I've been trying to find an answer, but to no avail. Is what we call a perfect interval somewhat arbitrary? It seems as if the modern definition is "perfect under inversion". I know the other thing people say is that it is consonant, but I can't find a rigorous definition of consonance.
Is there a solid definition of perfect intervals, lying around somewhere I just can't find?
My answer builds on the answer contributed by DR6.
Based on your reaction to other very good answers posted here already, your question seems to boil down to: "Why do humans innately feel that certain intervals are consonant". And so much so that they are willing to call them "perfect". Before getting to that question, let's look at why Western culture might consider them "perfect". My answer to your question will be rather freeform because the truth of the matter is there is not really good answer to your question outside the music theory-based explanations given above.
The modern Western music system has been inherited from some of the groundwork set by Pythagoras. It has been heavily modified to the point now that the modern 12-tone equal temperament we use now has the spirit of the original ideas from Pythagoras even if it differs greatly in many other ways. To Pythagoras, and possibly many Greeks at the time, certain intervals sounded very pleasing to the ear. Mathematically, these intervals are superparticular ratios [(n + 1)/n) or multiples [(x*n)/n]. For example, 4/3 is a superparticular ratio and 3/1 is a multiple. In other words, when the two frequencies resonate together and the ratio of the frequencies comes out in either of these forms many people in Western culture would agree they are pleasing. The perfect ratios display this quality in the best sense: 2/1 is an octave, 3/2 is a perfect fifth, and 4/3 is a perfect fourth. There is the least amount of conflict in the frequencies between the notes allowing for more complete symmetrical intersection between the waveforms. This is probably why Pythagoras liked these intervals - the Pythagoreans loved this kind of mathematical perfection. He liked it so much he tried to develop a tuning system out of it (Pythagorean Tuning) which ended being impossible without introducing a tuning error (the Pythagorean Comma).
I am not too clear on how Pythagoras's discoveries exactly carried over through time but his ideas were often used and cited by other musicologists through time. One example is Ptolemy who created scales based of Pythagorean tuning that included other less consonant intervals (thirds). What I am getting at here is that our assumption of the "perfect" intervals derives from the fact that the system's originator (and possibly his culture) deemed them to be perfect. It's hard to say why the name persisted through time but needless to say, thousands of tunings systems were developed after Pythagoras, most of which tried to preserve the perfect fifth, fourth, and the octave while allowing wiggle room for other intervals to fit together in the scales (I'm oversimplifying but that's the idea).
But is it pleasing to humans in general? That depends. Many cultures developed other systems that don't necessarily have this obsession with the perfect intervals or used many others equally. Other cultures (Persian music) have divided the octave into 53-tones, 24-tones (some forms of Indian music), and other divisions. One response to this is that the majority of non-Western cultures tended to develop music systems that were melodically complex: complex scales over a single droning note, but not harmonically complex like Western music. So perhaps they never needed to develop the notions of "perfect" in the first place. There is also the fact that in the modern era we have become increasingly attracted to dissonant or unusual forms of harmony. There is widespread interest in rock/metal which emphasizes distorting the sound wave to emphasis dissonant overtones (even if the intervals actually played are quite consonant). Dubstep is not exactly harmonically pleasing either but it is popular. Modern Jazz uses some complex and dissonant forms of harmony. A lot of 20th century classical music is also very dissonant. The question comes down to if it's a matter of taste, the unexpected (things that surprise us make things interesting, a change from regularity), culture/social norms, or if it's innate. There's also a difference between enjoying dissonant music and actually finding it pleasing. I love dissonant music but I don't really find it more "pleasing" than consonant music - I like it because it is jarring.
Music psychology and cognitive neuroscience has not come to a firm conclusion on this question. There have been a lot of studies on this topic but none are quite conclusive. One simple explanation is that evolutionarily, the human brain learned to find patterns and structure to apply semantic meaning. This means that we seek things that have regularity and predictability and attempt to assign meaning to things to help them to fit within these frameworks. Dissonant music deliberately goes outside predictable frequency ratios that line up, producing uneven sounds. Perhaps the aversion to these sounds is a by-product of the general manner in which the brain functions in the world.
But this is a post hoc explanation. Cognitive neuroscience has been asking these questions for a long time and modern advances in computational neuroscience may soon provide an answer. A simple look at this question can be found in this Nature article.
To summarize: We probably call it "perfect" because of Pythagoras and musicologists that came after him. We probably think it's "perfect" for cultural and social reasons. If it is really "perfect" to us innately is to be determined.
There are four types of perfect interval: perfect unison, perfect fourth, perfect fifth, and perfect octave.
These can be thought of as belonging to two groups. In the first group, all intervals of a unison or an octave are called perfect because the note is not changed. An octave is twice (or half) the frequency of the first note.
