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In music theory, the subdominant is unstable and need to be resolved to the mediant. Similar resolution happens to the leading note and other notes that are not in the tonic triad. (Of course, I am talking about the diatonic system)

By subdominant, I just mean the IV note, NOT a chord. The subdominant note usually resolve to the mediant, no matter whether it is a part of a dominant 7th chord or a part of the IV chord.

See here for an example.

Question: this doesn't seem to have anything to do with consonance or dissonance (because it is only a single note!), so why are they "unstable" and need to be resolved? Where does such lack of stability come from? Are there any theories of math and physics that explain this?

For example, dissonance can be explained in terms of beating. Can I explain the instability of the subdominant in a similar way?

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    “the subdominant is unstable and need to be resolved to the median” – a) I suppose you mean the mediant? b) it's unclear whether you're talking about a subdominant chord or just a Ⅳ note c) [citation needed]; without further context it's definitely just not true that a subdominant always needs to resolve to the mediant. At least there is no such rule in common-practice classical music theory. – leftaroundabout Jul 31 at 7:36
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    It's only theory, and lots of pieces have the subdominant note followed by others rather than the mediant. Yes, the mediant is a favourite, sonically, as it's only a semitone away. Just like the leading note to the tonic. Most pieces tend to gravitate towards their tonics - and when you consider that the tonic triad takes up three of the avaiable diatonic notes, any of the other four will be considered unstable to a degree. – Tim Jul 31 at 10:58
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    In Chinese music the subdominant does not even exist (pentatonic). Adds weight to your claim perhaps? – Rusi Jul 31 at 11:01
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    @leftaroundabout OK a mathocentric path. In math there are operations and inverses. Inverses are not always available and/or may be expensive. Eg in integers multiplication exists but it's inverse division does not. One can get it but at the cost of making integers more heavyweight - rationals. Likewise one can take adding a sharp (to a signature) as the operation. And adding a flat as its inverse. If you don't have the inverse you cant go C -> F. Though you can go C to every other diatonic note in Cmaj. – Rusi Aug 1 at 10:24
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    @Rusi now that's an argument I like... — The inverses are in this case utonalities. – leftaroundabout Aug 1 at 10:27
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The only really 'stable' thing in triadic tonal music is the tonic triad, which consists of the tonic, mediant, and dominant notes. The subdominant isn't one of these, therefore according to the common expectations around this kind of music, it's seen as 'wanting' to move somewhere at some point.

In terms of common notions of measured/calculated dissonance, the perfect fourth is actually more consonant with the root than the major third is - see e.g. http://sethares.engr.wisc.edu/consemi.html. So this notion of the fourth as unstable isn't so much about the sensation of dissonance as the expectation of resolution to the established tonic triad.

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    Exactly. It is built on expectations, in just the same way that you have been taught to expect me to finish the – Phil H Aug 1 at 10:16
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Context is really important with the subdominant role.

By subdominant, I just mean the IV note, NOT a chord.

I understand what you mean, but melody and harmony are inextricably linked. You can't really separate them. Importantly in tonal music, even if the music is entire a single melodic line, the tonic is a reference point and harmonic relationships are implied. Historically, these tendency tones ideas evolved from counterpoint and harmony.

Question: this doesn't seem to have anything to do with consonance or dissonance (because it is only a single note!)

In the vertical sense, yes, there is only one tone. But music is a temporal art! In time there are other notes. In other words, there are other notes linearly. This is where the sense of stability is created: linear movement to and from the subdominant degree.

Let's call the subdominant tone FA. How you move to and from FA is what creates the sensible of stability and resolution.

When FA moves down to MI the harmonic implication is MI is the mediant and a member of the tonic chord. The critical part is a downward move of a half step is regarded as FA to MI. In modern harmony we would regard FA to MI as the upper voice in V7 I or viio I. (If FA were in the bass, it would be V4/2 I6.) The various harmonic implications are a move to a tonic chord from a dominant harmony.

When the direction is reversed and MI moves up to FA the harmonic implication changes. MI up to FA can be re-contextualized as TI up to DO. In that case, where FA is treated as DO the move is to an implied stable chord. In modern harmony it might be something like I IV regarded as V I. I think it becomes clear when the tones are in the bass: I6 IV regarded as V6 I.

