Jean Louchet - s3.eu-west-3.amazonaws.com€¦ · 13.02.2020 · 4 Zarlino: coming closer to Physics The Pythagorean system focuses on the harmonic 3 and proudly ignores higher harmonics

THE FOUNDATIONS OF MUSICAL SYSTEMS

Jean Louchet

THE FOUNDATIONS OF MUSICAL SYSTEMS Jean Louchet 18th December 2012 Hogeschool Gent page 1/104

Summary

1 Introduction 32 Some physics: harmonics, partials and inharmonicity 43 Pythagoreas and the classical musical notation 104 Zarlino: coming closer to Physics 215 The genesis of Harmony: Musica Recta, Musica Ficta 316 Time-frequency diagrams and the analysis of music 367 The organ and the synthesis of sound colour 438 Stringed keyboards: from the harpsichord to the pianoforte 52


1 Introduction

Is there any serious tonal theory behind music?

or:

why are there 12 notes on each octave on a piano?

Many interval systems in the world

7 note gamut (Occidental system)in reality 12 notes are used in modern music but many more have a name!

European Medieval and Renaissance music use more refined scales.

Middle-Eastern musical systems: 24 notes (“quarter tones”)and many others...


2 Some physics: harmonics, partials and inharmonicity


2.1 What is an octave?

Sound ‘modulo...’ a frequency factor 2.

Our auditory cortex has got a built-in “logarithmic modulo 2”is it reasonably accurate?

General philosophy: just like with vision,the audio information that comes to the conscious level:

• is not the sound,• is not the frequencies,• it is actually an attempt at identifying the physical source of the sound.

Similarity with vision:

Human vision is the process of reconstructing a physical object from its image(s).

Human audition is the process of reconstructing an object from the sound(s) it produces.

Biological point of view...


2.2 Musical objects and the lattice of harmonics

Here, we assume a string is perfectly elastic.

Metal strings are relatively recent...

• Gut• Copper• Brass• Bronze• Iron• Steel• Nylon• PVF (PolyVinylFluor)• Self overspinning (gut, PVF)• Classical overspinning:

• metal (aluminium, brass, silver) over gut or PVF (plucked and bowed instruments)• metal over metal: annealed copper over iron or steel (pianos)• close or open


Fundamental frequency , harmonic # has frequency .f n nfDecomposition into a product of prime numbers:

n = 2u × 3x × 5y × 7z × 11t...Striking point + internal damping → reduce higher harmonicsHearing modulo 2 → eliminate 2u

The meaningful harmonics are given by

n = 3x × 5y × 7z × 11t...Internal damping, choice of striking points and initial conditions (e.g. piano hammer curvature) reducehigher harmonics.Algebraists call this a lattice structure.

[Electronicians also work on lattice structures: Boolean algebras are defined as ’distributive complementedlattices’. Karnaugh diagrams are a consequence of this lattice structure]

The useful harmonics are given by

n = 3x × 5y × 7z

and even, in most stringed instruments, by

n = 3x


2.3 String vibrating modestension T = π ρ d2 F2 L2

f requency F =1

d L

T

π ρFor a perfectly flexible string, partial #n has frequency:

Fn = n F =n

d L

T

π ρthus each string produces a series of partials with relative frequencies 1, 2, 3, 4, 5, 6, 7, 8... ‘harmonics’but in reality, due to string stiffness, the actual frequency of partial #n is higher than the harmonic:

f n = n F (1 + B n2)1/2

using the basic inharmonicity coefficient

B =π2 E S K2

T L2E is the Young’s modulus, only depending on the string’s material.For a round section string, (gyration radius) andK = d4

B =π2 E d2

64 ρ For example if the partials will beB = 0.005

1; 2 (1 + 4B)1/2 ≈ 2 × 1.01; 3(1 + 9B)1/2 ≈ 3 × 1.022; 4(1 + 16B)1/2 ≈ 4 × 1.039, etc.


Reference: Fletcher (Neville H.), Thomas D. Rossing,The Physics of Musical Instruments, 2nd edition,Springer1998-99.