The second group includes the perfect fifth or perfect fourth. Actually, traditionally the fourth was not considered consonant. However, since the fifth is perfect, and the inversion of the fifth is a fourth, then the fourth is exactly the same thing as a fifth and must also be perfect. These notes add a very slight amount of coloring but not really enough to constitute a harmony.
Rather than using dissonance or consonance (somewhat subjective terms), I prefer to think about it as adding harmonic content or not.
Take any root note, and add as many unisons, octaves, and fifths (or fourths, but please not both, because now these two will conflict with each other), and you have no real harmony. The unisons and octaves do not add harmonic content because they're the same note as the root. And the fifth doesn't add harmonic content because it is the strongest overtone in the harmonic series. In a nutshell, if you play the root note C, you are also to some extent playing a G because the G is audibly present in the harmonic series of the root note C. Whenever anyone plays a C, they're also playing a G, because physics. So whether you then use your instrument to play a second G or not, the G is present inside of the C anyways.
So perfect intervals are those which are so consonant that they don't add any harmony.
Note: edited for clarity due to a number of comments asking for clarification.
A “perfect” interval is one that has nice small integer frequency ratios in Pythagorean tuning. These are traditionally considered the most consonant intervals.
Major and minor intervals have more complex ratios:
(They are distinguished by major intervals having a power of 3 in the numerator, and minor intervals having a power of 3 in the denominator.)
Augmented and diminished ratios, being father away from unison on the circle of fifths, are more complex still.
This classification may not make as much sense in other tuning systems like 5-limit just intonation, which aims to make major and minor thirds more consonant by simplifying their ratios to 5:4 and 6:5, or to the now-ubiquitous equal temperament which abandons integer ratios altogether. But musical terminology is slow to change.
Perfect intervals are the ones that don't have two forms: major and minor.
C Db D Eb E F F# G Ab A Bb B C
root minor major minor major perfect tritone perfect minor major minor major octave
2nd 2nd 3rd 3rd 4th aug/dim 5th 6th 6th 7th 7th
4th/5th
The tritone is just an oddball from this (over-)simplified view.
All intervals can be turned upside down.(Called inverted). Thus a C-E as a major third, when played E-C becomes a minor sixth. There is a 'rule of nine'.Minors become majors, majors become minors, augmenteds become diminisheds, etc. The exceptions are the octaves, 4ths and 5ths. (Unison doesn't count !) Those do not change their identities. A 4th of C-F becomes a 5th of F-C, BUT, the interval stays as is - perfect. It hasn't changed.
Since this has come up in comments, I feel like maybe it's different enough information to write a separate answer for those interested in the history of the actual term "perfect" consonance.
While SyntonicC's answer rightly points out the root of this distinction arising partly from Pythagorean theory, the history is a little more complicated.
To the Pythagoreans, consonance was thought of melodically (rather than as simultaneous pitches). There's a lot of detail I'll gloss over, but briefly their symphoniai (things "agreeing in sound") encompassed intervals formed with ratios of the numbers 1 through 4 (symbolically represented in their system with the number 10 = 1+2+3+4). The symphoniai thus included the ratios 2:1 (perfect octave), 3:2 (perfect fifth), 4:3 (perfect fourth), 3:1 (perfect twelfth), and 4:1 (double octave). There were all sorts of mathematical and mystical reasons they gave as justifications for treating these numbers as special. (I would note that the "perfect" eleventh is notably absent here, despite being simply composed of a perfect fourth and an octave, a point of contention over the millennia both in ancient Greece and in medieval Europe.)
A lot of these ideas were inherited by medieval Europe, translated imperfectly (no pun intended) by Boethius and others. And there were lots of classifications on intervals, but the first use of term "perfect" (Latin perfectus) came in the early 13th century, where intervals were generally classified into three categories:
As for why the term perfectus was chosen, it likely had to do with the fact that unisons obviously enjoy a special status, and octave equivalence had become commonly accepted in the 11th and 12th centuries to the point that notes in different octaves were referenced with the same letter. (This is not an obvious development -- the original letter systems for pitches often began with A and just kept going through the alphabet in different octaves.) Hence, by around 1200, all notes we call "A" would have been thought of as equivalent in some respects, thus any unisons or octaves created by them would be "perfect" intervals.
Over the 13th and 14th centuries, the fifth was gradually elevated to the perfectus category, while the fourth became sometimes perfectus and sometimes a dissonance in practical counterpoint, which is still generally its status in modern music theory. It's likely that the elevation of the fifth and fourth to the perfectus category had something to do with the traditional Greek list of symphoniai intervals.