I may not be explaining it clearly. I'm trying to paraphrase an overview from: Gjerdingen, Music in the Galant Style where he provides an "...excursus on eighteenth-century solmization." I had to re-read it about twenty times - and look for it in real score - before I felt like I understood the concept.

The main point is:

Ascending and descending movements to and from the subdominant have different harmonic implications.

Those harmonic implications - even in an unharmonized line - create a sense of stability or in-stability for the subdominant degree.

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    Interestingly, Eric Dolphy (who played some pretty out there jazz horn) once said something along the lines of: I think of everything I play in terms of harmonic relationships. – ex nihilo Jul 31 at 15:47
  • Excluding speculation about how music evolved, my personal view is melody is derived from harmony. – Michael Curtis Jul 31 at 15:52
  • I don't like to think of melody as being that tightly coupled to harmony (i.e., derived from, not that it can't be a reasonable and useful point of view); it is a subtle and nuanced relationship, but certainly a relationship. – ex nihilo Jul 31 at 15:56
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The reason that the fourth of the tonality you're working in feels unstable is because of the products of combination tones. Say I'm playing a C4 and an F4 at the same time in a song in C major. For further simplicity, assume that the C is at 240 Hz, making the F's frequency 320 Hz. Arbitrary numbers, but correct ratios, so it's fine.

Now we find the combination tones generated by this interval. The combination tone's frequency is equal to the difference between the frequencies being played, or 320 - 240 = 80 Hz. This would put the combination tone at 1/4 the frequency of F4 as has been established, so two octaves down => F2. Subsequent combination tone calculations using the initial two tones in conjunction with the third tone we just found will result in more F notes in different octaves.

So why does this matter? It matters because the combination tones magnify the sound of the F, so that it is the root of the interval. This is something that Paul Hindemith talks about a lot in Volume I of The Craft of Musical Composition. Because the root of the fourth interval is the top note, if the bottom note is the tonic of the key you're working in, then it will naturally feel unsettled because it places weight on the F, not the C--whereas the combination tones generated by a perfect fifth (C & G) yield C notes, and those generated by a major third (C & E) yield a G, which consequently yields a C.

More on combination tones from Adam Neely:

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    If we do this with a minor third (ratio 6:5), our minor third (D#) has the frequency 288Hz, and the combination tone is 288-240 = 48 Hz, which (in your intonation), would be a G#, I think, not a C. Did I work that through properly? If so, that doesn't seem to support this theory, because a minor tonic chord can be seen as stable... – topo Reinstate Monica Jul 31 at 23:49
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    As Neely explains, it's actually not that easy to hear combination tones, so I question their importance in how people perceive harmony. – Your Uncle Bob Aug 1 at 2:53
  • @topomorto - that minor third is called Eb, not D#! – Tim Aug 1 at 10:03
  • @Tim Better to say "our minor third (Eb) has the frequency 288Hz, and the combination tone is 288-240 = 48 Hz, which (in your intonation), would be an Ab" ? – topo Reinstate Monica Aug 1 at 14:36
  • @topomorto - probably! It's just that the m3 of C is Eb rather than D#. Is it a guitarist thing?! – Tim Aug 1 at 16:01
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In terms of tendency note for the diatonic system, the fourth or subdominant is considered to be less stable, therefore it’s needed to resolve to the mediant.

The tendency note is considered by its natural sound (Harmonic partials) in terms of the tonic chord e.g C E G and this can be noted as: C is the strongest G is the second strongest E is the third strongest

Therefore, the leading tone could be: - Leading tone to Tonic - Subdominant to Mediant - Supertonic to tonic - Submediant to Dominant

The relationship between the subdominant and the mediant is only the half step far. Therefore, it has more tendency to be resolved. The subdominant could be resolved to the dominant but they are one step far, so in terms of voice leading the half step should be more meaningful.

However, the perfect fourth interval is unstable if formed with the bass.

