J.H.Padovani
Harmonia I [2018s1]

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Harmonia I
[aulas 2/3: os sistemas de afinação/temperamento e os primórdios do tonalismo]







José Henrique Padovani





. vibrações, modos de vibração, harmônicos


quando um sistema vibratório é excitado, certas frequências tendem a ressoar devido à formação de ondas estacionárias










. série harmônica


as ondas propagam-se ao longo do sistema vibratório criando padrões de ressonância e cancelamento que formam as ondas estacionárias.


. série harmônica


as ondas propagam-se ao longo do sistema vibratório criando padrões de ressonância e cancelamento que formam as ondas estacionárias.

em um sistema vibratório/ressonador homogêneo e ideal, portanto, as ondas estacionárias se formarão a partir de divisões do corpo vibrante (corda, coluna de ar, membrana etc) em partes proporcionais a números inteiros.


. série harmônica


as ondas propagam-se ao longo do sistema vibratório criando padrões de ressonância e cancelamento que formam as ondas estacionárias.

em um sistema vibratório/ressonador homogêneo e ideal, portanto, as ondas estacionárias se formarão a partir de divisões do corpo vibrante (corda, coluna de ar, membrana etc) em partes proporcionais a números inteiros.

o sistema vibra, nessas proporções inteiras, em diversos modos de vibração: com antinós (regiões em que o deslocamento de massa [ou amplitude] é o maior possível) e nós (regiões em que o deslocamento é o menor possível)


desses modos de vibração relacionados às características ressonantes de um sistema ideal (como uma corda ou uma coluna de ar), surge a série harmônica: uma série formada por uma frequência fundamental – f – e múltiplos inteiros dessa frequência (f *1, f *2, f *3, etc.)



...e se tentássemos jogar todas as notas dessa série harmônica com 32 notas para a mesma oitava (eliminando as notas repetidas)?



Enquanto na escala moderna (afinação igual), temos 12 notas separadas por 100 cents (100 centésimos de "semitom igual") –- distantes, portanto, de uma nota dó de referência a partir das distâncias de...

0 100 200 300 400 500 600 700 800 900 1000 1100 

... -– na escala obtida a partir dos primeiros 32 parciais harmônicos (rejeitando notas que seriam repetidas em diferentes oitavas), acabamos com 16 notas diferentes, com os seguintes valores em cents a partir da mesma nota de referência...


0 104 204 298 386 470 552 628 702 772 840 906 968 1030 1088 1146

. sistemas de afinação / temperamento


sistemas de afinação: sistemas que definem quais notas serão tocadas/entoadas em um instrumento ou prática vocal.


. sistemas de afinação / temperamento


sistemas de afinação: sistemas que definem quais notas serão tocadas/entoadas em um instrumento ou prática vocal.

e o que isso tem a ver com harmonia?




. sistemas de afinação / temperamento


sistemas de afinação: sistemas que definem quais notas serão tocadas/entoadas em um instrumento ou prática vocal.

e o que isso tem a ver com harmonia?

um sistema de afinação determina o repertório de notas/frequências à disposição de quem compõe/toca



. sistemas de afinação / temperamento


sistemas de afinação: sistemas que definem quais notas serão tocadas/entoadas em um instrumento ou prática vocal.

e o que isso tem a ver com harmonia?

um sistema de afinação determina o repertório de notas/frequências à disposição de quem compõe/toca
...e quais serão razões entre as vibrações de duas alturas



. sistemas de afinação / temperamento


sistemas de afinação: sistemas que definem quais notas serão tocadas/entoadas em um instrumento ou prática vocal.

e o que isso tem a ver com harmonia?

um sistema de afinação determina o repertório de notas/frequências à disposição de quem compõe/toca
...e quais serão razões entre as vibrações de duas alturas
...com isso, determina quais intervalos surgirão entre duas ou mais notas e, juntamente com valores culturais/estéticos e fatores timbrísticos, espectrais e psicoacústicos, determina quais dessas relações intervalares soarão como consonâncias ou como dissonâncias

. escala pitagórica

Construída a partir da superposição de 5as puras (razão 3/2) ou 4as puras (razão 4/3).

(IAZZETTA, Fernando. "Escala Pitagórica" In: Tutoriais de áudio e acústica. Disponível em: http://www2.eca.usp.br/prof/iazzetta/tutor/acustica/escalas/pitagorica.html. [Acesso em: 3/mar/2018])

Construção de uma escala cromática pitagórica no OpenMusic:

patch no OpenMusic: pitagoras-D.omp

. escala justa

Uma escala justa é uma escala construída a partir de razões entre números inteiros pequenos. Qualquer intervalo afinado desta maneira é denominado um intervalo puro ou simples.
(https://en.wikipedia.org/wiki/Just_intonation)