3 Pythagoras and the classical musical notation


3.1 Pythagoras and the harmonic 3We are working in frequencies ‘modulo a factor 2’Let us choose a frequency (say ) and work out its harmonics 3xF F = 1As we work modulo a factor 2, let us see how the numbers are spread on the scale of frequencies.3x × 2y

C D E F G B CA

The sequence of the first 7 Pythagorean harmonics:FCGDAEB (natural order), or ABCDEFG (frequency order): tonality of C.

This is one of the simplest possible musical systems. It is possible to make music using 7 tones. Their orderof appearance is F C G D A E B. Actually they are named after the order of their frequencies: A B C D E FG.The tradition in the Western musical systems is to call C (the second tone of the sequence) the Tonic. F theSubdominant and G the Dominant. If we take the tonic C as a point of departure, the interval widths are:large, large, small, large, large, large, small.Now it is of course possible to use another note than F as a point of departure of the sequence FCGDAEB.For example lets take C. The new sequence is: CGDAEBX where X is a new frequency. We have to give aname to X; as the name F is no longer used in this new system, we recycle the name F and write the sign #to avoid any confusion with the old F.This is how key signatures are made – and why there is one # in G major, two ## in D major, etc.

C D E F G B CA

X = F#

The new sequence using C as a point of departure (tonality of G)


Do Do# Re

Reb

Mi Fa Fa#

Solb

Sol Sol#

Lab

La#

Sib

Si DoRe#

Mib Reb

La Do#

Enriching the scale: more tones upwards and backwards

This is the basis of the standard musical notation and the infinite Pythagorean scale:

... Ebb Bbb Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# Fx Cx Gx ...tone of C: –––^––––––––––––––– no alterationtone of G: –––^––––––––––––––– one #tone of D: –––^––––––––––––––– ##tone of A: –––^––––––––––––––– ###tone of F: –––^––––––––––––––– one btone of Bb: –––^––––––––––––––– bbtone of Eb: –––^––––––––––––––– bbbetc.hence the classical ‘key signatures’ which tell the tonality: here are the most frequent ones:


The most frequent key signatures and the corresponding tonalities.


Tuning with pure "fifths" (harmonic no. 3) shows the Pythagorean comma: 312 / 2something


...and some flats. This may be pushed to infinity, an endless story...


The infinite system of music as it is written:


The Pythagorean scale is infinite. This is because the equation:

3x 2−y = 1has no solution with positive integers.However we should be so glad to be able to eventually fall back into our shoes rather than having aninfinity of keys at each octave – in other terms, are there approximate solutions to this equation?The first solution is .x = 7, y = 11

3x 2y =21872048

≈ 1.068

which is a very crude approximation. It would result in dividing the octave into 7 equal intervals.The second solution is .x = 12, y = 19

3x 2y =531441524288

= 1.0136

which is the basis for the ‘equal temperament’, a very old system (China, ca. 2000 B.C.?) that has re-appeared in Europe during the late 18th Century, as a simplification of the more refined previous systems.The third solution is x = 43, y = 68.

3x 2y =3.282569674 E202.951479052 E20

= 1.1122

and the fourth solution is x = 53, y = 84.

3x 2y =1.938324567 E251.934281311 E25

= 1.0021

The latter is remarkably accurate. It results into the octave being divided into 53 equal micro-intervals,called ‘Holder’s Comma’. The difference with the exact Pythagorean frequencies are smaller thananything audible.


C C# D

Reb

E F F#

Gb

G G#

Ab

A#

Bb

B CD#

Eb Db

A C#

The 53 interval system: a good approximation - one Pythagorean comma is not far from 1/53 octave(Holder comma).

If we express the Pythagorean system into Holder’s commas, we have:C - D 9 commasD - E 9 commasE - F 4 commasF - G 9 commasG - A 9 commasA - B 9 commasB - C 4 commas

The interval between a note and the homologous sharp or flat note (e.g. between D and D#, or between Dband D) is called the ‘chromatic semitone’ in the Pythagorean system. Its value in Holder’s approximationis 4 commas.

