Why are the perfect fifth and fourth called "perfect" in 12-ET when they differ by approximately 2 cents from just intonation? [duplicate]

Question

The terms "perfect fifth" and "perfect fourth" are commonly used in music theory to describe these intervals in both 12-tone equal temperament (12-ET) and just intonation. However, in 12-ET, the perfect fifth and fourth are slightly off from their pure just intonation counterparts, differing by about 2 cents. Given this discrepancy, why are these intervals still referred to as "perfect" in 12-ET? What is the historical or theoretical reasoning behind maintaining this terminology despite the small deviation?

Answer 1

The terms "perfect fifth" and "perfect fourth" are commonly used in music theory to describe these intervals in both 12-tone equal temperament (12-ET) and just intonation.

Actually, most music theory textbooks don't talk much at all about tuning or temperament, and many of them theorize about music that was written before equal temperament came into common use; 16th-century counterpoint and early 18th-century harmony were written by composers who preferred rather unequal temperaments because their ideal major third was in a 5:4 ratio, or about 386 cents. (And just intonation is not useful for playing harmonic music on keyboards or any other instrument that can't modify its pitch, so was never much used by anyone.)

"Perfect" doesn't mean "perfectly consonant" and in fact has very little to do with tuning. After all, we don't call a 5:4 major third a "perfect third."

"Perfect" means that it is an interval that doesn't come in major and minor varieties; an implication of this is that the inversion of a perfect interval is also perfect.

In the diatonic scale, the intervals larger than a unison and smaller than an octave are the second, third, fourth, fifth, sixth, and seventh. The second and third come in two sizes each, smaller, or minor, comprising one and three semitones respectively, and larger, or major, comprising two and four semitones respectively.

When you invert these intervals, the seconds become sevenths, the thirds become sixths, and the major intervals become minor intervals, and vice versa.

The fourth and the fifth, however, nearly always have five and seven semitones, respectively. "Nearly always" here means in six cases out of seven. The only exception, the interval between B and F, is so unstable as to be treated somewhat exceptionally. It is not therefore called a minor fifth nor, in inversion, a major fourth, but rather a diminished fifth or augmented fourth.

This remains true whether you use Pythagorean tuning, meantone temperament, or any other temperament. Many of these tuning systems have perfect fifths that are rather smaller than those of equal temperament, and some have a few that are larger than Pythagorean, yet they're still "perfect fifth" intervals -- they're just tempered somewhat differently.

Historically, the term used to denote an acoustically tuned interval is "just" (in a somewhat older sense that means "correct," more or less); this is the source of the term "just intonation."
A more modern term that one often hears to denote such intervals is "pure."

In addition to the fourth and fifth, of course, the unison is the third "perfectable" interval, along with the octave extensions of these intervals (octave, eleventh, twelfth, fifteenth, eighteenth, nineteenth, twenty-second, etc.) These intervals are always perfect, augmented, diminished, multiply augmented, or multiply diminished.

The imperfect intervals, by contrast (the second, third, sixth, seventh, and their extensions, namely the ninth, tenth, thirteenth, fourteenth, sixteenth, seventeenth, etc.) have two sizes between diminished and augmented instead of one, those of course being "minor" and "major."

Answer 2

"Perfect" is an attribute that is used with regard to the role of the interval in the scale; it has no bearing on intonation.

You may be confusing this with "pure". A pure (or just) interval is one that shows no beatings deviating from its defining fraction of frequencies.

So you have a pure (just) fifth with 3:2, a pure fourth with 4:3, a pure major third with 5:4, a pure minor third with 6:5.

Answer 3

The terms "perfect fourth" and "perfect fifth" describe pitch intervals on a music-theoretical level (i.e., the idea that such intervals play a particular role in composing melodies and harmonies). They're disconnected from both tonality (i.e., the idea that specific pitch ratios are more or less consonant, and in particular the idea that simple whole-number ratios are more consonant) and tuning theory (i.e., the idea that specific sets of pitches allow for acceptable approximations of tonal-ideal pitch ratios - or ones with a particular "character" or even deliberate dissonance - perhaps varying depending on the key signature). They're "perfect", as phoog says (and the proposed duplicate goes over) because they don't have major or minor equivalents. That's it.

They're "fourth" and "fifth" because of diatonic scale degrees - those terms don't mean anything to do with the fractions 1/4 or 1/5, either in pitch ratios or "fraction of an octave". This is all the necessary evidence of the disconnection in terminology.

A diatonic scale is any scale that spans an octave and is composed of 5 tones and 2 semitones (such that neither interval is unison, and a semitone is strictly "smaller" than a tone - but 2 semitones do not necessarily add up to a tone) and the semitones are properly spaced out.

12EDO is the unique special-case tuning that embeds, in every key, a diatonic scale where the tone "size" (i.e., logarithm of pitch ratio) is exactly twice that of the semitone. This requires the octave to be equal to 6 tones by that logarithmic measure (since the two semitones add up to a tone), i.e. it requires the tone to be a logarithmic pitch interval of 200 cents (equivalently, pitch ratio of 2^(1/6)); and thus requires the semitone to be equal to 100 cents. The "perfect fifth" is mapped onto an interval of 700 cents, being the closest that this pitch-set can get (regardless of key). Yes, that differs from an exact 3/2 pitch ratio. This is irrelevant because the terminology has nothing to do with that difference - in fact, it has nothing to do with either the exact 3/2 pitch ratio nor its approximation in 12EDO.

12EDO offers a few good quality approximations, key indifference and simplicity. Of course, the cost of this simplicity is a lack of complexity - i.e., richness. A variety of interesting JI intervals, therefore, may map to the same number of 12EDO scale degrees. (In tuning terminology: the "comma" consisting of the ratio between those intervals, is "tempered out".)

You can have a "perfect fourth" or "perfect fifth" in any tuning that embeds a qualifying diatonic scale (for example, all the various meantone tunings experimented with in earlier Western music; but also many EDO tunings - for example, 22EDO, using a semitone equal to 1/4 of the tone (although people who compose in 22EDO might prefer non-diatonic scales).