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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.

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.

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.

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:

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).

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.

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.

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".