The complement of a chromatic semitone to a tone (e.g. between D and Eb, or between C# and D) is calledthe ’diatonic semitone’. Its value in Holder’s approximation is 5 commas.The interval between a note and its close neighbour (as between Eb and D#) is called the ‘Pythagoreancomma’ or ‘enharmonic interval’in the Pythagorean system, ≈ 1.01364.312 / 219

Its value in Holder’s approximation is one Holder’s comma, or 1/53 of an octave, or 1.01316.21/53 ≈


Unfortunately, while the musical notation is entirely based on the Pythagorean system, most keyboardsonly use 12 keys per octave.

This does not necessarily imply that the instrument is tuned to the equal temperament!

Standard keyboard with x = 12


4 Zarlino: coming closer to PhysicsThe Pythagorean system focuses on the harmonic 3 and proudly ignores higher harmonics.This is a heavy restriction to musical composition, as intervals like C - E, F - A and G - B (called ‘majorthirds’) will sound badly on any instrument that produced harmonic 5 (i.e. about every instrument!)

On a Pythagorean scale, the ratio E : C is 34 / 26 = 81/ 64.The sound of C will contain its harmonic 5, which will beat with E as the ratio is 81/80 this is called thesyntonic comma.

Some European medieval systems consider the interval of third as dissonant and normally forbidden. This is aserious limitation.

To overcome this, Zarlino (1517-1590) introduced a slightly different scale, taking the harmonic 5 intoaccount.

F→C, C→G, G→D factor 3 (‘pure fifths’)C→E, F→A, G→B factor 5 (‘pure thirds’)

C D E F G A B C

Pythagorean scale 1 9/8 81/64 4/3 3/2 27/16 243/128 2

Zarlinian scale 1 9/8 5/4 4/3 3/2 5/3 15/8 2

Easy to tune by ear...


The Zarlinian scale can be extended: here are the most useful tones.

A→C#, B→D#, Db→F, Eb→G, E→G#, Bb→D→F# factor 5C C# Db D D# Eb E F F# G G# A Bb B C

Pythagorean 1 2187/2048≈1.0679

256/243≈1.0535

9/8=1.125

32/27 32/27 81/64 4/3=1.3333

1024/729≈1.4047

3/2=1.5

128/81≈1.5802

27/16=1.6875

16/9≈1.7778

243/128≈1.8984

2

Zarlinian 1 25/24≈1.0417

16/15≈1.0667

9/8 75/64 6/5 5/4 4/3 45/32≈1.4062

3/2 25/16=1.5625

5/3≈1.6667

9/5=1.8

15/8=1.875

2

With the Pythagorean scale, the Pythagorean ‘enharmonic’ comma (e.g. between C# and Db) is312 / 219 ≈ 1.01364.Here, we find a new interval between C# and Db: 1615 :

2524 = 109 ≈ 1.1111

As both D# and Eb are frequently encountered in musical compositions, many Renaissance instruments(especially at Zarlino’s time) had split accidentals. Same with G#/Ab.

Another interesting point is that flats and sharps are swapped:

• in the Pythagorean scale, enharmonic sharps are higher than flats: C# > Db• in the Zarlinian scale, enharmonic sharps are lower than flats: C# < Db.

This system is the backbone of most Mediterranean antique, medieval European and traditional Amerindianmusic systems.


However, complex polyphony arose from the late Middle Ages – Perotin and the Ecole Notre Damein the 13th Century, then Binchois, Dufay, Ockeghem, very probably as the development of a Christiantradition to symbolise the multiple voices inspired by the Holy Spirit at the Pentecost, all of this having aconsiderable development during the 16th Century – Thomas Tallis’ Guinness record motet ‘Spem in aliumnunquam habui’ with 40 singing voices is a telling example.

A 6-tone system, while well suited to plain-song, is a hard limitation when attempting to sing with more thantwo voices (cf. medieval Bicinis). It was necessary to find a way around this difficulty.