When pitch can be intoned with a modicum of flexibility, the term ‘just intonation’ refers to the consistent use of harmonic intervals tuned so pure that they do not beat, and of melodic intervals derived from such an arrangement, including more than one size of whole tone. On normal keyboard instruments, however, the term refers to a system of tuning in which some 5ths (often including D–A or else G–D) are left distastefully smaller than pure in order that the other 5ths and most of the 3rds will not beat (it being impossible for all the concords on a normal keyboard instrument to be tuned pure; see Temperaments, §1). The defect of such an arrangement can be mitigated by the use of an elaborate keyboard.
(LINDLEY, Mark. "Just [pure] intonation". In: Grove Music Online. [Acesso em: 3/mar/2018])


a ideia geral na construção da escala justa é manter o máximo de relações intervalares puras
(i.e., baseadas nas razões das primeiras alturas resultantes da série harmônica)

para a obtenção de uma escala diatônica justa, é possível utilizar uma altura principal de referência e a 5ª acima e abaixo dela como frequências a partir das quais se obterá as respectivas 5J e 3M puras/justas:



(https://en.wikipedia.org/wiki/Just_intonation#Diatonic_scale. [Acesso em: 3/mar/2018])

(IAZZETTA, Fernando. "Escala Justa" In: Tutoriais de áudio e acústica. Disponível em: http://www2.eca.usp.br/prof/iazzetta/tutor/acustica/escalas/justa.html. [Acesso em: 3/mar/2018])

. escala mesotônica

Let's start with Europe's most successful tuning, if endurance can be equated with success. Meantone tuning appeared sometime around the late 15th century, and was used widely through the early 18th century. In fact, it survived in pockets of resistance, especially in the tuning of English organs, all the way through the 19th century. No other tuning has survived in the west for 400 years. Let's see what meantone offered.

Every elegant tuning has a generating principle. The generating principle behind meantone was that it was more important to preserve the consonance of the major thirds (C to E, F to A, G to B) than it was to preserve the purity of the perfect fifths (C to G, F to C, G to D). There are acoustical reasons for this, namely - though I wouldn't want to go into the math involved - that the notes in a slightly out-of-tune third, being closer together than those in a fifth, create faster and more disturbing beats than those in a slightly out-of-tune fifth. (...) The aesthetic motivation for meantone was that composers had fallen in love with the sweetness of the major third, and were trying to get away from the medieval austerity of open perfect fifths.
(GANN, Kyle. Disponível em: http://www.kylegann.com/histune.html#hist2. Acesso em: 6/mar/2018.)

In a purely consonant major third, the two strings vibrate at a frequency ratio of 5 to 4. For example, if

A
vibrates at
440 cycles per second,
then
C#
vibrates at
550 cycles per second.


Or if G vibrates at 100 cycles per second, then B vibrates at 125, and so on. (If you'd like this explained in more detail, visit my Just Intonation Explained page.) The size of a pure 5:4 major third is 386.3 cents, a cent being one 1200th of an octave, or one 100th of a half-step. Since an octave is 1200 cents, by definition, it is easy to see that three pure major thirds (3 x 386.3 cents = 1158.9) do not equal an octave. That's the whole problem of keyboard tuning, where you're limited to 12 steps per octave. Where do you put the gaps in your chains of perfect major thirds?
A pure perfect fifth is a 3 to 2 frequency ratio; if

A

vibrates at
440 cycles per second,

then
E
vibrates at
660 cycles per second.



A pure perfect fifth should be 702 cents wide, which is just about 7/12 of an octave; our current equal-tempered tuning accomodates perfect fifths (at 700 cents) within 2 cents, which is closer than most people can distinguish, but the thirds (at 400 cents) are way off, and form audible beats that are ugly once you're sensitized to hear them.

(GANN, Kyle. Disponível em: http://www.kylegann.com/histune.html#hist2. Acesso em: 6/mar/2018.)



Let's look at the meantone solution. There was no one invariable meantone tuning; before the 20th century, tuning was an art, not a science, and each tuner had his own method of tuning according to his own taste. The following is a chart of what was initially the most common form of meantone, called 1/4-comma meantone, first documented by Pietro Aaron in 1523, though he didn't draw it out to all twelve pitches:

Pitch:CC#DEbEF F#GG#AA#B C
Cents:076.0193.2310.3386.3 503.4579.5696.8772.6 889.71006.81082.91200

(I adapt this chart, and ones following below, from an invaluable book, the bible of historical keyboard tuning: Owen Jorgensen's Tuning: Containing the Perfection of Eighteenth-Century Temperament, the Lost Art of Nineteenth-Century Temperament, and the Science of Equal Temperament, Michigan State University Press, 1991.) Now let's look at the sizes of the major thirds and perfect fifths on each pitch:

Major thirdCentsPerfect FifthCents
C - E386.3C - G696.8
Db - F427.4Db - Ab696.6
D - F#386.3D - A696.5
Eb - G386.5Eb - Bb696.5
E - G#386.3E - B696.6
F - A386.3F - C696.6
F# - A#427.3F# - C#696.5
G - B386.1G - D696.4
Ab - C427.4Ab - Eb737.7
A - C#386.3A - E696.6
Bb - D386.4Bb - F696.6
B - D#427.4B - F#696.6

A major third and perfect fifth on the same pitch, of course, make up a major triad, the most common chord in European music from 1500 to 1900 - the meantone era. Let's look at what kind of major triads we have in meantone tuning.
(GANN, Kyle. Disponível em: http://www.kylegann.com/histune.html#hist2. Acesso em: 6/mar/2018.)




