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# Aftermath - Temperament

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## Paul Thomas

on 2 February 2016

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#### Transcript of Aftermath - Temperament

MU 5933 STYLES IN MUSIC
SPRING 2016

JANUARY 25
MUSIC IN THE AFTERMATH OF
TEMPERAMENT
The Fluid Piano
Comparing just intonation to equal-temperament
A Cent is a standard unit for measuring the size of an interval between two frequencies. Each half-step (i.e. C to Db) is equal to 100 cents. Thus, a minor third is 300 cents and a perfect fifth is 700 cents and so on.
Pythagoran Tuning
M2: 9/8
M3: 81/64
P4: 4/3
P5: 3/2
M6: 27/16
M7: 243/128
Just Intonation
M2: 9/8
M3: 5/4
P4: 4/3
P5: 3/2
M6: 5/3
M7: 15/8
Developed by Ptolemy in the first century CE, Just Intonation is a tuning system based on ratios of small whole numbers.
12-ET
100
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500
600
700
800
900
1000
1100
1200
Just Intonation
111.73 +11.73
203.91 +3.91
315.64 +15.64
386.31 -13.69
498.04 -1.96
582.51 -17.49
701.96 +1.96
813.69 +13.69
884.36 -15.64
996.09 -3.91
1088.27 -11.73
1200 0
Interval
m2
M2
m3
M3
P4
TT
P5
m6
M6
m7
M7
P8
Pythagorean Tuning
90.22 -9.78
203.91 +3.91
294.13 -5.87
407.82 +7.82
498.04 -1.96
588.27 -11.73
701.96 +1.96
792.18 -7.82
905.87 +5.87
996.09 -3.91
1109.78 +9.78
1200 0
Tuning Comparisons
In Bach's day, equal (or well) temperament referred to the fact that you could play in all 12 keys equally well - not that the pitches were equally spaced. Bach's Well-Tempered Clavier was written to capitalize on the differences in keys, not because the differences did not exist.
The Wolf Tone
Pythagoras (570-495 BCE)
The Monochord
So what's the problem with Pythagorean tuning? The problem is that octaves (2:1) and fifths (3:2) are prime numbers and different prime numbers can never be equal (they are incommensurate).
Because Pythagorean tuning is based on fifths, the resulting fractions can become rather large, and dissonant.
Just Intonation
Pythagoran Tuning
M2: 9/8
M3: 81/64
P4: 4/3
P5: 3/2
M6: 27/16
M7: 243/128
Meantone Temperament
Tuning built on tempered fifths (slightly lowered) and just thirds.
Two pitches that are almost in tune.
While a C major chord sounds great in just intonation, play a D major chord and the result is painful. That's because the P5 between C-G is 701.96 cents while the P5 between D-A is 680.45. This dissonant note was called a wolf tone.
Werckmeister III tuning
C 0.0
C# 90.225
D 192.225
D# 294.135
E 390.225
F 498.045
F# 588.27
G 696.09
G# 792.18
A 888.27
A# 996.09
B 1092.18
C 1200

Well-tempered DOES NOT mean equal-tempered.
Yes, I just said that.
The most famous type of well temperament in Bach's day was by the German organist Andreas Werckmeister (1645-1706).
The perfect fifths found in this tuning are all between 696 and 702 cents (a just P5 is 701.96), allowing for all twelve keys to be used. Most of the M3 are also closer to just tuning as well (386.31)
Equal-temperament
0
100
200
300
400
500
600
700
800
900
1000
1100
1200
Equal temperament is based on the equal division of the octave into 12 semi-tones (increments of the 12th root of 2).
Contrary to our modern sensibilities, many 19th-century musicians where slow to accept equal-temperament, rejecting the sameness equal-temperament bestowed on every key.