• the first route was a development of the Zarlinian system, leading to the multiple Hexacord systemand Musica Ficta – an intonation system born in the 12th and 13th Centuries with l’Escole Notre-Dame (Maître Pérotin), further developed by the medieval poet and musician Guillaume de Machautthen through all the Renaissance period and the early Baroque (Monteverdi), ca. 1610.

• the second route, born in Italy during the late Renaissance (ca. 1560) abandoned the refined accuracyof Musica Ficta in favour of a circular system, at the cost of the importance of the term – at least intheory. It only became the dominant system from the mid 19th Century.

5y

• Most of the baroque, classical and early romantic music uses a mixture of the two approaches, throughcircular temperaments. J. S. Bach’s Das Wohltemperirte Klavier is an outstanding illustration of howit is possible to write into all possible tonalities while using an unequal circular temperament (this hasoften been misunderstood). In these temperaments, some thirds are pure or acceptable while otherones are terribly false, the trick is to use them only on extremely short values.


A 16th Century Italian harpsichord (virginal) with split accidentals. St Cecila’s hall collection, Edinburgh/


This is D#, not Eb.


Guarracino’s Renaissance ‘enharmonic’ keyboard with (in fact only 19 keys per octave). It isplayable!

x = 53

The Danikey system: 22 keys per octave


August Forster’s enharmonic piano with 36 keys per octave.


harmonic 3

harmonic 5

small diesis=enharmonic comma

syntonic comma=Zarlinian comma

Pythagorean comma

schisma

diaschisma

~11 schismas

~12 schismas

Reference

-5 32 x5 = 125/128

2 x3 x5 =2025/2048-11 4 2

2 x3 x5=32805/32768-15 8

2 x3 x5 =81/80-4 4 -1

2 x3 =531441/524288-19 12

Approximate meeting points and micro-intervals. Red = too high, green = too low.


The Meantone systems: some pure Thirds, honour to the fifth harmonic! - the wolf tone.

The meantone temperament above (mesotónico) is but one among many practical solutions.

Well tempered ≠ equally tempered!


5 The genesis of Harmony: Musica Recta, Musica Ficta


5.1 Solmization in theory

Mi contra Fa, diabolus in Musica

tetrachords, hexachords, solmization


D E F G A B B C

ut re mi fa sol laut re mi fa sol la

ut re mi fa sol la


5.2 Solmization in practiceExample taken from William Byrd, O God that guides the cheerful sun.


6 Time-frequency diagrams and the analysis of music

Intensity and time-frequency diagram of a spoken sentence.


Voice pitch and harmonicsPitch frequency: male 100-130, female 180-250

Vocal tract shape and formants – the envelope of the harmonics of the pitch frequency.


“goes out as a great champion”itre.cisupenn.edu/~myl/languagelog/archives/002353.html


Characterising vowels by their first two formants.


DIDEROT & D’ALEMBERT WERE WRONG!

THE RATIOS OF HARMONICS DO NOT MAKE THE SOUND COLOUR

THE FORMANTS MAKE THE COLOUR


Broman’s ‘monster harpsichord’ made after Diderot’s principles.

• one string material• one string diameter• one string tension

don’t make the sound timbre homogenous. Fortunately, the case gives its own formants.


‘Clavecin brisé’ de Jean Marius: three cases, three different sound colours

TRY A TAPE RECORDER...


7 The organ and the synthesis of sound colour


7.1 How an organ works• Flue pipes and reed pipes• Pipe scaling and denomination• Bellows, windchest, pipework, console.• Positive, great [organ], recit / swell, echo, pedal• Swell and expression.

First understand the ‘real’ pipe organ.

Flue pipes: a whistle (similar to a recorder’s) and a pipe. The pipe length determines the pitchReed pipes: use a resonating reed (similar to the clarinet or saxophone)

At the heart ot the instrument: the WINDCHEST (‘sommier’) is basically a matrix.Air supplied by a bellows or electric pump.Air pressure stabilised by a bellows system. *tremblant

The windchest matrix is organised into

• columns: one column per key• rows: one line per register.

