# (1) What are the difference tones of these intervals? (2) How do I know which interval they are?

The image above (coming from my MUS 204 course) shows three intervals and three difference tones, the latter being what I must calculate. I am not asking for the answer to each, I only would like to know how we would know the first interval (far left) is a perfect fifth and the second, a major third, and how to find the last interval. The former two are what I believe my professor labeled them out to be, but I do not know why.

Also, I would like to know how to find the difference tone. The first interval is an octave down from E4, so the difference tone is E3. However, why would I find the difference tone this way? How would I apply finding it to the other intervals?

• Did you skip a course with a number like MUS 103 or 104? Normally by the time you’re taking a 200 level course, you would have been taught how to recognize intervals. Commented Oct 10, 2021 at 18:15
• @ToddWilcox For MUS 204, one of the pre-requisites was playing a musical instrument for three years, which does not translate, for me, into knowing the intervals. I had not taken any MUS course before, and taking one was not required if you played an instrument for three years. Commented Oct 10, 2021 at 18:53
• I'm curious to know the actual course title for MUS 204. Based on the questions you're asking, it's unlike any music theory course I've had the opportunity to take. Commented Oct 10, 2021 at 19:42
• @Aaron Sure, it's "Musical Instruments, Sound, Perception, and Creativity". Commented Oct 10, 2021 at 19:44
• It’s too bad the teacher didn’t explain how to do the assignment before they assigned it, given the fact that there’s no prior music theory required for the course Commented Oct 11, 2021 at 1:47

These are two quite unrelated questions. Concerning recognizing intervals I would suggest you reading a textbook about basics of music theory, or perhaps asking a separate, very focused question about what you don't understand about it.

Concerning difference tones (often called combination tones): you find their frequency by taking difference of frequencies of the notes.

For a perfect fifth, if the frequency of the lower note is `f`, frequency of the higher note is `3/2f` (just intonation). The difference between the two is `3/2f – f = 1/2f`, that is half of the lower note frequency, thus the combination tone is an octave below it.

In just intonation frequency of major third is `5/4f`. The difference `5/4f – f = 1/4f`, that is a note two octaves below the lower note. However in 12-TET size major third differs quite much from that in just intonation, it is `2⁴⸍¹²f ≈ 1.26` rather than `5/4 = 1.25`. The pitch of the difference of the frequencies will be then:

``````1200·log₂[(2⁴⸍¹²f – f)/f] = 1200·log₂[2⁴⸍¹² – 1] ≈ –2333 cents
``````

2333 cents below the lower note, or 67 cents (more than a quarter tone) above the note that is two octaves below the lower note. This demonstrates how pitch of the combination tones is very sensitive to small differences in frequencies of the input notes.

• Your explanation is helpful! Let's say we were to identify the note, instead of how many cents the combination tone is. After self-learning intervals, I believe the third one is a minor third, thus 6/5f - f = 1/5f. How would we find the note? Commented Oct 10, 2021 at 23:52
• @Renée put the numbers in the formula: 1200·log₂[1/5] = –2786 cents, that is a note 2 octaves and a major third below the lower note, +24 cents. Then try the same with 12-TET minor third and you find a semitone lower note! Commented Oct 11, 2021 at 1:11

"The former two are what I believe my professor labeled them out to be, but I do not know why." - Intervals are measures of the "distance" (both in frequency, and on keyboard instruments [most notably], physical) between two notes. Western music has divided the octave (the "distance" between two notes whose frequency ratio is exactly 2:1) into twelve "steps" (which, in modern equal temperament, are all equal). By counting the number of such "steps" between each note, you can identify which interval it is (see here for a table). Eventually, with familiarity and practice, this becomes something you don't have to think of. This website (among many others) can be used if you want practice.

As to the second question, that seems to have been answered; although I'll try a simpler answer: a difference is the result of a substraction (source); a "difference tone" is the tone at the frequency found when you calculate the difference between the highest and the lowest note (in your first example, this would indeed be E3). This page goes to more details.

• 'and on keyboard instruments... - on my piano, physically, E to G# is 57mm! Not sure that's what you meant..!
– Tim
Commented Oct 11, 2021 at 6:35
• Interesting! Wiki appears to say something different - about a P5 interval, where the notes are part of a harmonic sequence, there's no difference tone. But isn't that exactly what violinists are listening for when tuning?
– Tim
Commented Oct 11, 2021 at 10:12
• @Tim Much as with a unison, there's sure a difference tone (or noticeable interference pattern of "beats," if not an actual tone in audible range) if one of them is off by a few cents! Commented Oct 11, 2021 at 12:29

'Difference tone' is maybe a term that hasn't travelled well across the Atlantic, but here goes.

Having played an instrument for >3 yrs doesn't mean knowing about intervals! It's maybe not the best criterion. Your prof. really should have explained why E-B is P5, rather than merely telling you. Many, many people who've played for way more than 3 yrs wouldn't know that, and for most of them, there'd be no reason to know, either.

The questions appear to be about intervals, but there really isn't any need to involve any more than simple maths.The reason E>B is P5 is twofold: E F G A B = 5 'steps' (including the 1st). Now why P? P = perfect, and means the two notes are the same - either both ♮, both ♯ or both ♭. Had just either been one of those, the interval would still be 5, but not P5. Still 5, due to E F G A B. That's a cunning plan, only to be spoiled by Bs and Fs...) A safer plan is count the semitones between - P5 always has 7 (without mentioning d6 - oops).

2nd one is, as you both say, M3. Major third. E F G - 3. 4 semitones. By decreasing the space by one semitone, in your case G♯ to G♮, that interval is now m3 - minor third.

Difference tones (also Tartini tones, after Giuseppi, who 'discovered' them) are the pitches heard when two notes are heard - and another note is also apparent. Maths alert! Say one pitch is 1500 Hz, the other 2000Hz. The difference is 500Hz, and were the two notes sufficiently loudly played, a pitch at 500Hz would be heard. Interesting, though - if both notes are part of a harmonic series (i.e. P5, no extra sound will be heard. (Consider Q1). If both pitches are quite close, a different effect will be produced, a sort of warbling, not unlike tremolo, as used by many guitarists when checking tuning.

• The 'warbling' that guitarists use to tune is exactly the same effect, but far below audible frequencies (i.e. less than about 20Hz). Commented Oct 11, 2021 at 8:57
• @PiedPiper - and as the two strings align, it should get down to 0Hz.
– Tim
Commented Oct 11, 2021 at 9:18