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I'm reading a book where the augmented 3 interval(two note) is considered a dissonant. However being that its enharmonic equivalent is a perfect fourth, I can't see how this can be so.

I've done a little digging and some say that it has to do with the context of the music. So I'm presuming that if perhaps a composer for example drifted into a different mode/scale he may spell the perfect fourth interval as a Aug 3rd (and therefor a dissonant) in this new context. Depending of course on the scale.

So for example this interval would be written in as a Aug 3 if the composer was writing in c lydian; the e sharp clashing with the f sharp of the lydian mode.

Am I right in this thinking

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    As a clarification, you are referring to the interval (two notes) of an augmented third? (As opposed to and augmented triad (e.g., C E G#).)
    – Aaron
    Jul 22 '20 at 0:47
  • Yeah, so C E# . Sorry I should have made that clearer Jul 22 '20 at 0:49
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    Depending on the "language" (in the sense of style and other details), even the perfect fourth is considered dissonant if it's "alone", with no other voice to put in a context. Many examples of two-voice counterpoint that produce a perfect forth are considered like this. When a third voice comes into play, things change for the perfect fourth, and also put in different "situations" the resulting voice-conducting for the augmented third. Also, if you consider it as a melodic interval (in a harmonic context), it's usually difficult to sing.
    – Pipetus
    Jul 22 '20 at 1:32
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    Actually many people hear the P4 as dissonant. So it is not clear that your question makes sense in this context. Perhaps a little more reading on this topic would help.
    – user50691
    Jul 22 '20 at 15:28
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    The key takeaway here is that dissonance is not a purely acoustical property.
    – phoog
    Aug 12 '20 at 21:40
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All augmented and diminished intervals are considered dissonant. The interval C-E# is only equivalent to C-F in a tempered scale (particularly the equal-temperament that most keyboards now use.) On stringed and wind instruments as well as with voices, C-E# is different from C-F.

Augmented intervals generally expand and diminished intervals contract. Perfect, major, and minor intervals have their own voice-leading procedures.

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    "On stringed and wind instruments as well as with voices, C-E# is different from C-F": this is hardly universal truth. Most musicians I know would have no clue about how or why they should treat them differently, and consequently would not treat them differently.
    – phoog
    Jul 22 '20 at 4:48
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    phoog's comment is right, but still +1 because C-E♯ is conceptually not a 3:4 fourth. If the score says C-E♯ but the musicians intuitively intonate it as a fourth and the listener hears that as well, it's probably the composer's fault for not clearly bringing out the true intend in the context. Jul 22 '20 at 10:26
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    @leftaroundabout you assume that the composer intends there to be a difference, but what if the composer is one of those musicians who isn't aware of the difference? What if the composer's intent is that the E sharp be tuned as a 16:15 half step higher than E and that E is a 5:4 major third above C? What if the composer's intent is that the piece be performed in some 12-tone temperament? You should not draw conclusions about a composer's intent solely from the composer's choice of a particular enharmonic spelling option.
    – phoog
    Jul 22 '20 at 10:45
  • @phoog I didn't draw any conclusions about what the composers intend is, but I stand by it that the intend isn't an audibly consonant fourth. That doesn't mean the actual frequency ratio can't be very close to 4:3, as it is for the 12-edo augm3, but then the context should divert attention away from that. If none of that applies, then I would indeed claim that it's a mistake by the composer. Either make the augmented third really dissonant to the listener, or just write a perfect fourth instead. Jul 22 '20 at 11:01
  • @leftaroundabout 364 days later, I have another thought: the conceptual difference is more important than tuning. I am coming to the conclusion that people are using the (possible) tuning differences as an easy way to explain the difference, but it doesn't really do the job. Literally decades before I understood anything about unequal temperament, I knew that enharmonic spelling mattered, but I wasn't terribly good at explaining why. Even in equal temperament, the diminished fourth feels different from a major third, for example melodically in iv7 V7 i.
    – phoog
    Jul 21 at 17:36
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Upon first glance a perfect fourth and augmented third seem for all intents and purposes equal...

enter image description here

...they are enharmonically equal. On a piano they are literally executed with the same keys.