This two-fold classification of perfectus vs. imperfectus in consonances basically survives to the present day: i.e., "perfect" consonances are unisons, octaves, perfect fifths, and perfect fourths (and their compound intervals), while thirds and sixths are "imperfect" consonances.
Ultimately, the definition is somewhat arbitrary -- for the Greeks it had to do with the integers up to 4 (the tetractys) and their mystical appreciation of the number 10. For medieval folks, as they were trying to shuffle the fifth into the "perfect" category, they hedged about the fourth, as it already was causing counterpoint problems and being treated as dissonant sometimes. And then they started dealing with the practicalities that thirds and sixths sounded pretty good too, which led to more debates.
For a more detailed introduction to the historical issues, I might suggest starting with James Tenney's A History of Consonance and Dissonance.
"Is there a solid definition of perfect intervals, lying around somewhere I just can't find?"
Yes. A "perfect" interval is an interval that is not one of minor, major, diminished, augmented.
All the rest have answered in terms of high-level music theory concepts, but I think it can be interesting to look at the intervals as raw coefficients instead. Harmonic intervals between notes are the intervals that can be expressed with simple rational numbers, where a "simple" rational number is one with a small amount of small prime factors.
For example, the distance between two tones (let's say, 440Hz and 880 Hz) is an octave if the frequency of the second tone is exactly two times the frequency of the first: 2 and 1/2 are the simplest rational numbers possible after the unison.
As our ear detects two tones that only differ by an octave as the "same" tone, multiplying or dividing by 2 an arbitrary number of times doesn't make intervals less simple. This makes 3 the simplest "significant" prime number. A fifth is an interval of 3/2, and a fourth is an interval of 2/3*, so we may conclude that a perfect interval is an interval that contains at most a single 3 as a prime factor and no other prime factor(as I said, we don't care about 2s).
* Technically, in the equally tempered scale this is not literally true: a fifth is 2^(7/12), which sliightly differs from 3/2, but our brain can't tell the difference.
I like @Dan04's answer re. simple ratios, but the other ones are very dense. I want to add a more straight forward answer:
The distinction is based on how the interval classes relate to the tonal center.
Keep in mind notation and enharmonic spellings make a difference. A minor seventh and augmented sixth are the same distance, but they are "spelled" differently in notation and those enharmonic spellings are used to make the harmony clear in a score.
Tritone is an alternative term for augmented fourth or diminished fifth. Do not use it if you want your enharmonic spelling to be clear.
This wikipedia page covers a lot of this in detail https://en.wikipedia.org/wiki/Interval_(music)
I mostly agree with the answers given here and elsewhere on the site, and in particular, the answer here correctly states that:
The minor intervals are not minor because they are found in the minor scale and the same goes for major intervals. The intervals are ... based on ... and absolute distance in semitones.
In other words: when Western music theory decides that there's two versions of the same note, the sharp one is called "major" and the flat one is called "minor." The "perfect" notes are traditionally thought of as those that don't have different flavors.
In more detail: the chromatic scale is traditionally broken up into adjacent notes that are called "minor something" and "major something" respectively. The pattern breaks down at the middle, and this is where the perfect notes are found. In particular, we have:
Unison / Minor Second, Major Second / Minor Third, Major Third/ Perfect Fourth / A weird note that doesn't fit comfortably into traditional music theory / Perfect Fifth / Minor Sixth, Major Sixth / Minor Seventh, Major Seventh / Unison
However, these are historical comments. From a future-oriented perspective, the question is really whether we ought to introduce the notion of a perfect second (for example).
I would argue that we should.
From a JI perspective, the major second really splits into two notes, namely 9/8 (which is to be found at about 2.04 semitones above the tonic) and 10/9 (which is to be found at about 1.82 semitones above the tonic). Under 12-tone equal temperament, both these notes are given the same pitch - namely, they're both treated as being exactly 2 semitones above the tonic. However, you can add sweetness and sophistication to your music by ensuring they're treated differently. The question then arises of how to distinguish these notes terminologically.
I'd argue that 9/8 should be referred to as the "perfect second", while 10/9 should be referred to as the "major second." By adopting these conventions, we ensure that the three most important chords in the major scale have exactly one occurrence of a "major" note, which is always the middle note:
I = Unison, Major Third, Perfect Fifth
IV = Perfect Fourth, Major Sixth, Unison
V = Perfect Fifth, Major Seventh, Perfect Second
More generally, my position is roughly that "perfect" ought to mean Pythagorean, which means a note whose ratio only involves the prime numbers 2 and 3. The most important examples are:
1/1 (unison) 9/8 (perfect second) 4/3 (perfect fourth) 3/2 (perfect fifth) 16/9 (perfect seventh).