The major thirds that are about 386 cents wide will be sweet, consonant, attractive. Eight pitches have virtually perfect major thirds on them - all except Db, F#, Ab, and B, whose major thirds are all about 427 cents.

(...)

All of the fifths are about 696 cents except for one, that on Ab, which is 737 cents and sounds terrible. The fifths would sound better at 702 cents, but at 696 or 697 you don't really notice the difference, especially if the chord is filled in with that perfect major third to smooth over the discrepancy. This is where the practice originated in European music of never having an open fifth sounding by itself without a third filling it in: the spare perfect fifth isn't quite consonant, and that fact becomes obvious if the third isn't there.



So meantone tuning gives us eight usable major triads: on C, D, Eb, E, F, G, A, and Bb. If you're writing a piece in meantone, those are the major triads you have available. Look through some 16th-century keyboard music: how many F#-major and Ab-major triads do you see? Probably none, and if you do see some, it means the composer was counting on a meantone tuning centered around some pitch other than C.

If you want to use I, IV, and V chords in your piece, you can write in the keys of C, D, F, G, A, or Bb major.

If you're writing in A major, you can't go to the V/V chord (B major), because it sounds awful. Renaissance and early Baroque music tends to be in a few keys grouped (in the circle of fifths) around C, usually C, F, G, D, Bb, or A.

Ever wonder why Palestrina and Orlando Gibbons and Heinrich Schutz didn't get around to composing in F# major or Ab major? They couldn't, it sounded terrible in their tuning. (There were a few purely vocal early works that went through triads in diverse keys, such as Josquin's motet Absalon fili mi and Di Lasso's Prophetiae sybyllarum, the tuning and even notation of which have been subjects of much 20th-century controversy.)

Before we leave the subject of meantone, lets look at the available minor triads:

Minor thirdCentsPerfect FifthCents
C - Eb310.3C - G696.8
C# - E310.3C# - G#696.6
D - F310.2D - A696.5
Eb - Gb269.2Eb - Bb696.5
E - G310.5E - B696.6
F - Ab269.2F - C696.6
F# - A310.2F# - C#696.5
G - Bb310.0G - D696.4
G# - B310.3G# - D#737.7
A - C310.3A - E696.6
Bb - Db269.2Bb - F696.6
B - D310.3B - F#696.6


A pure minor third is supposed to have a frequency ratio of 6:5. For example, if C# vibrates at 550 cycles per second, E should vibrate at 660. A 6:5 ratio interval is 315.64 cents wide. None of the minor thirds in this meantone are quite that wide, but most of them are 310 cents, which is, pardon the expression, close enough for jazz. (Actually, a narrow 7/6 minor third, often used by La Monte Young, is 266.8 cents, invitingly close to that 269; but 7/6 is an interval that was never recognized by European theory, though used in jazz and Arabic music among others.) Therefore the minor triads on C, C#, D, E, F#, G, A, and B are acceptable. (Not the one on G#, despite its OK minor third, because it has that wildly beating fifth.) If you think about it, these triads define the relative minor of the major keys implied by the major triads above:

Major:CDEbEFG ABb
Minor:abcc#d ef#g





These 16 triads, 8 major and 8 minor, constitute the harmonic vocabulary of Renaissance and early Baroque music. Don't believe me? Look through a 16th- or 17th-century keyboard collection, such as the Fitzwilliam Virginal Book.

One important keyboard work from the early 17th century (a real masterpiece, in fact) is Orlando Gibbons's Lord Salisbury Pavane. It's in A minor. If you look at it (it's in the Historical Anthology of Music), Gibbons several times goes to the major triads on F, G, and C (which are in A natural minor), E (in the harmonic major), and D (not in A minor). He never, however, uses a B major (V/V) or F# major (V/ii) triad, even though V/V is not rare and V/ii not unthinkable in a minor key. He avoids them because they don't really exist in the tuning of his harpsichord. Had Gibbons begun in the key of C minor, he would have had to write a different piece, because instead of moving from A minor to F major, he would have had to move from C minor to Ab major, and Ab major, strictly speaking, didn't exist on his harpsichord.




[material a ser expandido futuramente:
- a afinação da terça justa e da terça a partir das quintas pitagóricas
- coma sintônica
- a afinação mesotônica de quarto de coma
- aplicativos/sites para escutar exemplos (SCALA, exemplos específicos em áudio, etc.)
- exemplos iniciais de De La Motte em diferentes afinações/transposições]


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