But by definition augmented and diminished intervals are considered dissonant.

Why? I think the reason is clear when we consider whether the modified interval is diatonic or chromatic.

Play something to just establish a diatonic setting...

enter image description here

...then play an augmented third that is not enharmonically equal to a diatonic perfect fourth...

enter image description here

...the sound is pretty jarring, because it's outside the key. It's a sort of dissonant against the diatonic context.

I don't know if that is the origin of the theory that all augmented and diminished intervals are dissonant, but it's my way of making sense of it.

One final thought going back to the diatonic, enharmonically equal example. If we add a note to create a triad, we get...

enter image description here

If the context was clearly a C chord, like a tonic in C or dominant in F, then spelling it with a B# is aggravating. It doesn't make a good example of an augmented interval as a dissonance. It's just an example of bad harmonic spelling.

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At first sight some augmented intervals could be transformed in consonant intervals considering the semitones. So it might seem that mathematically this dissonance is only theoretically and supposed by the notation image.

But, as Pipetus says in his comment even the fourth is dissonant.

And yes, all augmented and diminished intervals are considered as dissonant. (Ttw).They are stressed, containing lead tones, leading to the nearby next tone.

https://en.wikipedia.org/wiki/Augmented_third

So we could also compare the augmented second with a minor third:

The augmented interval is dissonant with a tension of resolving into a nearest note step, defined by the melodic and harmonic context ( sharps leading up, flats leading down).

On a piano these dissonances are not always audible. (Bartok Mikrokosmos vol. IV, nr. 111

enter image description here

Read more about Unusual Transformations in Bartok's "Minor Seconds, Major Sevenths”

https://www.esm.rochester.edu/integral/wp-content/uploads/2019/06/INTEGRAL_12_gollin.pdf

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  • The fourth is treated carefully in voice leading. It's treated as a dissonance against the bass (like in a 6-4 chord) but as a consonance against upper voices(like in a 6-3) chord.
    – ttw
    Jul 22 '20 at 14:03
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    Yes, we should say the fourth is considered as dissonance as suspension. As an inversion of the 135 triad it is consonant of course. Jul 22 '20 at 14:15
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    The augmented second is an excellent example because it's such a characteristic feature of the harmonic minor scale. You simply cannot mistake that oversized step on the scale for a minor third. Jul 23 '20 at 7:54
  • @ttw or we could just say the fourth is consonant but the eleventh is dissonant. Jul 23 '20 at 8:19
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Some of the answers focus on tuning here, but as Michael Curtis points out, that's not an issue on many instruments (and -- as phoog notes in comments -- isn't even an issue for many performers on instruments with more flexible tuning unless they're particularly sensitive to how they tune unusual intervals). Historically, the distinction was partly wrapped up in tuning issues, but there's more to why ultimately all diminished and augmented intervals are declared "dissonant," even if they have enharmonic "consonant" equivalence.

And that has to do with function, something hinted at in a few answers. Functionally, if an augmented or diminished interval is spelled correctly, then it is necessarily also occurring in a situation that would be considered melodically (and possibly harmonically) dissonant. And any melodically dissonant leap would have additional contrapuntal rules placed upon it, such as whether it was possible to leap to or from those notes, or how that melodic leap should be used or resolved in subsequent melodic context.

This is easier to see by analyzing intervals that are more likely to occur in practice, like a diminished fourth. Enharmonically, a diminished fourth on a piano is the same interval as a major third, but whenever a diminished fourth is properly spelled within a key, it acts as a dissonance.

For example, consider C minor. If one leaps from E♭ to B♮, that is utilizing common notes within the minor scale, going from the normal third scale degree to the leading tone. But that's a "charged" interval. If you play for a few bars in C minor and then play that interval, you'll feel like it has to "resolve" (similar to how a tritone leap feels like it should "resolve"), and your ear will likely recreate a feeling of dominant harmony under the B, even if you don't play the chord. The B♮ generally must resolve upward to C in a C minor context, particularly after a leap like that.

However, now reimagine the same keys on the piano played in the context of B major. Now it's a leap down from D♯ down to B, and both of those notes feel like "stable" parts of the tonic chord. There's no need for resolution.

