My question comes from my MUS 204 course, which is the following: How many steps in 15-EDO should be used to approximate just perfect fifth?

I do not know how to calculate the number of steps. However, I know how to calculate the step size by dividing 1200 by 15, which is 80 cents. I also know just perfect fifth (ratio 3/2) is 702 cents.

What steps do we use to find the correct answer to the question above? (The answer, according to my answer sheet, is 9 steps.)

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    Well, you want ≈700 g of chocolate. You have chocolate bars weighing 80 g each. How many chocolate bars do you need? Oct 9 '21 at 22:49
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    @leftaroundabout The number of chocolate bars I need does not relate to the number of grams I want.
    – Aaron
    Oct 9 '21 at 23:02

As with How many cents comprise a quarter-tone in 15-EDO? the answer lies in multiplication and division.

There are 1200 cents in an octave, and a 15-EDO step (semitone) is 1200/15 = 80 cents.

We want to know what multiple of 80 is closest to a perfect fifth, which is defined as 702 cents.

80 × 8 = 640; 702 - 640 = 62
80 × 9 = 720; 720 - 702 = 18

Clearly 9 semitones in 15-EDO is closer to a perfect fifth than is 8 semitones.

  • 1
    Why is '*' deemed to make any sense - I always used 'x' when I did my sums. Maybe I'm just behind the times?
    – Tim
    Oct 10 '21 at 8:03
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    @Tim the preferred symbol is actually × (U+d7 MULTIPLICATION SIGN), or alternatively (U+22c5 DOT OPERATOR), but * isn't exactly new either, having been used in pragramming since the 50s. Whether it makes “sense”? – well, all of these are just conventions. At any rate x (U+78 LATIN SMALL LETTER X) should not be used, because that's a common variable name. (Arguably, the correct symbol for the variable name is actually 𝑥 (U+1d465 MATHEMATICAL ITALIC SMALL X), but almost nobody uses that and operating systems do not consistently support it.) Oct 10 '21 at 9:55
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    BTW the multiplication sign can be typed in StackExchange answers as ×. Oct 10 '21 at 10:00
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    The first subtraction should be 702 - 640 = 62.
    – phoog
    Oct 11 '21 at 10:38
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    Note, however, that defining P5 as 9/15=3/5 of an octave means that the intervals M3=P4 and M7=P8 become indistinguishable.
    – dan04
    Oct 11 '21 at 22:52

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