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Ok, so I am very curious about why the major scale 4th and 5th are deemed "perfect" while other intervals are "major." This may just be an arbitrary thing based on sounds we find pleasing, but I've been at least trying to find something more rigorous. I did notice one thing.

If you take the C major scale, C D E F G A B C, we know there are 12 semitones from C to C. The 4th, F, is five semitones from C, and the 5th, G is at 7 semitones from C. 5 + 7 = 12. In other words, if you choose either one of those notes and find its distance from C in semitones, then it turns out the remaining number of semitones to reach 12 is the position of a note that is IN THE SCALE AS WELL.

If you check D, the 2nd, it's at 2 semitones, and you'd need 10 to get to 12, but the scale doesn't have a note 10 semitones above C. Similarly for E, the 3rd; it's 4 semitones from C, and there is nothing in key at 8. F and G, the 4th and 5th, are unique by this criterion.

Ok, so far so good. I then said to myself, "Ok, the major scale is the Ionian mode. What about the other modes?" So I proceeded to walk through all seven of them, and the answer varies from mode to mode. Specifically, you get this:

Ionian: 4th+5th

Dorian: EVERY SINGLE NOTE

Phrygian: 4th+5th

Lydian: 4th only (it's 6 semitones above the root, so it's its own partner)

Mixolydian: 4th+5th and 2nd+7th

Aeolian: 4th+5th and 2nd+7th

Locrian: 5th only (same story - it's 6 semitones above the root)

So this is all very interesting. But really I'm just a mathematically-inclined engineer with a strong curiosity about music theory. I'm primarily self-taught using the internet. So my question is simple:

Does what I laid out above actually mean anything? Or is it just a curious coincidence? I don't really believe in coincidences - usually when something this intriguing shows up there's a reason for it.

As a possible motivator for insights, the Aeolian mode is the natural minor scale, and in addition to 4th+5th that one has 2nd+7th. So is there any existing knowledge out there about the 2nd and 7th intervals taking on any new prominence in the natural minor scale?

One more interesting thing. If you use the circle of fifths to "order" the modes, you get this order:

Lydian: 4th only

Ionian: 4th+5th

Mixolydian: 4th+5th, 2nd+7th

                                 MAJOR ABOVE

                                 MINOR BELOW

Dorian: Every note

Aeolian: 4th+5th, 2nd+7th

Phrygian: 4th+5th

Locrian: 5th only

There's structure and pattern there...

marked as duplicate by Dom theory Jan 28 '18 at 20:00

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To your first question, an interval is named perfect when inverting it results in another perfect interval. This is a source of confusion and mystery for no small number of musicians, but it's actually fairly straightforward.

In all cases, the logic is commutative.

When you invert a perfect interval, it becomes perfect. When you invert a major interval, it becomes minor. When you invert an augmented interval, it becomes diminished.

It breaks down like this:

Augmented - Major - Minor - Diminished

Augmented - Perfect - Diminished

Fourths, fifths, unison, and octaves are all perfect intervals.

What your analysis of the existence (or non-existence) of a note at 12 semitones minus the number of semitones from the root away is basically doing is getting to this point in a very... erm, roundabout way.

The second is a very important note in the minor scale and the seventh is a very important note in every septatonic scale, but on this subject one could write a book.

  • My concern with that definition is that it seems arbitrary - you've picked both of the perfect intervals, and they're inversions of each other. – Kip Ingram Jul 28 '17 at 19:26
  • Ooops - didn't realize it was going to submit on enter. Continuing, if the inversion also has to be in the scale, that's getting a bit more demanding. – Kip Ingram Jul 28 '17 at 19:26
  • Interval names have nothing at all to do with scales, really. It is nevertheless true that if both notes of an interval are in a key, then the inversion of that interval is in that key. I'm not sure how that's relevant, however. There are, by the way, far more than two perfect intervals. There's an infinite number, and it is still true that a twelfth -- a perfect interval -- inverts into a perfect interval and is therefore so named. – Fugu Jul 28 '17 at 19:28
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    Do you see what I meant about it seeming arbitrary? C-G is a 5th. G-C is a 4th. So the definition seems to be saying that the 5th is perfect because the 4th is perfect, and the 4th is perfect because the 5th is perfect. It's circular. If I just decided to call the 3rd C-E and the 6th E-C perfect, then the same conditions would be satisfied. I'm trying to find something inarguably unique about perfect intervals. – Kip Ingram Jul 28 '17 at 19:35
  • You can't call C-E perfect because when you invert it the corresponding interval isn't also perfect. There is E-C# to consider. – Fugu Jul 28 '17 at 19:36
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An interval is perfect when it can be derived unambiguously as a Pythagorean ratio, i.e. when the frequency ratio is f1f0 = 2 k · 3 j for some k, j ∈ ℤ. That includes beyond doubt

  • Octave: f1f0 = 2, which is a span of 1200 ct
  • Fifth: f1f0 = 32, or 702 ct
  • Fourth: f1f0 = 43, or 498 ct

...and compounds thereof. Now you may say, but why isn't every interval perfect then – can't you just traverse the circle of fifths..? – that's why I wrote unambiguously. You can indeed stack those perfect fifths, but that's not compatible with post-1600 Western harmony, which also makes use of frequency ratios including the number 5.

For all of the intervals from Western standard theory save the ones above, there are thus at least two intonation ratios available, and it's not possible to say universally which one is right: it depends on the context (and to some degree, the performer) which one is chosen.

  • Major third: either 54 (386ct) or 8164 (408 ct)
  • Minor third: either 65 (316ct) or 3227 (294 ct)
  • Major second: either 98 (204ct) or 109 (182 ct)

and so on.

  • I'm not convinced there is any historical justification for this logic. It doesn't fit with two inconvenient facts: (1) the catholic church decreed than 4ths and 5ths were "perfect" and 3rds and 6ths were "imperfect," and (2) the same church, at the same time, decreed that the size of a major third should be 81/64 and nothing else - tuning systems using the 5/4 ratio were forbidden. Of course they had a 1,000 year track record of making dogmatic pronouncements before they made those two, so nobody much argued (except the peasants who carried on using 5/4, but they didn't count for anything!) – user19146 Jul 28 '17 at 20:09
  • Well, I think your answer is certainly mainstream, but it does bring us back to a human judgement call - we're giving preference to the tones that arise from the smallest integers in the calculations. A line was drawn. And that's fine - it may be that the ultimate answer is "because those sound special to us, and we decided." I still think the pattern I observed is pretty interesting, though. :-) – Kip Ingram Jul 28 '17 at 20:19
  • Note that many modern players use 12-TET, so each semitone is actually just 1/12 of an octave higher. Unless you're a stringed, fretless instrumentalist or a vocalist, your instrument (mostly) determines your tuning for you. (Yes, I know how wind instruments can bend their pitch and those which are tuned to a key exist.) – CAD97 Jul 29 '17 at 1:12
  • @CAD97 apart from the fact that you could easily turn that statement around (“unless you're a keyboard-, fretted-string or valved/keyed wind instrumentalist...”) it's doesn't refute my point. In 12-edo, the impure intervals are literally imperfect in the sense of always being significantly out of tune. – leftaroundabout Jul 29 '17 at 6:42

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