Chapter 5

Pitch: The Simplest Musical Implication of Characteristic

Oscillations

Characteristic Frequencies

• Here we will look at some classes of instruments and some of the frequencies they emit.o Metal Barso Wooden Barso Chimeso Bellso Strings

Scientific Process

• This chapter presents experiments with various instrument groups.

• Many times conclusions are not reached.

• Note the differences between the frequencies recorded and the perception of pitch.o A future chapter will explain

Rectangular Bars• Consider a glockenspiel (orchestra bells )

o Freely floating barso 2.5 – 3 octave range starting at C6 o Played by striking bars with a hammer

Frequencies of Longest Bar

P = 1046.5 Hz Q = 2810 Hz R = 3906 Hz S = 5494 Hz

Benade often will label different frequencies with a letter if he has need of the label again.

Frequency Ratios

Observed• P/P = 1.000• Q/P = 2.68• R/P = 3.73• S/P = 5.25

ComputedP/P = 1.000Q/P = 2.756R/P = S/P = 5.404

Simple model, but note that frequency R is not predicted

Differences

• The discrepancy between the model and P/P, Q/P, and S/P is due mostly to the mounting technique and tuning.

• The missing R/P ratio in the model is an oversight of the model.

Wooden Bars - Xylophones

• The damping times are different than metal bars.

• There will be other frequencies in the hardwood bars due to the graininess of the wood.

Small Clock Chimes

Characteristic Frequencies of One Rod

Name Frequency Amplitude

P 5 to 10 Inaudible

Q 180 1

Ra 525 10

Rb 530 6.3

S 1063 22

T 1772 44.7

Observations

• Rod generally sounds either just above F3 (near Q) or just above C5 (near the R’s) - depending on who is listening.

• Each group hears the other tone as well as the high-pitched T, but they differ in their assignment of it. But nobody picks a pitch at S or T.o Look at the amplitude columno Decay times about the same

• We’ll come back to these questions

Q (180) R (525, 530) S (1063) T (1772)

Bells

• Classic study is Lord Rayleigh on the Terling Bells

• System of eight bells of different tones.

• In order to compare we normalize the frequencies so that the lowest frequency on each bell is what the listener calls middle C (261.6 Hz).

Terling Bells

BellNumber

Lowest(P)

2nd (Q) 3rd (R) 4th (S) 5th (T)

1 278 467 620 786 1046

2 286 458 590 832 982

3 267 532 641 764 1071

4 275 512 620 796 1026

5 272 544 622 819 1033

Notes• All of the P tones are above middle C, even

though the listener called them C.• The range of the Q tone is very large

o Could some of the Q’s be actually misidentified C5? But C5 is 523 Hz – not close

• R’s only roughly agree and S and T vary widely

• Changing the amplitude has little effect on the pitch assignment.

Plucked Strings

• String stretched tightly between supports is struck or plucked and allowed to freely oscillate.

• piano, guitar, harpsichord, harp

• Again we normalize so that the lowest frequency is 300 Hz.

Plucked Strings

String Pitch Lowest          

Number Name (P) 2nd (Q) 3rd (R) 4th (S) 5th (T)

1 E4 300.0 600.9 900.2 1200.0 1500.9

2 B3 300.0 599.2 900.0 1200.1 1500.0

3 G3 300.0 602.0 902.8 1204.6 1504.1

4 D3 300.0 600.6 900.0 1204.5 1508.2

5 A2 300.0 595.4 897.0 1198.1 1500.0

6 E2 300.0 603.7 900.0 1201.9 1500.0

Notes

• The frequency ratios are very nearly whole numbers

• Look at the A string

P/P = 1.000Q/P = 1.985R/P = 2.99S/P = 3.994T/P = 5.000

Differences from Whole Number Ratios

• Influenced by guitar, strings, and weather

• Same for piano – differences small in the middle of the keyboard – greater at the ends

Instruments with Integer Harmonics

• Upper frequencies are integer multiples of the fundamental

• Voice, strings, woodwinds, brass

• All are capable of sustained sound

fn = nf1

Integer Multiplesfo

2fo

3fo

4fo

Sum

Harmonics

1st

2nd

Sum

Assigned Pitches

• Our ears assign pitches based on any whole number sequences they can find.o Assigned pitch is the lowest frequency that gives

harmonics

• Some harmonics may be missing, but the pattern is recognizedo Tone knob on a stereoo Perfect tone suppressiono Cheap radio

Chime Pitches

• Earlier we saw that experienced people disagree on the pitch of clock chimes. Some got F3 and others C5

Terling Bell #1

• C4 has a fundamental of 261.5 Hz and D5#has a fundamental of 622.25. Look at the first several harmonics of C4

C4 2*C4 D5# 3*C4 4*C4 2*D5#

C4 - D5# 261.5 523 622.25 784.5 1046 1244.5

Terling Bell #1 278 467 620 786 1046  

Bell Harmonics

Find the Hidden Tune

  1st Tone 2nd Tone 3rd Tone

1st Note 392 1176 15682nd Note 1176 1568 1960

3rd Note 880 1320 22004th Note 1482 1976 none5th Note 392 784 19606th Note 494 1482 24707th Note 440 1760 22008th Note 1172 1465 none

  1st Tone 2nd Tone 3rd Tone Harm. Harm. Harm. Fund. Note

1st Note 392 1176 1568 1 3 4 392 G4

2nd Note 1176 1568 1960 3 4 5 392 G4

3rd Note 880 1320 2200 2 3 5 440 A4

4th Note 1482 1976 None 3 4   494 B4

5th Note 392 784 1960 1 2 5 392 G4

6th Note 494 1482 2470 1 3 5 494 B4

7th Note 440 1760 2200 1 4 5 440 A4

8th Note 1172 1465 None 4 5   293 D4

Review

Characteristics of Impulsively produced sounds:

•Made up of damped sinusoids

•Frequencies characteristic of the object

Classes of Instruments

• Sustained Toneso Voiceo Bowed Instrumentso Orchestral Wind instruments

• Ringing Tones that die awayo Bells o Strings that are plucked or struck

String pulled to one side and released (harpsichord, guitar, harp)

String struck by a hammer (piano, cimbalom, clavichord)

Sequence of events for stringed striking or plucking

String vibrates drives bridge drives soundboard drives air in the room

drives ear