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    Relating the tonic triad to harmonic partials makes a lot of sense for major, but doesn't work so well for minor - according to en.wikipedia.org/wiki/…, you're up at the 19th harmonic before you see a harmonic that relates to the minor third. – topo Reinstate Monica Jul 31 at 10:14
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    @topomorto Yes, I agree!! – user506602 Jul 31 at 11:01
  • @topomorto I really like the way you’ve contributed to this forum overall. You are very knowledgeable and polite! – user506602 Jul 31 at 13:22
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    Well, thank you... I do try to suppress my inner rage - some days maybe I succeed :) I think there are lots of quite nice people on this SE, who mostly manage to be civil despite the fact that SE encourages people to fight for their right to be right! – topo Reinstate Monica Jul 31 at 13:37
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    In the context of I6 IV the subdominant is surely stable. The sense of stability is related to the direction of the movement and the implied harmonies. – Michael Curtis Jul 31 at 15:20
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In terms of the "math and physics" you ask about, in fact dissonance arises in one way and one only, namely two frequencies that are close to one another but not the same. The peaks of the waveforms are out of phase, creating a clashing effect. This occurs in the chromatic scale whenever two tones are a semitone apart, or their harmonics are (which explains the strong dissonance of the tritone interval, since the tritone is one semitone away from the fifth, which is in 2:3 resonance with the root).

Now with respect to the subdominant, the fourth is a semitone away from the major third, hence dissonant with respect to the root major triad. As others point out, this is implicitly present to the listener's ear when listening to tonal music in the major scale. The "dissonance" between the expected major third and the fourth is what creates the "suspended" effect in a suspended chord.

So there is a combination of actual dissonance and convention/expectation at work here.

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    Love to know the reason for the downvote... – see sharper Aug 1 at 2:25
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    Reasons for dvs are rarely given, we never know why! – Tim Aug 1 at 9:57
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    Re downvote: I'm going to guess someone decided "dissonance arises in one way and one only, namely two frequencies that are close to one another but not the same" was wrong and skipped over the rest of the paragraph which explains "or their harmonics are". It's easy for people to misread stuff like that. – npostavs Aug 1 at 12:15
  • I didn't downvote. Bot I probably should have! The 'clashing frequencies' model is all very well on the laboratory bench. But it falls down when considering real-life instruments and real-life tuning systems. Just about EVERYTHING in our equal-tempered musical world would be a clashing dissonance. – Laurence Payne Aug 2 at 11:46
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    @LaurencePayne the ear/auditory system has some capacity to reconcile slight frequency variations, such as occur in equal tempered systems, so the more complete answer would be that dissonance occurs when the difference between two frequencies is large enough to overcome that effect, but small enough to create a clashing impression. I don't think that omitting that nicety merits a downvote, but hey, downvoting is everyone's right! – see sharper Aug 3 at 3:00
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We are very accustomed to hearing the 4th as part of a 4-7 tritone in the dominant 7th chord. This learned behaviour may have a lot to do with the assumption it will resolve downward to the 3rd rather than up to the 5th. (Which it actually often does, particularly when it's the bass note.)

I wouldn't get too tied up with analysing dissonance as beating between harmonics. It's an attractive idea, until you look at the actual tuning schemes of most of the music we play and hear (not to mention the actual overtone structures of the actual instruments we use).

  • I'd find it less confusing to say “accustomed to hearing the Ⅳ note as part of a Ⅳ-ⅶ tritone”. There's not actually a fourth involved there (well, not a pure one at any rate). – leftaroundabout Aug 1 at 8:30
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the simple answer is -- because people always resolve it.

After a while, you get used to the fact that other composers always resolve it. When you hear a subdominant chord, you expect it to resolve, hence you perceive is as "unstable".

This is simply a conventional thing.

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because it is only a single note!

No it's by definition not a single note. The concept of a subdominant is referred to as a "function", which is the what we call the relation between a chord (usually) and its key. You can't talk about subdominants without being in a key, because there can't be a subdominant. Without it, the chord could have any function.

A fourth interval, which the IV note of course is, in relation to its root, is certainly dissionant compared to intervals like octave & fifth, and historically the fourth was grouped together with seconds and sevenths as dissionants.

In modern music however (especially pop from the 90's and later) the fourth is usually not treated as a dissionant, and there are many examples of songs that use the subdominant chord as a "landing point".

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