what's the purpose behind these alternate namings for intervals?
I was studying music theory and I came to an interval called an Augmented unison which is a half-step distance from the first note and is equivalent to a minor second. Then the book mentions intervals like the diminished fourth and augmented fifth which are equivalent to the major third and minor sixth, respectively. There are also terms like diminished and augmented octave, so what's the purpose of such namings, and are they of use in our current music system?
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What's the purpose of such namings, and are they of use in our current music system?
I strive to write relatively consonant 12 tone music, and try to reflect the tonal nature of the harmony by using standard chord spellings.
However, in order to keep the chord spellings meaningful, the melodies often end up with the awkward intervals you asked about. This often happens in heavily chromatic music.
The example below illustrates how a diminished 4th and diminished octave can come about. Measure 1 starts with a Dmin triad and an AbMaj7, followed by an C#min and BMaj7 in measure 2. The bass is { D, Ab, E, B } and the melodic intervals between those notes are an augmented 4th, diminished 4th, and Perfect 5th.
The suprano is { F, G, Ab/G#, D#, A# } and the melodic intervals are Major 2nd, minor 2nd, Perfect 4th, Perfect 5th.
For the base in measure one, I could have used G# instead of Ab, renaming the augmented 4th to a diminished 5th, and the diminished 4th to a Major 3rd. However, then we would have G in the suprano against G# in the bass, giving a diminished octave!
In the end, these strange intervals come about because our staff and interval naming was based on the diatonic system, consisting of only 7 steps, while modern music uses all 12 intervals.
If you are working with fully chromatic music (for example, 20th-century 12-tone music) then there is absolutely no difference because the intervals are enharmonic.
If you are dealing with tonal music, those intervals may seem the same to you but they are different with respect to diatonic function and voice leading. For example, a dominant 7th chord (a.k.a. major-minor chord with C E G and Bb) has exactly the same notes as a German augmented sixth chord, but the minor 7th tends to want to close in/get smaller while the augmented sixth resolves by expanding outwards. The former is a dominant function chord while the latter is a predominant chord that typically introduces a significant musical event with a half-cadence (e.g. is always used to introduce a cadenza in Classical sonatas, for example). (Greatest example of augmented 6th chord, ever, at 0:47, on the lyric "homo").
Also, in terms of performance, top-tier musicians do not play notes that are exactly in tune with equal temperament. They will make thirds sharper so they are properly major, make fifths slightly smaller so they are more perfect, etc., continuously listening and compromising so that the frequency ratios of their harmonies work out to whole numbers (instead of being slightly off if they were equally tempered). For those musicians, the difference between a diminished fourth and a major third can be huge, even if it is just a few cents. With regards to your specific example, an augmented unison implies upward motion and would be played a bit sharper than a minor second.
The augmented unison is useful when transposing music (say you want to transpose E flat major music to E major--yup, you need to shift the piece up an augmented unison). The diminished octave behaves similarly.
To illustrate what a difference a harmonic context can make, here's a simple experiment you can do with a piano. It will give you a context to hear the rather startling difference between an Augmented 5th and a minor 6th. This looks like a lot of text, but it will only take you a moment to play, and it will be totally worth your time.
Play the following notes together, which I've written here from bottom to top: C, E, G. Hey, it's C major! Notice that there's no strong tension anywhere.
Now, play an inverted A minor slightly higher, starting from the same E: E, A, C. We've just made a nice, darkly-tinged transition.
Now, let's do E Major, for reasons that will be made clear shortly: E, G#, B. Once again, there is no strong tension anywhere. In fact, let's illustrate that we're hearing this chord as a first, third, and fifth by strumming these 5 notes in a row: E F# G# A B. It should jive perfectly with the chord you were just playing.
Now, play that C Major chord again. In fact, play it a few times to reset the palate and get your ears ready to hear the next chord in context. So, C E G, here we go. When you've really settled into that sound, move to the next chord.
Next step, let's play an inverted F minor chord: C, F, Ab. Beautiful and dark, that high Ab kinda wants to fall gently back down to the G... so let's let it.
Back to C E G. Did you feel the small release of the Ab on its way down? Beautiful interplay of tension and release. Play this a few times to really get the sound into your ear - you're going to need to be settled in to it.
Finally, C augmented. We're going to use almost exactly the same notes as our F minor chord, but instead of F, we'll play E, and the Ab will now be spelled G# (for reasons that will be made clear). Go ahead and play it: C E G#.
Bam! What in the world just happened??
When we played C-Ab in our F minor chord, it was relaxed, but here C-G# is strident as can be. Could the tension come from the E-G#? Not likely, because we already played exactly that combination of notes in E Major, and it sounded nothing like this.
What's going on is that the E is forcing you to hear G# instead of Ab. This should make some intuitive sense - heck, it did the same thing in E Major, right? We heard the first (E), third (G#), and fifth (B) of the chord. But C to Ab is a minor sixth, it was dark and lilting. C to G# is... well, it's what you just heard.
The takeaway
Enharmonic intervals (such as augmented unison vs minor second) communicate the harmonic relationships between notes. When you find the rarer versions out in the wild, if you take the time to listen to them very carefully, you will find that, like the augmented 5th, they each have their own very interesting behaviors.
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