Why is a major third considered more consonant than a perfect fourth?
The "perfect" quality implies it should be as consonant as a perfect fifth, but that doesn't seem to be true. Moreover, a perfect fourth has the slightly nicer ratio of 4:3 compared to a major thirds 5:4. Is the naming just for historical reasons?
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Looking backwards, one can justify the situation by arguments about the harmonic series, as in the other answers, but there are other considerations, for example:
In equal temperament, and most "unequal" temperaments where all 12 major and minor keys are useable, the tempered major third is a long way from the just-intonation 5:4 frequency ratio. In equal temperament, it is almost 1/6 of a semitone too wide. Asking why most listeners to western music accept this as "in tune" is a question about culture and learned musical experience, not about the harmonic series.
In the first well-defined temperament system that was used in music that was recognizably "western" (around 1000 AD), specified by the catholic church for all religious music, a major second was defined as a frequency ratio of 9:8 and a major third was defined as two major seconds, i.e. a ratio of 81:64 compared with 80:64 for a just intonation third. An 81:64 third sounds "out of tune" even to modern western ears that are accustomed to the "out-of-tune" thirds in equal temperament.
At the beginning of the era of common practice harmony (about 1700-1750 AD) the perfect fourth was considered a consonant interval except when the lowest note was in the bass part. The reason may have been because of the "difference tones" that are heard (because of nonlinear effects in human hearing) between the two notes.
For a fifth, the difference tone between frequencies of 1 and 3/2 is 1/2, which is an octave below the bottom note and therefore reinforces the bass of the harmony. But for a fourth, the difference tone between 1 and 4/3 is 1/3, which is two octaves below the top note and therefore de-stabilises the bass.
In CPP harmony, the fourth is dissonant against a bass but not in upper voices. Supposedly it's because a fourth against a bass can be heard as a 6/4 chord which is a unstable (if treated as a cadential 6/4 that is followed by a 5/3 on the same bass). In other cases, the fourth may not be so dissonant. Much of consonance vs dissonance (especially with the fourth) depends on context.
The key to understanding consonance and dissonance lies in understanding the natural harmonics of most vibrating systems (including the ear and its components), and the relationship between these harmonics of different notes in an interval.
Many instruments have a natural harmonic sequence related to the fundamental pitch that you are playing. If f0 is the frequency of the fundamental then the sequence is n*f0. When you play or sing a note a combination of these harmonics are created.
Consonant intervals will have more harmonics lining up (matching). Dissonant intervals will have harmonics that don't line up.
As has been pointed out the third and the fifth are natural harmonics of any note. So in a sense when you play one note you are generating the major triad! You can't stop it from happening. Typically the higher harmonics are low amplitude and we don't perceive them as distinct pitches, they contribute to the tone of the note. But when superimposed onto another note the harmonics of each will interfere with each other. This was studied in great detail in the late 1800's by Herman Helmholtz, and published in the text "On the Sensations of Tone". It's a heavy read.
When the harmonics of each note line up or are distinguishable to produces a "pleasant" sound. When the harmonics don't line up and begin to clash the overall sound gets muddied be the interference. This is perceived as dissonance.
A good book for musicians on this is Rigden's "Physics and the Sound of Music".
Both a 3:2 perfect fifth and a 5:4 major third appear in the harmonic series in a way that the lower note of the interval is in the same pitch class as the fundamental. By contrast, in a 4:3 ratio perfect fourth, the upper note is in the same pitch class as the fundamental. At least to my ear, if the pitch class of one note is recognized before the other, the latter note will sound consonant if it's in the harmonic series of the former note's pitch class and dissonant otherwise. Thus, a descending perfect fourth is (at least to my ear) a consonant interval since the lower note is an the harmonic series of the upper note's pitch class, while an ascending or simultaneously-played 4:3 perfect fourth without anything else to guide the ear is more dissonant because the upper note is not in the harmonic series of the lower one's pitch class (and a 21:16 is further away from a proper perfect fourth than an equal-tempered major third is from a 5:4 one).
I have to disagree with Todd Wilcox on this. The fourth as an interval is present in the overtone series, lower than the major third. It is the interval that exists between the 3rd and 4th harmonics. We don't have a major 3rd in the harmonic series until we allow the 5th harmonic to be considered.
So, purely on the basis of the overtone series, the fourth is quite consonant. And in fact for centuries (when Pythagorean tuning ruled the day, and fifths and fourths were tuned to perfect ratios) the fourth was considered far more consonant than the major third. The third is actually quite removed (about 22 cents sharp) from being an integer ratio relationship with Pythagorean tuning, and thus is a sour interval in this tuning system.
