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I stumbled on this question while checking a grade 7 mathematics book. It grabbed my eyes having been introduced to some of music's amazing topics.

I know that harmony ( or more accurately, consonance) occurs when two pitches vibrate at frequencies in small integer ratios e.g., 2:1, 3:2, 4:3, 5:4.

However this problem that you see in the screenshot claims a much simpler case. it states that if the ratio of the frequencies of two notes can be simplified, the two notes are harmonious. I am thinking about other possible numbers. Apart from the E and G in the example, imagine a 445 and 315 Hz notes. They make a ratio that can be simplified to 89:63. Would you call that harmony? 414 and 500 make up a 207:250 ratio. Would you consider the notes as harmonious?

I'd refute this claim according to my humble knowledge. However, I am not sure. it might just be true. Therefore, I'd like to ask your opinions about it.

Many thanks!

A screenshot from a grade 7 Maths book

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  • See also Equal temperament
    – Karlo
    Commented May 10, 2017 at 12:58
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    To me this is like a math textbook that says something about having some number of cows on some amount of land and an actual cattle farmer reading it and thinking, "There's no way 50 cows can survive on a single acre!" or "With that density this farmer will be out of business in two years max!" Math textbooks tend to fail often when they try to create real world examples of problems.
    – Todd Wilcox
    Commented May 10, 2017 at 15:06

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This is flabbergastingly in the "not even wrong" category. Consonance in relation to pure intervals is often defined as a frequency ratio of small numbers (I think the limit is usually considered to be around 6 but don't take my word for it).

However, pure intervals occur only in just intonation, and just intonation only occurs relative to particular scales (like C major).

In equal temperament, by far the most common temperament in use these days, the relation of the frequencies of a major third is 2^(1/4), an irrational number (about 1.19). A pure minor third is indeed 6/5, namely 1.2.

Pythagorean tuning, one "just" temperament with rational ratios, has G/E turn out as 27:16 instead.

So basically, this is one of the rather common math text questions which bungle an actual truth or insight to the point of incomprehensibility and turn it into a mathematical task with at best a rather tenuous relation to the real-world relations having inspired it.

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    Don't understand the '(like C major)' part. Once an instrument is tuned to just intonation, it would appear that it sounds best in one key only. That key may or may not be C. It's the relative pitches against each other, not the key itself, am I wrong?
    – Tim
    Commented May 10, 2017 at 15:40
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    The OP forgot the word small in "small numbers". In this context "small" typically means "less than about 6", at least for must humans who have been listening to Western music since birth. In some other cultures, all the unwritten assumptions in the question are just wrong. Also, hearing is not mathematically exact - if a ratio of 396:330 "sounds harmonious", then a ratio of 396.12345: 329.8754 won't sound much different, for most listeners. Also, in his/her examples, the OP might have made the assumption that all frequencies are integers, which of course is not correct.
    – user19146
    Commented May 10, 2017 at 17:36
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If the ratio of the frequencies of two notes can be simplified, the two notes are harmonious

Taken literally, This is incorrect, because (as you say) you could take a ratio that relates to a dissonant interval - say 64:45 (which is one possible ratio for a Tritone), double each number (to 128:90) - and then the above statement would imply that because 128:90 can be simplified, that a tritone is harmonious.

However, perhaps a charitable interpretation of the statement would be

If the ratio of the frequencies of two notes can be simplified to a ratio involving only small numbers, then the notes are harmonious.

This would be true, but only because it's basically just a restatement of the idea that:

frequency ratios involving small numbers correspond to harmonious intervals.

So the original statement is not incorrect if you interpret it 'charitably', but it's still saying something trivial.

Of course it's actually not only literally small-number ratios that are consonant, but also ratios that are close to them as a fraction (which may themselves be very large number ratios!). We rely on this fact for equal temperament to 'work'.

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