Mathematics and Piano Tuning

1
Mathematics and Piano Tuning

• By Renee Thrash, Ashley Spain, and Emily Palma

2
Pythagoras Theory

• Pythagoras discovered 32 ratio
• Numbers are linked to musical intervals and
  harmony.
  
• Discovered the relationship between simple ratios
  and musical intervals
  

3
History of Stuart Piano

• Before 1800 links of strings on a piano were the
  same as a harpsichord.
  
• Wayne Stuart invented clamp to keep vibrations
  vertical.
  
• Robert Stuart was able to demonstrate how clamps
  make piano sound more like singing.
  

4
More on Stuart Piano

• This discovery lead to up right pianos in homes.
• The strings are coupled with the sound board.
• Stuart pianos use a vertical zigzag system

5
Harmonics

• Harmonics is the study of mathematical roots of
  harmony
  
• Pythagoras discovered the octave, the perfect 5th
  and the perfect 4th .
  
• The different pitches that are heard when one
  single note is played are harmonics.
  
• The second harmonic always has twice of the
  frequency of the first, the third harmonic has 3
  times of the frequency of the first, and this
  pattern continues.

6
Harmonic Ratios

• Unison- 11
• Semitone- 1615
• Minor tone-109
• Major tone- 98
• Minor third- 65
• Major third- 54
• Perfect fourth- 43
• Augmented fourth- 4532
• Diminished fifth- 6445
• Perfect fifth-32
• Minor sixth-85
• Major sixth- 53
• Harmonic minor seventh- 74
• Grave minor seventh- 169
• Minor seventh- 95
• Major seventh- 158
• Octave 21

7
Musical Intervals

• The difference in musical pitch between any two
  notes is the musical interval.
  
• In Western music if you play 1,3,5,6,8,9,10,12,
  and 13 it is the major scale
  
• A half step or a semitone is the interval between
  adjacent keys on a piano.
  
• A octave is the musical interval of 8 notes.

8
Theory of Temperament

• The temperament theory is based on the theory
  that musical intervals and consonance are based
  on whole number ratios.
  
• The temperament theory is about whole number
  ratios fitting together
  
• Tempering is a process where musical intervals
  are tuned

9
More on Temperament

• The temperament theory is the basis for piano
  tuning
  
• Temperaments and their mathematics are idealized
  theory
  
• Temperament is a scientific theory

10
Circle of Fifths
11
Information on Circle of Fifths

• The interval of a fifth extends over 3.5 whole
  steps.
  
• Explains the relationships between notes in the
  diatonic scale.
  
• 7 half steps or the distance between the 1st and
  8th keys on a piano is a perfect fifth.
  
• It is most often used to help people identify key
  signatures.