Harmonic function is a fancy way of saying the purpose of each note in solfege. Its the reason we say Do Re Mi Fa Sol La Ti Do instead of 1 2 3 4 5 6 7 8 or something else. Its also the reason I sing minor keys Do Re Me Fa Sol Le Te Do instead of doing La based minor.
The gist of it is, each solfege unit has a specific purpose. A "feel" so to speak and purpose that it carries out. Musicians have named each of these, so I will go through each in the major scale.
Do = Tonic
The tonic is the fundamental note in the key. If I'm playing a song in C major I can essentially play a C at any point and it will work pretty well in the chord. Additionally, whenever you play a C it feels somewhat like a return to home or has a grounded feel. This is assuming you are not playing a chord borrowed from another key or a secondary chord. Finding the tonic by ear while listening to a song is usually not that difficult as its the note that has the most definition.
Re = Supertonic
Super in this context meaning above. Essentially the supertonic feels like a passing note between the root of the key and the middle note of its chord. Even the chord based on Re has a somewhat passing feel. If you look at the Kodaly hand sign for Re you can see it is designed around this fact. The note and chord are a bridge between two sounds note necessarily leading up or down.
Mi = Mediant
The mediant of the key feels like a middle ground despite it being only the 3rd of 7. You can see in the hand sign that it sits moderately grounded as it doesn't lead up or down, but at the same time is centered between two and thus not totally stable.
Fa = Subdominant
The subdominant has a feeling of slightly leading downward. Its similar to Ti's desire to lead to Do but rather has a desire to fall to Mi. The hand sign shows it leading downward but not so strongly that it cannot go to sol easily. It's still the 3rd most stable chord as shown by it being mostly a fist. Finally, it has a close relationship to Sol as it can feel like a descended version of sol.
Sol = Dominant
The dominant is the second most stable chord. The note itself leads primarily to Do as shown that the hand could easily close to a fist, but is also quite foundational. When listening to a song in C major, the G note should stick out as the second most prevalent note in the key and in some cases even more prominent than Do depending on the goals of the composer. Having difficulty finding sol by ear in a key can often lead to an inability to stay tuned on the subdominant and the mediant which both have a direct relationship to it.
La = Submediant
Very similar to the mediant, but slightly less stable. Has a similar middle feeling while wanting to both lead into Ti and fall to Sol. Because of its desire to fall to the dominant, it is often seen like the 4th most stable chord in the key.
Ti = Leading Tone
Ti is very obvious about its purpose. Ti leads to Do. Play the scale starting at Do up to Ti and pretty much anyone can very clearly feel its desire to go up. While Ti can go down, it almost feels like a betrayal to do so. Ti loves going up. If singing Ti doesn't feel like going up, you're not in tune.
This is a general run down of the harmonic function of each note and chord. It can dramatically help with tuning to know these purposes as well as contribute greatly to song writing. If you need help with this kind of thing sign up for lessons by clicking here!
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