What is the origin and original meaning of "tonic", "supertonic", "mediant", etc.?

Question

Each of the pitches in the diatonic scale has a "name":

tonic
supertonic
mediant
subdominant
dominant
submediant
subtonic

I was first introduced to tonic, dominant, and subdominant in the context of "dominant chords lead to tonic chords, and subdominant chords lead to dominant chords". My own logical inference suggested that the "sub" in "subdominant" meant "less important" given its role in relation to the dominant.

Later, I learned that mediant and submediant are a third above and below the tonic, respectively, and, similarly, the supertonic and subtonic are a second above and below the tonic.

That leaves me with some confusion about dominant and subdominant. Subdominant could mean "one step below the dominant", but that would be inconsistent, since the others are relative to the tonic. Or it could mean that dominant and subdominant are a fifth above and below the tonic, but this is also inconsistent: why would scale degrees 2, 3, and 5 be named relative to the tonic rather than 2, 3, and 4?

My vague understanding is that these terms predate the diatonic scale, finding their origins in earlier (chant/modal) theories, and perhaps even as terms borrowed or adapted from ancient Greek theory.

Where do these terms come from, what were their original meanings, and what was their musical significance?

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This question has its origin in the discussion in the comments of this post: Answer: What are the degrees of a pentatonic scale?

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UPDATE

Wikipedia's "Submediant" entry contains this interesting tidbit:

In French and Italian, a conception with two centres, subtonic (sous-tonique, sotto-tonica) and supertonic (sustonique, sopra-tonica) on both sides of the tonic, subdominant (sous-dominante, sotto-dominante) and "superdominant" (sus-dominante, sopra-dominante) on both sides of the dominant – and the mediant left alone between the two.

The English/German terminology was used in constructing the present question, in which the super/sub designations relate to position relative to the tonic. But the French/Italian terminology supports the intuition that "subdominant" should mean "below the dominant" (see first paragraph above).

subtonic
tonic
supertonic

mediant

subdominant
dominant
superdominant

Answer 1

This is a little complex to answer, because "dominant" has been used to mean different things over time. In the (late) Middle Ages, "dominant" actually just meant "reciting tone," which would be a fifth above the final of authentic modes and a fourth above the final for plagal modes.

The first appearance of "mediant" that I'm aware of was in a treatise in 1639 by Antoine Parran, where he both fixes the note dominante to be the fifth above the final and introduces the mediante, the point between the final and the note dominante.

As far as I can tell, it was Jean-Philippe Rameau (1683–1764) who introduced "subdominant." Quoting Joel Lester in his chapter ("Rameau and eighteenth-century harmonic theory") in the Cambridge History of Western Music Theory (p. 768):

In practical matters, there were several important additions to Rameau's theory after the Traité [de l'harmonie of 1722]. Among the most important are the introduction of the subdominant (sous-dominante)...in the Nouveau système (1726)...which became central to his theory of tonality in the Génération harmonique (1737), even though [it is] already implicitly present in his Traité. Beginning with the Génération harmonique, he posits the tonal organization of a key as the tonic surrounded by an upper and lower fifth—a dominant and subdominant.

Thus the subdominant is clearly understood as a fifth below tonic, not the scale degree immediately below dominant. (And note, by the way, that before Rameau, the "dominant" was actually called the dominante-tonique!)

As for the remaining terms—supertonic, etc.—I admit I'm not certain. My guess is somewhere between Rameau and Riemann, but I'm currently not able to verify this. But I think what's important is that these terms were added (and, in some cases, their definitions were changed to match their current usage) over time and by various different authors.