[In ancient Greek theory,] pertaining to harmonia and harmonike. When applied to chords or scales, it indicates that they are a section of the harmonic series.
The mathematical mean of two tones which produces intervals in the reverse order of the arithmetic mean is called the harmonic mean by Archytas because it was felt that this sequence was the more musical.
In Western European theory, harmonic refers to harmony.
... it really bugs me that so many music-theorists and historians are so quick to believe that "harmony" in music, meaning simultaneous combinations of different pitches, was "invented" by the Franks in the 800s.
certainly, there was a long time where the paradigm was monophonic "Gregorian" chant. but what about the period before that? Gregorian chant developed after a period of extreme social upheaval in Europe, namely, the Germanic migrations of the 400s-500s.
actually, i believe that the ancient Romans had some form of polyphony, even if it was basically this note-for-note style. Boethius in c. 505 specifically says that a "consonance" is the blending of two or more *simultaneous* sounds, and Nicomachus said the same thing c. 100 AD, when Greece was part of the Roman Empire. see the evidence i present below. note the emphasis on sounds which are "struck at the same time".
----- below: quotes from Boethius and Nicomachus ----
Quae sit natura consonantiarum.
XXVIII. Consonantiam vero licet aurium quoque sensus diiudicet, tamen ratio perpendit. Quotiens enim duo nervi uno graviore intenduntur simulque pulsi reddunt permixtum quodammodo et suavem sonum, duaeque voces in unum quasi coniunctae coalescunt; tunc fit ea, quae dicitur consonantia. Cum vero simul pulsis sibi quisque ire cupit nec permiscent ad aurem suavem atque unum ex duobus compositum sonum, tunc est, quae dicitur dissonantia.
ENGLISH [Bower, p 47]
28. What the nature of consonance is.
Although the sense of hearing recognizes consonances, reason weighs their value. When two strings, one of which is lower, are stretched and struck at the same time, and they produce, so to speak, an intermingled and sweet sound, and the two pitches coalesce into one as if linked together, then that which is called "consonance" occurs. When, on the other hand, they are struck at the same time and each desires to go its own way, and they do not bring together a sweet sound in the ear, a single sound composed of two, then this is what is called "dissonance".
In a footnote, Bower [p 47, footnote 130] says:
"Compare this definition of consonance (and dissonance) with Nocomachus Enchiridion 12 (JanS. 262.1-6); an important element in these definitions [by Boethius] and those found in Nicomachus is the phrase "struck at the same time" (simul pulsae, a translation of the Greek ανα ξρουσθηντησ [hana xrousthentes]). ..."
here is the passage from Nicomachus (c. 100 AD), chapter 12:
A system is a combination of two or more intervals. But no note among the intervals is consonant with the one immediately following it, but is completely dissonant with it. Among the systems, however, some are consonant, others dissonant. Systems are consonant when the notes comprising them, though they be different in compass, commingle with one another when played together or are somehow sounded simultaneously, in such a way that the sound produced from them is of a oneness like a single voice. Notes are dissonant, however, when the sound emanating from both of them is heard to be disparate in some way and unblended.
Consonae quidem sunt, quae simul pulsae suavem permixtumque inter se coniungunt sonum. Dissonae vero, quae simul pulsae non reddunt suavem neque permixtum sonum.
ENGLISH [Bower, p 116]
Consonant pitches are those which when struck at the same time sound pleasant and intermingled with each other; dissonant pitches are those which when struck at the same time do not yield intermingled sound.
[Boethius describes the construction of a device and labels parts of it with letter-names. Then...]
... atque alterutra vicissim .EK. et .KF. plectro adhibito pellantur, diatessaron distantia consonabit, sin vero simul utrasque percussero, diatessaron [-349-] consonantiam nosco.
ENGLISH [Bower, p 160]
I ... using a plecturm prepared for this purpose, strike EK and KF. If I strike them one after the other, the interval of a diatessaron ["4th"] will sound, but if I strike both at the same time, I come to know the consonance of the diatessaron. [emphasis added by Monzo]
Bower has this to say in a footnote [p 160, footnote 92]:
The concept of consonance as simultaneous sounds found in this passage is wholly consistent with that found earlier in the treatise -- e.g., at 1.28 and 4.1. The distinction between consonance and interval, the one struck at the same time (simul), the other alternately (alterutra), is unique to this passage.
Thesaurus Musicarum Latinarum
Boethius, Anicius Manlius Torquatus Severinus. c. 505.
Nicomachus of Gerasa. c. 100 AD.