The harmony books I have seen have given the fourth an ambivalent evaluation, as being either consonant or dissonant depending upon context. The fourth specifically is considered dissonant is if it appears as the lowest interval in a chord. The reason for this is due to the strength which the interval evokes the overtone system in which the lower note would be the fifth degree.
One tends to hear a chord progression that ends on the V as leaving us hanging, waiting for a resolution. The same pertains to the interval as it strongly suggests the lower note is a V that is mostly likely going to be "resolved".
As soon as you add a third or root below the perfect fourth, it becomes a more consonant structure. This also suggests the nature of the dissonance is not about the frequency ratio of the fourth itself (which of course doesn't change by adding a lower note to the mix). Rather, it is about the structure not having the same drive to resolve when the lowest notes are the root or third (1st inversion) as it does when the lowest note is the fifth of a chord.
The octave, the fifth, and the major third are all low-order members of a harmonic series. We can generate a harmonic series by multiplying a frequency by successive positive integers (1, 2, 3, 4, 5...). After multiplying a frequency, we can divide by a power of two to lower the octave of the new frequency. All multipliers that are powers of two (2, 4, 8, etc.) are just octaves above the original frequency.
So let's build some intervals based on the harmonic series:
Unison
Perfect Octave
Divide by 2 for a ratio of 3/2 and we get a perfect fifth
Two octaves
Divide by 4 for a ratio of 5/4 and we get a major third
(omitted)
Divide by 16 for a ratio of 21/16 and we get close to a perfect fourth
So one answer is the major third is earlier in the harmonic series than the perfect fourth. And the "perfect fourth" generated above when inverted is about 30 cents wider than the perfect fifth generated by the harmonic series. So even if we go to the 21st member of the harmonic series, we don't really get a usable perfect fourth. The fourth we use is generated by inverting the fifth, so it's not really part of the harmonic series.
See: en.wikipedia.org/wiki/Harmonic_series_(music)
I'll put it another way. Let's coin the term harmonic ratio to mean a frequency ratio that is part of the harmonic series, but perhaps changed in octave. That means all harmonic ratios will have a whole number (a harmonic number) for the top number, and a power of two (octave shifting) for the bottom number. 4/3 is therefore not a harmonic ratio, since the bottom is not a power of two. Because of their relationship to the harmonic series, our ears perceive harmonic ratios to be more consonant than other ratios. And 5/4 is a harmonic ratio, because the bottom is a power of two.
Because the inversion of the fifth, which we call the fourth, has a ratio of 4/3, and because that's a 3 on the bottom, there is no harmonic ratio that is exactly a fourth, so the fourth will never be as consonant as major thirds or perfect fifths. We can get arbitrarily close to a 4/3 fourth by going up the harmonic series (21/16, 43/32, etc.), but as we go to higher harmonic numbers, the consonance declines because those higher numbers are less resonant with natural overtones in the lower note of the interval.
I don't much like Todd's explanation as it stands, however his point is indeed indirectly relevant.
My stance is: a perfect fourth is not a dissonant interval at all. However the perfect eleventh is, and in common practice it's usually allowed to stretch intervals by an octave. For fifths and major thirds, this only makes them more consonant, if anything:
P5: 3:2 P12: 3:1
M3: 5:4 M10: 5:2
But not so with the fourth, nor, incidentally, with the minor third:
P4: 4:3 P11: 8:3
m3: 6:5 m10: 12:5
(Tangent: it just occurs to me that this might be the real reason why minor keys tend to feel sadder! A major key is stable in an optimistic, wide voicing, whereas a minor key becomes painfully yearning when the voicing goes wide, and is most stable in a contemplative / introvert close voicing.)
So, while the fourth itself isn't dissonant, you can easily find fourth-like intervals that act dissonant. And thus the generalising rule that's often taught for common practice, to consider the fourth as a dissonant-ish interval.
Having read the answers and thought about it, it seems to me that the solution to this is complex and to some extent cultural. I agree with leftroundabout that the fourth is not dissonant, at least in a mathematical sense. 4/3 is a very simple ratio. It's of course the fourth interval between two consecutive steps of the harmonic series, after the unison, octave, and fifth. The difference between the fourth and the octave, fifth, and major third, is that the fourth in the harmonic series has the fundamental tone above and not below.
This does not affect the sound of the interval per se, but it does affect its perceived tonality: it's hard not to place the interval as 3 to 4 in an harmonic series, and to hear 4, the upper tone, as the tonic. And we expect to hear 4, or 2, or 1, the basis of the harmonic series, as the bottom note in an interval or chord that represents the tonality, and is thus stable.
Thus, I would argue that the perfect fourth, 4/3, is more consonant than a major third 5/4, in a purely physical way, but is less stable, because its bass note is not the perceived tonic.
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