The truth is, the keys chosen by composers like Mozart and Beethoven were chosen because of the unique characteristics of the individual keys.
There is evidence of an equal temperament system used in China in the 16th century as well as a similar system devised by Vincenzo Galilei during that same period. It was not until the nineteenth-century that tuning gradually moved more toward equal temperament. One obstacle was that it was one of the most difficult temperaments to tune, as it was not until 1917 that an instrument could be tuned exactly to equal temperament.
For some instrument makers and composers, it was not enough to divide the octave by twelve notes. Renaissance theorist Nicola Vicentino created archicembalos with 31 and 36 keys to the octave.
Other Tunings
Vicentino's Enharmonic Keyboard (1555)
Syntonic Comma
(Comma of Didymus)

4 voice madrigal played on a 24 key to the octave harpsichord
Nicola Vicentino (1511-1572)
Musica prisca caput
JANUARY 25
MUSIC IN THE AFTERMATH OF
THE FORTEPIANO
Clavichord
The earliest stringed keyboard instrument. The clavichord was popular in the late middle ages and was played up until the early 19th century. This instrument was Intended for private use, not an instrument for public performance.
Clavichord - how it works
Harpsichord
The harpsichord was popular from the end of the 15th century up until the 1790's when it was eventually replaced by the fortepiano. Strings were plucked by quills and used as a solo instrument and as a continuo in chamber and orchestral settings.
Harpsichord - how it works
Clavichord how it sounds
Clavichord from c. 1620
Zuckermann clavichord built from a kit
Bartolomeo Cristofori (1655-1731)
Fortepiano
Action!
Cristofori action (1726)
German action (1773)
Modern grand piano action
43'-49'
A closer look
Florentine harpsichord maker who invented the first fortepiano in 1700. Three of his fortepianos dating from the 1720's are still in existence.
The arrival of the fortepiano did not instantly change the face of music. Rather, it was not until German instrument makers got a hold of the instrument that it began to take off.

Gottfried Silberman, Conrad Graf, and Anton Walter, out of Vienna, and many others developed new features to the instrument, this included modifying the action and escapement, inventing the damper pedal, and expanding the range (compass).
J.S. Bach plays a Silbermann fortepiano in 1736 but is unsatisfied with the treble register and its heavy action. In 1747 Bach plays an improved Silbermann fortepiano at the court of Frederick II of Prussia and is impressed, eventually becoming a commercial agent for the instrument maker.
Wolfgang Mozart acquires a fortepiano by Anton Walter of Vienna in 1781 for personal and concert use. Still periodically performs on the harpsichord as late as 1787.
Haydn buys a Shrudi Harpsichord in 1775. Fifteen years later in 1790, Haydn urges his friend and patroness to replace her harpsichord with a fortepiano.
A clear break between when composers stopped writing for the harpsichord began writing exclusively for the fortepiano does not exist. Many keyboard pieces even up to the mid-19th century did not specify piano only.
It took years for all the technological challenges of making a hammered stringed instrument to be "hammered out." The resulting instrument finally gave composers the unprecedented dynamic flexibility they desired, a flexibility the "plucky" harpsichord just could not match.
Six sonates pour le clavecin, op. 1 (1763)
Sonata II, E major
12 Sonate da cimbalo di piano e fort, detto volgamente di martelletti
Beethoven wrote pieces that were beyond the sturdiness and register (compass) of the current fortepianos of his time.

During Beethoven's life, the keyboard expands from 5 octaves to over 6 1/2 octaves. With the Waldstein Sonata (1804), Beethoven wrote for a six-octave instrument.

The current standard of 88 keys (over 7 octaves) was not arrived upon until approximately 1890.
A Peaceful Transition
RCA Mark II Sound Synthesizer (1957)
Garageband on a Macbook
Beethoven Op. 31 sonatas performed on a fortepiano
Lodovico Guistini (Florence, 1732)
Johann Gottfried Eckard
Contrast between clavichord and harpsichord
C.P.E. Bach Wurttemberg Sonata No 1 in A minor
Sonata in D major, K. 311 (1777)
Mozart
Haydn
Andante in F minor (1793)
Waldstein Sonata (1804)
Beethoven
Thanks Cristofori!
MUSIC
IN THE AFTERMATH
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