History Of Music Theory 1

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6th Century treatise that brought Greek theory to the Latin west

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De Instiutione Musica (The Fundamentals of Music)

by Boethius

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Boethius

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ca.480-ca. 524, Roman writer and statesman, wrote about music as "a part of the mathematical arts"

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Boethius's Three Kinds of Music

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Mundana (cosmic),
Humana (human), and Instrumentalis (that which rests in certain instruments)

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Boethius's three classes of musical art

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Also three classes of musical art- those who perform on instruments, those who compose songs, those who judge instrumental performance and song…the third being the true musician because he could weigh, with reason and sound judgement, the quality of songs and compositions.

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Five Classes of Ratios (from Fundamentals of Music)

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Multiple, Superparticular, Superpartient, Multiple superparticular, Multiple superpartient

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Ratio: Multiple

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large number contains within it the whole smaller number within itself twice, three times, four times and so on.
Ex. 2:1 4:2 3:1 6:2 25:5 27:3

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Ratio: Superparticular

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large number contains the whole smaller number plus a single part [factor] of it
Ex. 3:2 5:4 4:3

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Ratio: Superpartient

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large number contains within it the whole smaller number plus several parts of it
Ex. 5:3 9:5 11:6 13:7 10:6

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Ratio: Multiple superparticular

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large number contains within itself the smaller number either twice or three times, plus one other part of it.
Ex. 5:2 7:2 10:3

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Ratio: Multiple superpartient

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large number contains the whole smaller number more than once, plus more than one single part of it.
Ex. 8:3 16:6 11:4

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Seven liberal arts

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rhetoric, dialetic, grammar, arithmetic, geometry, music, astronomy

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How to find the sum of two intervals whose ratios you know

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Take both ratios in their lowest terms and multiply the larger number of one of them by the larger of the other and the smaller by the smaller, and the ratio of the products of these two multiplications will be the ratio of the aggregate of the two aforesaid ratios.

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How to find the ratio of the difference of two intervals whose ratios you know

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Take both ratios in their lowest terms and multiply the larger number of each of the the ratios by the smaller of the other and the ratio of the products of these two multiplications will be the ration remaining when the subtraction has been made.

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Greater Perfect System

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Made of Tone, Tetrachord, Tetrachord, Tone, Tetrachord, Tetrachord.


Tetrachord hyperbolaion
Tetrachord Diezeugmenon
Tone
Tetrachord Meson
Tetrachord Hypaton
Tone

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Tetrachord

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a system of four notes, contained within the limits of a perfect 4th.

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Tetrachords, disjunct and conjunct

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disjunct (don't share a pitch) or conjuct (share a common pitch).

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Types of Tetrachord

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Diatonic- semitone, Tone, Tone
Chromatic- semitone, semitone, minor 3rd
Enharmonic- quarter tone, quarter tone, Major 3rd

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Hucbald of St. Amand

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c. 850- 930, Benedictine monk, theorist and composer. wrote "De Harmonica Institutione" (known in "De Musica" in Gerbert's 1784 edition of the text).

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De Musica (formerly De harmonica institutione)

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by Hucbald, a practical handbook for the education of young monks in the proper performance of psalmody: constructed of a series of propositions and concepts that are closely linked