28

As cents

Cents diffs

Ratios for degree 29

As cents

Cents diffs

Ratios for degree 30

As cents

Cents diffs

Ratios for degree 31

As cents

Cents diffs

Ratios for degree 32

As cents

Cents diffs