The Oxford History of Western Music, Pt. 6

Ars Nova theorists Philippe de Vitry and Jehan des Murs wrote treatises on the theoretical aspects of music. They sought to reconcile the rhythmic character of the twelfth-century motet, where a "longa" was equal to twice the "breve," and the thirteenth-century, "Franconian," motet, where the "longa" equalled a "perfection" of three "tempora."

From a mathematical perspective, the innovations that reconciled those rhythmic irregularities were the by-product of the theory of exponential powers and the theory of "harmonic numbers."

In basic terms, it was proposed that the breve could be broken into 3 (perfect, deriving from the holy trinity) or two (imperfect) semibreves. Semibreves could then also be broken into perfect or imperfect parts. There are four combinations of these:

1. Breve division: 3      Semi-breve division: 3

2. Breve division: 3      Semi-breve division: 2

3. Breve division: 2     Semi-breve division: 3

4. Breve division: 2     Semi-breve division: 2

The breve was roughly the equivalent of what we might now consider a bar. Therefore, combination 1 might represent 9/8 time, combination 2 3/4 time, combination 3 6/8 time,and combination 4 2/2 time.

One main difference between this and our notion of time signatures is that the above division do not dictate an accentuation scheme.

The Oxford History of Western Music, Pt. 5

Hermannus Contactus, a monk in the Swiss abbey of Reichenau, proposed that the six-note chord, C-A sums up, with greatest possible economy, the "sound" of Gregorian chant. Therefore the "finals" (see last post), where extended downwards to include the C. The tetrachords beginning on those finals (now C, D, E, and F) now begin to point to what we now think of as major and minor tonality.

The Italian monk, Guido of Arezzo, proposed, around 1030, placing nuemes on the lines and spaces of a ruled staff to define their precise pitch. He used colours, and then clefs, to denote "key lines"; points of reference to be used to find the pitches of the surrounding notes.

A specialist in ear-training, Guido chanced upon a tune (the hymn Ut queant laxis) that not only contained each diatonic interval up to and including a fifth, but where each half-line began on each of the notes proposed by Harmannus. The syllables at these points (corresponding with C, D, E, F, G, and A) are Ut, re, Mi, fa, Sol, and la. What we now know as sol-fege.

The full major scale came about when singers in the early seventeenth century began adding an extra note (si, later becoming ti) at the top of Hermannus's hexachord. Guido achieved something similar by transposing and superimposing the hexachord back onto itself, thus creating the gamut: a full range of pitches that were later represented on the Guidonian hand. Placing this pitches on the hand was a technique derived from public speakers and the like, and acted as a mnemonic aid. It is also where such expressions as "At one's finger-tips" or "rule of thumb" come from (the thumb on Guido's hand contains the first, most basic note of the gamut).

The Oxford History of Western Music, Pt. 4

A tonarius was a kind of index that grouped antiphons according to their psalm tones. They were created to assist finding applicable pieces for various church services, but also led to early theorists and analysts identifying the various ways the finalis (final note) and reciting tone (usually a fifth above) could be "filled-in" using step-wise motion. This led to four "species," each with it's own combination of semitones and tones to create a pentachord. The ending notes of these four species were dubbed the "four finals": D, E, F and G. The pentachord A-E was considered a doulbing of the first (D-A), while the one begining of B was deemed unusable due to it's including two semitones.

Chants ending on the four finals were then also categorised into two families. One where the final was at the bottom of the range of the melody, and one where it was in the middle. Analysts then categorised the different way the four species extended up (to the octave) and down (to the fifth) in the antiphons. The former were named "authentic" and the latter "plagal."

Thus the modes as we now know them (and named after tribes in Greece as well as Asia-minor) were first used as a method of categorising Gregorian chants, and only later became used as a basis for composition; chants composed after tonaries had been assembled demonstrate more clearly, in their melodic turns as well as their structure, the relationship between the final and reciting tone.

The Oxford History of Western Music, Pt. 3

• The Kyrie is a chant from the Franco-Roman Mass, derived from earlier forms of Congregational singing. The earliest sources of Kyries were little books called Kyriale. They contained two versions of each melody, a syllabic version along with a melismatic version. There is evidence to suggest that both verions were in usage, rather than one developing out of the other. If this is the case, it seems likely that the syllabic version was used as a mnemonic device to remind singers of the correspondence between notes and syllables, while the melismatic version was a guide to the intricate melody of the chant. It was this need to convey all of this information at once that could well have led to the development of the staff.