HARMONY

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INDICE

The cosmogonic vision of music

Diapason

Harmonic Arithmetic and Geometric mean

Cadenze
Autentic or harmonic cadence

Annotation: Zarlino's writing  have been resumed from this sites:

http://www.examenapium.it/STUDI/Dizionari/zarlino.htm

Thesaurus musicarum italicarum

THE COSMOGONIC VISION OF MUSIC

Cosmogony (from Greek kosmos world and gonè generation, exposure of  universe's theories)

http://it.wikipedia.org/wiki/Pitagora

Pitagora (in Greek Πυθαγòρας - Pythagòras -, from πεἰθω - pèithō- = to persuade + ἀγορά - agorà- = public square, lit.  who persuades the public square; Samo, 575 a.C. - Metaponto, 490 a.C) has been a mathematician, legislator and Greek philosopher.

Some historians doubt the historical veraciousness of such personage. Ancient biographers attributed him a semidivine nature, that allowed him to accomplish prodigies including  to heal from diseases. He founded the homonymous philosophical school but, being convinced of the superiority of oral tradition regarding writing, did not leave writings. Moreover, since he prohibited his followers to write and speak with strangers of own theories, it  result impossible to assess which ideas were one’s own and which of his followers.

http://www.liceoquadri.it/alberodeisaperi/didattica/Zanella Pitagora.pdf

Since celestial bodies, with their regular motion, perform numerically expressible  movements, the Pythagoreans reach to support the existence of harmony's celestial spheres, not afferrabile from human eye. Aristotle is the one who testifies such thought of Pythagoreans and their cosmology.

We find trace of the Pythagorean theory in several works: for example  Platone in Republic resumes the theory of Pythagorean origin of Er's myth: eight celestial spheres rotate through a spindle, hold on knees by Ananke (Necessities); eight Sirens, in correspondence of the eight spheres and their movement, produce eight various musical notes.

Pythagoreans developed many experimental searches in musical field. With tense strings of equal length and fixed in a datum point by wedges situated between the extremities, they achieved  results which established the bases of music.

The three fundamental two-note chord of eighth, fifth and fourth were discoveries and after all the others. The relationships, between the string parts lengths  to the right and to the left of the wedge in the three cases were respectively: 1:2; 2:3; 3:4, that is the numbers of the tetraktys.

DIAPASON

Octave represents the range in which several degrees are ordered. Octave or diapason is subdivided in several intervals, such a model can be replied as towards acute voices as towards low-pitched one.

The word diapason is composed by: dia (through) and pason (all), implying chordon (string). It therefore represents a range which extremities are characterized by two  degrees themselves various but representing the same structural aspect, that is degree transported towards acute voices or the low-pitched one.

Zarlino thus describes  it and explains how from its division the other intervals are born.

LA SECONDA PARTE Delle Istitutioni harmoniche DI MAESTRO GIOSEFFO ZARLINO DA CHIOGGIA.

Che è più ragioneuole dire, che gli Interualli minori naschino dalli maggiori; che dire, che i maggiori si componghino de i minori; et che meglio è ordinato lo Essachordo moderno, che il Tetrachordo antico. Capitolo 48.

(...) dalla diuisione della Diapason habbiano origine tutte le consonanze, et gli altri interualli musicali quantunque minimi: imperoche veramente ella è la prima in tal genere, et è la cagione de tutti gli altri interualli, et la loro misura commune; et ciò conferma Marsilio Ficino nello Epinomide di Platone, quando parla della Forma di tal consonanza, et dice; che la Dupla è riputata esser proportione perfetta; primieramente perche ella è la Prima tra le proportioni, generata tra la Vnità, et il Binario: dipoi, perche mentre pare, che si habbia partito dalla Vnità, restituisce tale Vnità raddoppiandosi. Oltra di ciò dice, che contiene ogni proportione in se: conciosia che la Sesquialtera, la Sesquiterza, et le altre simili, sono in essa come sue parti. Et tutto questo si verifica della Diapason nella Musica: la cui forma è essa Dupla: percioche è la più perfetta di ogn' altra consonanza, et non patisce mutatione alcuna delli suoi estremi: et mentre pare, che si parta da vna certa vnità de suoni, restituisce tale vnità raddoppiandosi nelle sue parti. Similmente contiene in se (come hò detto) ogni semplice consonanza, et ogni minimo interuallo. Onde non è marauiglia, se tutti li Greci, di commune parere, la chiamarono [Dia pason]

HARMONIC ARITHMETIC AND GEOMETRIC MEAN
 

The first forms of  diapason's division are represented by arithmetical  geometric and harmonic mean.

In music there are three proportional averages : the first one is arithmetic, the second one is geometric one, third one is harmonica.

Arithmetic mean it has when three terms follow exceeding each other of a same amount that is, as the first exceeds the second, as the second exceeds the third

a-b = b-c

[b = arithmetic mean, strings lengths 4 - 3 - 2]

Geometric mean  it is had when the terms are in this way: as the first one is to the second, thus the second is to the third.

a:b = b:c

[b = geometric mean,  strings lengths 4 - 2 - 1]

Harmonic mean  it has when the terms are in this way: how much part of himself the first term exceeds the second, as much part of the third the second exceeds the third.

(a-b):a = (b-c):c

[b = harmonic mean, strings lengths 6 - 4 - 3]

The first form of diapason's division is represented from the harmonic mean.

Zarlino Gioseffo Le istitutioni harmoniche Parte 1 (1558)

LA DIVISIONE, ouero Proportionalità harmonica si fa, quando tra i termini di alcuna proportione si hà collocato vn Diuisore in tal maniera, che oltra le conditioni toccate nel capitolo 35. tra i termini maggiori si ritrouino le proportioni maggiori, et tra li minori le minori: propietà che solamente si ritroua in questa proportionalità; la quale è detta propiamente Mediocrità: imperoche ne i suoni, la chorda mezana di tre chorde tirate sotto la ragione delli suoi termini, partorisce con le sue estreme chorde quel soaue concento, detto Harmonia.

As it appears from the down below figure, the harmonic mean (harmonic or authentic cadence) it divides octave and it is represented from the diapente (lit. through the fifth) and diatessaron (lit. through the fourth) defined from the numerical proportion 6-4-3.

Diapente with numerical proportion 6-4, sequialtera (half, implied more, than other, 2 + 1 = 3) and diatessaron with numerical proportion 4-3,

sesquiterza,  (half, [always refired to the ratio 3-2, that is 1] implied more, than other, 3 + 1 = 4)

We must consider three strigns whose the central one determines the diapente, its relation with the first is in the proportion 3-2. Considering proportion 3/2 as the ratio of frequency in relation to the first string we will have 1, 3/2, 2, in which the first string it will has frequency 1, the second 3/2 and the third 2.

 AUTHENTIC CADENCE
 

(harmonic)

We define autentic or harmonic cadence when the fifth degree the dominant moves for main step to the first degree both to the fundamental state.

It is represented by harmonic division of diapason ascending fourth (diatessaron) or the descendant fifth  (diapente).

Moving bass for main step and leading tone resolving to the tonic give a conclusion feeling, used to the term of a phrase, period etc.