Each register can be switched ON (pulled) or OFF (pushed).Depressing any key opens a valve which provides air to the pipes at the intersections of the depressed key(columns) and the pulled registers (rows).


Cut view of an organ windchest (tracker action)..


Register, stop, rank

Windchest and pipework: general view.

Pipework and denominationsA row of pipes is described by a name and a number.The number is the length of the lowest pipe of the register (in feet).A 8ft stop is a register at the normal octave.a 4ft stop will sound one octave higher.a 16ft stop will soud one octave lower.etc.Usual range: 4 octaves (Renaissance/Early baroque) to 5 octaves (modern)Pipe lengths: on a 4 1/2 octave 8ft register, from 240cm to 20cm.


7.2 Organ architecture

Cesar Franck at the console of Sainte Clotilde organ, Paris: 3 manuals, 1 pedalboard. THE FOUNDATIONS OF MUSICAL SYSTEMS Jean Louchet 18th December 2012 Hogeschool Gent page 47/104

Several organs into one organ1 to 5 manual keyboards, 0 to 1 pedal keyboard.One windchest for each keyboard

Continuo organs: 1 manual, 3 registersSmall church organs: 1 or 2 manuals, 1 pedalboardMajor church organs: 3 manuals, 1 pedalboard, couplers, swell, 40 to 70 registersMay go up to 5 manuals, 1 pedalboard and 120+ registers


buffet du

grand-orgue

buffet dupositif

Grand-orgue

positifde dos

soufflerie

Le buffet du Clicquot de Souvigny (Allier), d'après un relevé de B. Aubertin. On distingue bien le positif dugrand-orgue. A droite, un orgue français (D. Bedos).


7.3 Register families and registrationUnisons and octaves (flue pipes)

Principals, flutes and diapasons: 16’, 8’, 4’, 2’, 1’also called ’montre’, ‘prestant’, ’doublette’ (2’), ‘whistle / sifflet’ (1’), ‘cor de nuit’ (4’), gambas, etc.special case: stopped pipes. A 4’ long pipe will sound 8’: stopped diapason, bourdon, stopped flute etc.Stopped pipes don’t produce even harmonics.This is the backbone of the art of registration.

Simple mutationsSimilar pipes, but synthesizing the other harmonics to the 8’ fundamental:2’2/3 ‘fifth’, ‘quint’ or ‘nazard’ 1’3/5 ‘Third’ 1’1/3 ‘duodecima’ or ‘larigot’

Composite mutations (mixtures)Multirank stops, containing a combination of octaves and simple mutations, from 3 to 6 ranks:4’, 2’2/3, 2’, 1’3/5, 1’1/3, 1’with different balances, e.g.:

• ‘plein jeu’ emphasis on 4’, 2’, 1’• ‘fournitures’, ‘cymbals’, ‘ripieno’,• ’cornet’ (emphasis on 1’3/5 )

Some 19th century organs also include composite registers like salicional, unda maris, vox celeste etc.that combine slightly detuned or heterogenous pipe pairs.

Reeds‘solo’ stops, often inspired by ‘real’ instruments: crumhorn, dulcian, oboe, trumpet, gemshorn, regal, etc.


7.4 A classical organ: François-Henri Clicquot, Souvigny (France) 1783

Positif Dessus de flute 8’Prestant 4’Doublette 2’Plein-jeu VBourdon 8’Nazard 2 2/3’Tierce 1 3/5’Cromorne 8’Trompette 8’

Récit (C3 - D5) Bourdon 8’Cornet IVHautbois 8’

Pédale (F0 - A2) Flute 8’Flute 4’Trompette 12’Clairon 6’

Grand-orgue Montre 8’Prestant 4’Doublette 2’Plein-Jeu VIBourdon 8’Nazard 2 2/3’Quarte 2’Tierce 1 3/5’Cornet V (C3)Trompette 8’Clairon 4’Voix humaine 8’


8 Stringed keyboards: from the harpsichord to the pianoforte


8.1 The clavichord

Minimalist

Schéma de principe du clavicorde. La corde vibre entre la tangente et le chevalet.

