Why is the sixth considered consonant, but the second is not?
Question: Why is the sixth considered consonant, but the second is not? Harmonically the second is more closely related to the tonic than the sixth (i.e. C G D A etc. in the harmonic cycle). I am hypothesizing that maybe the tonic and second are too close and the ear notices the differences more than the beat matching. Then, I would guess that the ninth maybe would be a consonance... but it isn't. If you know the answer, please enlighten me. - M.S.

Answer: Enlightenment is what we're all about here. The question of relative consonance is one that inspires lots of low-level warfare among musicologists, but an answer based on acoustics accords pretty well with historical practice:

Consonance is not really the result of a close connection with the tonic; it relates to potential agreement among harmonic overtones (partials) of any two tones, which add to the sense of stability of the combination. You mention "beats" - beats are what it's all about. The phenomenon is discussed in more detail in the appendix of Exploring Theory with Practica Musica (with pictures!), but here's the gist of it: every musical tone consists of a complex of component tones, its partials, which for musical sounds are close to being even multiples of the basic frequency. The note C, for example, also is sounding a series of higher pitches that correspond to the octave of C, then G, C, E, G, and the pitches get higher and closer together as the partial number increases. If partials from two sounding tones are almost a match they will go in and out of phase with each other, making a "beating" sound. We're used to hearing beats and don't mind them - until they get too fast to count but not fast enough to sound like a new note. That's when they begin to be perceived as merely wrong. The interval seems "out of tune" unless the context is so strong or the duration so brief that we can ignore it. But when tuned well the tones with matching partials seem to reinforce each other, which is why the major triad has its unique position of strength; all its notes repeat partials of its root. We sense that even when the tuning is slightly off. A pretty good working definition of consonance might be "that quality obtained by two or more tones that have the potential for agreement among one or more of the most audible of their partials."

The sense of consonance has changed over time, largely due to changes in tuning systems. Tuning and temperament (adjustment of tuning) make a large topic: basically the problem is that if you want to tune notes acoustically exact with regard to their most audible partials the keyboard won't "come out even" if it's limited to 12 claves per octave. So there have been lots of different compromises, leading to the current equal temperament system in which all the fifths are close to exact and all the major thirds are somewhat sharp.

However, determinations of consonance are based on the ideal, and an ideally tuned acoustic minor 6th has a frequency ratio of 5:8. The practical meaning of that ratio is that there is a potential agreement between the 8th partial of the lower note and the 5th partial of the upper one. This is the exact inverse of the acoustically exact major third, ratio 5:4. These two have about the same sense of consonance whether upside-down or not; both are present simultaneously in purely tuned consonant major triad if you add an octave to the root. That chord's notes all seem to reinforce each other and it sounds smooth and strong. So a minor sixth shares the consonance of the major third.

A major second, on the other hand, doesn't have such an easily heard potential agreement between partials. A major second can be considered exactly tuned acoustically if the 9th partial of the lower tone agrees with the 8th partial of the upper one. This is the second that arises naturally when you tune an instrument by successive upward perfect fifths: C, G, D (there's also another possible acoustic major second with the ratio 9:10, but that's even worse harmonically). The partials involved in the second are getting very high and very faint; their potential for reinforcement being so dim this interval has always been considered dissonant or unstable, which is one reason its exact size can vary without much trouble. But what's worse is that if you turn this second upside-down it isn't an acoustically exact seventh. A true acoustic seventh has the ratio 4:7 (7:8 when inverted to a second). And if you tune the seventh to be acoustically exact then the resulting note doesn't agree with any other notes in the scale. The inverted 8:9 second makes a seventh that measured in cents (1200ths of an octave) is 996 cents. The acoustically exact 7th is 969 cents, a big difference. So neither the second nor its inversion can be considered a consonance.

Your question recalls a famous one [well, famous in very limited circles] asked by Roger North in early 18th century. He wrote to the noted composer and theorist Johann Pepusch as follows:

"The sound arising by the abscission of 8/9ths [of the string] is a tone, and more remote from perfection of consonance than that of 7/8th; Why then is the former accepted in music, and not the latter, which is abhorred?..."

Pepusch answered that the 8/9 tone worked because it could agree with other notes of the same scale, whereas the 7th tuned 7/8 could agree with no other note in the key:

"Considering only the numbers, it is true that 7/8 is nearer to concordance than 8/9; but as they are both discords, 8/9 is allowed, having a natural and immediate relation to the concords, which 7/8 having not, is absolutely rejected..."

In other words, an 8/9 whole tone above the tonic also makes an excellent fifth above the 5th degree of the scale, etc. But a 7th tuned acoustically pure with the tonic can be in tune with no other note in the scale, so it's the odd one out... Am I wandering off topic now? Your question was pretty much taken care of a few paragraphs ago, so I'll put on the brakes...

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