Of course, the note 16/9 (which is about 9.96 semitones above the tonic) is usually referred to as the minor seventh, but in my opinion it's better to reserve this name for the note 9/5 (which is about 10.18 semitones above the tonic). This doesn't quite accord with the historical meaning of the words "major" and "minor"; nonetheless, I think it significantly clarifies the underlying theory. In particular, referring to 16/9 as the "perfect seventh" ensures that the hree most important minor chords in the minor scale have exactly one "minor" note:
I = Unison, Minor Third, Perfect Fifth
IV = Perfect Fourth, Minor Sixth, Unison
V = Perfect Fifth, Minor Seventh, Perfect Second
For these reasons, if you're interested in microtonal music or just intonation, my position is that it's best to declare that "perfect" roughly means "pythagorean."
Other answers are about diatonic (major, minor…) intervals, but the concept can be extended to other moment-of-symmetry scales and rank-2 temperaments. Every such scale is generated by stacking an interval (generator) and repeating at an other interval (period). In short, the generator, the period and their inversions are perfect; all other intervals can be major or minor. It happens that the tone generated at the end doesn't line up with the tone generated at the beginning; that is where an augmented or diminished interval appears.
For example, diatonic scales are generated by tempered fifths and repeat at octaves. That makes fifths, octaves, fourths and firsts. C major has a diminished fifth between F and B.
A different example is the porcupine temperament generated by tempered seconds and repeating at octaves. The 7-tone porcupine scale has perfect seconds, octaves, sevenths and unisons. The interval between the last generated and the first generated tone is an augmented second.
Where exactly the augmented or diminished interval appears depends on the mode, that is choosing where in the scale you start, which is equivalent to choosing how much you generate down or up. So, in diatonic scales, Lydian has the diminished fifth at the end (from the tonic down) because it is generated by fifths always up. Locrian has the diminished fifth at the beginning (from the tonic up) because it is generated by fifths always down.
If you have a temperament of a rank higher than 2, you have more generators, so you have more different perfect intervals. But that can get messy quickly.
My understanding, and I don't remember where I learned this, is that the early Catholic church at first forbade harmony of any kind, then finally allowed only limited harmony with intervals that the church fathers considered "perfect" in the eyes (ears?) of God. This is why organum uses only perfect intervals.
The name "perfect" may be a reference to a numerical coincidence, which makes the interval of 7 semitones very close to the ratio 3:2 of frequencies.
27/12 = 1.4983...
3 / 2 = 1.5000...
Major and minor intervals are less precise:
24/12 = 1.2599...
5 / 4 = 1.2500...
which may make them annoying to the sensitive ear, as if e.g. your guitar is slightly out of tune.
This is only true for equal temperament tuning.
The fifth divides the octave with a fourth remaining above. The fourth divides the octave with a fifth remaining above. That is to complete the octave. Playing Perfect intervals that suggest no harmonic content and adding harmonic content is a'sound' approach to discovering the answer to the perfect interval question. All answers have certain validity. I think the best approach is the practice itself, which of course is music and musical instruments and listening.
Perfect intervals aren't simply there because they are the most consonant or stable or whatever. They are there because they have to be for it to even work in the first place and their presence helps define a lot of the music theory that we know today.
I'm going to take a different approach to explain this: proof by contradiction. Let's try to make a system of only diminished, minor, Major and Augmented intervals and see what we come up with.
We start out with some issues from the start. Prime = M1 is reasonable, but a m1 on B??? K, whatever, let's press on
Ah, this makes sense. m2 on C#, M2 on D, everything right where we want it
Woah, woah, hold on! m4 on F and M4 on a tritone!? I'm getting dizzy...
Ok, d5 on tritone, that's cool...m5 on G? ehhh...I guess that's ok...maybe?
This is weird, but I guess we could get used to it...
M7 = Octave!?
An octave is diminished 8!?!? nope nope nope nope nope
The DEFINITELY didn't work...Let's try something else
Ok, prime = P1, that's perfect! (...)
Ah, this makes sense. m2 on C#, M2 on D, everything right where we want it
Now, to avoid the issues from before, we'll put P4 on the most stable...
...and P5 on the other most stable
Back on track
See, isn't this nice? :)
Aaaaand back to an octave on P8 sigh of relief
Another interesting feature of the system we use is symmetry. The axis of non-perfect intervals is half way between Major and minor so, when flipped over the root, Major becomes minor and minor becomes Major (i.e. C-up->E = M3, C-down->E = m6). The axis of Perfect intervals, however, is on the Perfect itself so flipping a perfect over the root gives another perfect (i.e. C-up->G = P5, C-down->G = P4). Aug and dim intervals also flip with each other regardless of whether their midpoint is on a Perfect or between Major and minor. (see chart below).
Inverted Intervals