Fretted or unfretted?


Fretted clavichord

Expressiveness

Plus faict doulceur que violence (Arnold Dolmetsch)


8.2 The harpsichordPlucked strings


Italian action.


French action: shift coupler (’accouplement à tiroir’)


English action: dogleg.


Registers and registrationThe instrument is built around its stringss steel, iron, brass, bronze, red copper, gut?The Lautenwerk


Clavicytherium-lautenwerk (1490), copy by Boinnard. Original: Royal College of Music, London. THE FOUNDATIONS OF MUSICAL SYSTEMS Jean Louchet 18th December 2012 Hogeschool Gent page 62/104

The harpsichord in all shapes:

Spinets: Italian virginal, Flemish virginal, Flemish muselaar, “mother and child” muselaar, Englishbentside (Hitchcock).


Large harpsichords: clavicytherium, Italian and Flemish harpsichords.

North vs. South: the bottom line... plane.

• Southern (Italy, Spain, Portugal, Austria) built on the bottom plank.


• Northern: Flanders, Germany, France, England, re-Germany, Scandinavia first gets an internalbracing then a bottom plank.


PowerInharmonicitySound quality


Inharmonicité et qualité de timbre

La décadence: le clavecin expressif.


8.3 Il clavicembalo con piano e forteThe pionneer: Bartolomeo Cristofori, 1710.The follower: Gottfried Silbermann, 1740


8.3.1 Cristofori and the hammered harpsichord

Cristofori’s pianoforte


Cristofori’s action (copy) THE FOUNDATIONS OF MUSICAL SYSTEMS Jean Louchet 18th December 2012 Hogeschool Gent page 71/104

Rough model of a push action (Stoßmechanik) by Cristofori.


Stop action (Prellmechanik) by Zumpe & Buntebart (London)


Square piano by Zumpe (ca. 1770)


Silbermann’s inverted stop action, close relative to the (more recent) Wiener action.


8.3.2 Play closer: the escapement (“English action”), Clementi, Broadwood

English action: Broadwood 1777. Invention of the pushing jack and the escapement dolly.


Steinway 1860: an English action with a repetition lever (unconvincing).


8.3.3 Wiener/German action: Stein, Walter, Streicher, Schantz, Graf...compulsory lightness.

Stein 1779,the first Wiener action, evolve from Silbermann’s stop action.


Walter’s piano. The first Wiener pianos (with Stein) – and Mozart’s choice.


A modern copy of Schantz ca.1820 - lthe second Wiener school.

Turqueries to forget... THE FOUNDATIONS OF MUSICAL SYSTEMS Jean Louchet 18th December 2012 Hogeschool Gent page 80/104

Una corda, céleste, drommel, storm, forte (undamping).


8.3.4 Repetition actions: Sebastien Erard and the birth of the modern piano

‘Double échappement’ by Erard, invented 1819, patented 1821. Invention of the repetition leverasembly, the full development of Cristofori’s ideas!


8.3.5 Piano und übermenschen: when metallurgy helps.String tension and acoustical powerFrame strength and string tensionsHammer mass and string tension


8.3.6 String structureString vibration

T = 4 µ F2 L2

F =1

2 L

Tµ

We need heavier strings in the bass:

tension par note

clavecin

piano ancien

piano moderne

grave aigu

Tensions par note typiques sur un clavecin, un piano romantique et un piano moderne. Lesdiscontinuités à la basse correspondent aux passages tricordes-bicordes-monocordes. Le graphique est

qualitatif et ne représente pas des instruments précis. THE FOUNDATIONS OF MUSICAL SYSTEMS Jean Louchet 18th December 2012 Hogeschool Gent page 84/104

Elastic limit, Young’s modulus and inharmonicity

f n = n F (1 + B n2)1/2

B =π2 E S K2

T L2for a simple string,, K = d / 4It is necessary to reduce – using materials or structures?K


Hand spinning a piano bass string (usine Pleyel, Saint-Denis, 2008)


English style overspun string loop (Broadwood 1853)


Broadwood concert grand, 1853


A double overspun string.

