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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...

Dom
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Kip Ingram
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2 Answers2

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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.

Fugu
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  • 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
  • Well, that seems to bring me to what I talked about in the original post. I see that C-E is 4 half steps, and E-C is eight half steps. And eight half steps from the root isn't an interval that's in my scale. When you say "there is E-C# to consider," what exactly do you mean? I'm sorry - I'm not trying to argue but just to understand. I appreciate the time you're taking to answer. – Kip Ingram Jul 28 '17 at 19:39
  • As I said in my answer, this is a very common blind spot in theory knowledge for a lot of people, so I absolutely don't blame you. There is E-C# to consider because this is is also a sixth; the argument you set forth in your previous comment is that the naming is arbitrary because you can call any two set of intervals perfect, but you can't: If you call C-E a perfect third and E-C a perfect sixth, what do you call E-C#? An augmented sixth? Is C#-E now a diminished sixth? What of C#-Eb and its inversion? This comment is getting long so I'm going to break it up. – Fugu Jul 28 '17 at 20:04
  • The way that you're trying to analyze interval names is very divorced from the actual process of naming intervals. As I said previously, interval names have nothing at all to do with scales and I'm not totally sure why you're trying to analyze it from that perspective. If I understand what you're doing correctly, the correlation between an interval being named perfect and satisfying the criteria you've set up is simply because the major scale, like most commonly-used septatonic scales, puts the non-octave perfect intervals right in the middle. – Fugu Jul 28 '17 at 20:10
  • @Fugu you are just arguing about names, not things. Nothing in *music* would change if C-E was called and "Ermyntrude third", and C#-E a "Zebedee third", and C-G a "Dougal fifth." (Google "Magic Roundabout characters" if you don't get the references.) The only justification for a particular name is that everyone knows what it means! –  Jul 28 '17 at 20:15
  • ... there is no more logic behind the use of "perfect" and "imperfect" than the names used for rhythms in mensural notation (before 1600) when rhythmic divisions into three parts were called "perfect" (by analogy with God and the Holy Trinity) and divisions into two were called "imperfect". When religious dogmatism is sufficient reason for something, you don't need logic! –  Jul 28 '17 at 20:19
  • Sure, it doesn't matter whether they're called perfect or kdfjhdfkgd, but that's not the point: The point is that perfect is distinct from major and you can't conflate the two without losing the meaning of one or both terms. – Fugu Jul 28 '17 at 20:27
  • So, maybe this will help me, Fugu. Do you always call the 4th and 5th in a scale perfect, or is that term reserved for intervals of 5 and 7 semitones, respectively? – Kip Ingram Jul 28 '17 at 20:41
  • Because those are the ones that correspond to the ratios 4/3 and 3/2 in terms of frequency. – Kip Ingram Jul 28 '17 at 20:42
  • The fourth and the fifth of a scale are not always perfect intervals. As I said before, there is no connection between interval terminology and scales. Take a look at C major pentatonic: C D E G A. The fourth degree of this scale is a perfect fifth, but the fifth degree is a major sixth. Here's another scale, the rather obsucre B super locrian diminished: B C D E F G Ab. Here the fourth is perfect but the fifth is diminished. It's no accident that the major scale (and most commonly used septatonic scales) have perfect fourths and fifths from the root in them, but it's not why they're named. – Fugu Jul 28 '17 at 20:48
  • Ok. Then the perfect intervals are those where the upper tone is 4/3 or 3/2 of the lower one, or some octave derivative of those. – Kip Ingram Jul 28 '17 at 21:44
  • Or 1/1 or 2/1 (to get unison and octave). – Kip Ingram Jul 28 '17 at 21:44
  • Like leftaroundabout said. – Kip Ingram Jul 28 '17 at 21:45
  • Sure, but that's a happy accident and not why they're called what they are. That has to do with inversions. – Fugu Jul 28 '17 at 21:49
  • Alright, I had some time to think about this, and this is the best way I can explain this: Let's say we're no longer going to drive a distinction between major/minor and perfect. So now we have a major octave instead of a perfect octave. What happens when we invert the major octave? Because major intervals, when inverted, become minor, we now have to call this a minor octave, even though it's the same two notes. We now have two names for the same interval, which is impossible: Fundamentally, each set of two notes can be described by only one interval name. – Fugu Jul 28 '17 at 22:15
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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 f1⁄f0 = 2 k · 3 j for some k, j ∈ ℤ. That includes beyond doubt

  • Octave: f1⁄f0 = 2, which is a span of 1200 ct
  • Fifth: f1⁄f0 = 3⁄2, or 702 ct
  • Fourth: f1⁄f0 = 4⁄3, 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 5⁄4 (386ct) or 81⁄64 (408 ct)
  • Minor third: either 6⁄5 (316ct) or 32⁄27 (294 ct)
  • Major second: either 9⁄8 (204ct) or 10⁄9 (182 ct)

and so on.

leftaroundabout
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  • 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!) –  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 01:12
  • @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 06:42