I've done this demonstration many times for musical beginners to explain why spelling of intervals is important. Playing in a C minor context, they feel like the E♭-B♮ diminished fourth "needs to resolve" melodically, but then I'll immediately follow that by playing the same notes where the D♯-B are in a B major triad, and suddenly they become confused and say, "huh, that's consonant and stable."

(The same would apply if that diminished fourth would be used as a harmonic interval. In the scenario above, I suppose that could occur in the context of a dominant chord in C minor with an added minor thirteenth. That's not a stable interval in classical harmony. It would be treated as a dissonance. Either the minor thirteenth is going to have to resolve down to become consonant with the rest of the V chord, or the chord is going to have to change -- say dominant to tonic, where the leading tone resolves and again gets rid of the diminished fourth interval.)

The same thing goes for an interval like an augmented third. The problem is that it's quite a bit harder to demonstrate a situation where one would spell an augmented third melodically that way in a tonal context. I've tried a bit to think of one that didn't seem contrived, but it's difficult. In most contexts where I think this would likely occur, the augmented third would be part of some figure where the destination note is obviously a kind of chromatic appoggiatura that would be required to resolve.

Just to try to imagine a harmonic scenario, I suppose we might start in C minor again. One might, I suppose, have a D♭, as the root of a Neapolitan chord, and then leap to F♯ as the third of a V/V chord. This would generally be a very bad idea contrapuntally, but if you did it, you'd be obligated to resolve the F♯ upward to G as the leading tone to scale degree 5. The melodic interval would be considered "dissonant" and require resolution. If the harmonic context changed in such a way that you could actually respell the interval as a perfect fourth (e.g., a D♭-G♭ leap in the context of a modulation where the Neapolitan is used to modulate to the key of G♭ or something), the note leapt to would generally not require "resolution."

This is the sense in which "melodic dissonance" is meaningful: one or both notes of the interval are tendency tones that want to resolve in a particular direction, and after the leap, one is usually obligated contrapuntally to do so. But that obligation on an augmented third would not generally apply to a melodic leap of a perfect fourth.

While the same concept would apply to a harmonic interval, the chances of encountering a legitimate correctly spelled augmented third in a tonal piece are possibly even lower than finding a melodic one. I suppose one might consider a V7 chord in C. Now imagine that the leading tone B uses a neighbor tone figure moving B-A♯-B. That would create a "dissonant" interval of an augmented third between F and A♯ which is corrected ("resolved") by the A♯ moving back to the chord tone B. But here the "dissonance" is mostly created by the context, just as it was in the diminished fourth example in C minor discussed above. If we heard a passage with a bunch of chromatic lower neighbor tones in a piece, we'd know the A♯ needs to move back upward to the leading tone. Out of context, the F-A♯ harmonic interval could be lots of things (and could sound like the stable perfect fourth F-B♭, and it might even create a minor-minor seventh chord in a different context). But in that motivic/harmonic context, it's unstable and the A♯ must move to the B (which itself is part of a tritone that will then resolve as well).

In essence, spelling matters, even if the tuning of two enharmonic intervals is the same. Enharmonically equivalent intervals are functionally quite different -- in one context, they may be stable and "consonant," but in another context (and with a different spelling), they may require resolution and in any reasonable harmonic context will sound locally "dissonant." Pretty much any scenario you can come up with involving a harmonic or melodic use of an augmented third within a major or minor key and assuming that that spelling is correct is necessarily going to be unstable and require resolution.

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Unusual augmented and diminished intervals are there to create color in the harmonic progressions in pieces where they appear and are applied.. They are normally a mixed structure, so to speak and don't appear in your regular diatonic contexts. Context differences are key here. There are many possible combinations of a respective augmented third; (let's take the basic C to E sharp) The C can belong to any key containing the note C, and the E sharp belong to any key that contains the note E sharp. I am referring to regular scale keys here.

An example: The C can belong to F major which would be the dominant scale degree in F major; whereas the E sharp can belong to f sharp harmonic minor, where the E sharp is the raised leading note in f sharp harmonic minor.

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