Calculating a double overspun string requires the rsolution of a 6th degree polynomial equation:

ρ (∆2 − D2) (D + 0.1d) (D2 + 2.2dD + 0.2d2) =(dD2 + Dd2) (D2 + 2.2dD + 0.2d2) + σπ (2D3d + 12D2d2 + 16.8Dd3 + 2.4d4 + 0.8d5D ) (D + 0.1d)


8.3.7 The piano structureSquare piano and ‘harpsichord-shaped’ piano

Erard “en forme de clavecin” vers 1820 (Musée de Bruxelles), Erard carré grand modèle 1850,(Conservatoire de Tours)


Console piano (Jean-Henri Pape), Swedish cabinet piano, giraffenfluegel (Schimmel)

Piano frames: reinforcing barsThermostable frames: William Stodart


William Stodart compensation frame


Broadwood with assembled frame (‘cadre serrurier’, locksmith’s frame) and parallel string s– ca. 1860


Erard’s parralel strung piano with an assembled frame.


Cross strung Pleyel with an assembled frame


Single-piece cast iron frame sacrificing efficiency, sacrificing polyphony for more power.Cross-strung frame: sacrificing polyphony for more power (Babcock patent 1859).

Pleyel’s overstrung cast iron frame - small concert grand, ca. 1900.


Pleyel factory, Saint-Denis 2008: a pre-strung frame.


8.3.8 Repetition“downstriker actions”: Jean-Henry Pape, Nanette StreicherVariants of English actionThedeath of the square piano and the paradox of the upright piano’s survival.


8.3.9 Side inventionsFelt hammers (Pape)Agraffes (Erard)

Agrafes Erard, premier type.

Harmonic bar (Erard) - aka capodastro bar.The tonal pedal (Steinway)Aliquot segments and adjustable roofs


Aliquot strings (Blüthner)

The Fourth Pedal (Blüthner)


8.3.10 The future did not happenYamasteinway, or the compulsory uniformisation

The Steinmaha. A good instrument in spite of its totalitarian option!


Malmsjö et le piano-bananeLe Pleyel double

Duett piano: Pleyel’s double, collection of usine Pleyel, Saint-Denis


Creation of a 1825 Viennese style piano, by Christopher Clarke (octobre 2009)


Between Schantz and Graf... (Christopher Clarke)


THE FOUNDATIONS OF MUSICAL SYSTE...Summary1 Introduction2 Some physics: harmonics, partia...2.1 What is an octave?2.2 Musical objects and the latti...2.3 String vibrating modes

3 Pythagoras and the classical mu...3.1 Pythagoras and the harmonic 3

4 Zarlino: coming closer to Physics5 The genesis of Harmony: Musica ...5.1 Solmization in theory5.2 Solmization in practice

6 Time-frequency diagrams and the...7 The organ and the synthesis of ...7.1 How an organ works7.2 Organ architecture7.3 Register families and registr...Unisons and octaves (flue pipes)Simple mutationsComposite mutations (mixtures)Reeds

7.4 A classical organ: François-H...PositifRécit (C3 - D5)Pédale (F0 - A2)Grand-orgue

8 Stringed keyboards: from the ha...8.1 The clavichord8.2 The harpsichord8.3 Il clavicembalo con piano e f...8.3.1 Cristofori and the hammered...8.3.2 Play closer: the escapement...8.3.3 Wiener/German action: Stein...8.3.4 Repetition actions: Sebasti...8.3.5 Piano und übermenschen: whe...8.3.6 String structure8.3.7 The piano structure8.3.8 Repetition8.3.9 Side inventions8.3.10 The future did not happen

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Jean Louchet - s3.eu-west-3.amazonaws.com€¦ · 13.02.2020 · 4 Zarlino: coming closer to Physics The Pythagorean system focuses on the harmonic 3 and proudly ignores higher harmonics