Chapter 15

Successive Tones: Reverberations, Melodic Relationships,

and Musical Scales

Audibility of DecayingSounds in a Room

The first of the tone we hear is the directly propagated wave. Because of the precedence effect, the

direct wave will combine with the most direct reflections (within 30 to 50 milliseconds) and be perceived as one.

Picture of a Clearly Heard Tone

Attack – heard as one because of Precedence Effect

Decay – similar to the attack

Reverberation Time

The time required for the sound to decay to 1/1000th of the initial SPL

Audibility Time Use a stopwatch to measure how long

the sound is audible after the source is cut off

Agrees well with reverberation time It is constant, independent of frequency,

and unaffected by background noise

Why does Audibility Time Work?

Threshold of hearing temporarily shifted to 60 dB below a loud tone? 60 dB is 1000 times in SPL which then

matches the definition of Reverberation Time

Measurements show that this happens, but only for a few tenths of a second Not long enough to make audibility time

work

Why does Audibility Time Work?

The ear is responding to the rate of change of loudness? Look at example on next slide

Advantages of Audibility Time

Only simple equipment required Many sound level meters can only

measure a decay of 40-50 dB, not the 60 dB required by the definition

Instruments assume uniform decay of the sound, which may not be the case

Device to Study Successive Tones

Tone Generator 2

Tone Generator 1 21

SwitchAmplifier Speaker

Notes on Tone Switcher

Tone generators produce fundamental plus a few harmonics to simulate real instruments

Switching cannot be heard Reverberation time at least ⅓ sec.

Experiment

Start with TG1 on C4

Switch to TG2 and adjust

At certain frequencies the decaying TG1 will form beats with the partials or heterodyne components of TG2 The beats will be most audible when the

amplitudes are equal.

Using Reverberation

These experiments show that we can use reverberation as an aid in performing It is easier to perform in a live room

(shower) Noise can mask the decaying partials

and make pitch recognition more difficult

Conclusions We can set intervals easily for successive

tones (even in dead rooms) so long as the tones are sounded close in time.

Setting intervals for pure sinusoids (no partials) is difficult if the loudness is small enough to avoid exciting room modes.

At high loudness levels there are enough harmonics generated in the room and ear to permit good interval setting.

Intervals set at low loudness with large gaps between the tones tend to be too wide in frequency.

The Beat-Free Chromatic(or Just) Scale

We will use the Tone Switcher to help find intervals that produce beat-free relationships to the fundamental. The fact that the frequency generators

contain harmonics makes this possible Notice that the octave is a doubling of

the frequency and the next octave would be four times the frequency of the fundamental

First Important Relationship

Three times the fundamental less an octave 3f/2 or an interval of 3/2 or a fifth Fundamental will have harmonics that

contain the fifth Five such relationships can be found in

the first octave

Just Intervals (with respect to C4)

Chromatic ScalesListed Interval Interval Computed Cent

Frequency Name Ratio Frequency Difference

(equal-tempered) (beat-free)

C 261.63

E 329.63 3rd 5/4 327.04 14

F 349.23 4th 4/3 348.84 2

G 392.00 5th 3/2 392.45 -2

A 440.00 Major 6th 5/3 436.05 16

C 523.25 octave 2/1 523.26 0

Relationships Among Five Principles

NoteFrequency

(equal-tempered)IntervalRatio

IntervalName

ResultingFrequency

Note

F 349.23 3/2 5th 523.85 C

E 329.63 4/3 4th 439.51 A

G 392.00 4/3 4th 522.67 C

F 349.23 5/4 3rd 436.54 A

E 329.63 6/5 Minor 3rd 395.52 G

A 440.00 6/5 Minor 3rd 528.00 C

Finding the Missing Steps

Notice the B and D are not harmonically related to C

Finding B A fifth (3/2) above E gives 490.56 Hz A third (5/4) above G gives 490.00 Hz Difference is 2 cents – sensibly equal

The Trouble with D

A Fourth (4/3) below G gives 294.34 Hz

A Fifth (3/2) below A gives 290.70 Hz Difference is 22 cents or 1¼%

Sounded together these “D’s” give clear beats

Intervals with B and D

5th

CGDC E F A B

4th

5th

3rd

Filling in the Scale

3rd

3rd

3rd 3rd

4th

Minor 6

G CDC E F A B

Notice that C#, Eb, and Bb come into the scheme, but Ab/G# is another problem.

Putting numbers to the Ab/G# Problem

From at Interval Ratio Giving

E 327.04 Third 5/4 408.80

C 523.26 Third 5/4 418.61

The Problem with F#

3rd

3rd

min3

CDC E F A BG

Other discrepancies exist but these highlight the problem.

Saving the Day

As the speed increases discrepancies in pitch are more difficult to detect.

The sound level is greater at the player’s ear than the audience. He can make small adjustments. He is always better tuned than the audience demands.

Working Toward Equal Temperament

The chromatic (Just) scale uses intervals which are whole number ratios of the frequency. Scales have unequal intervals

E 327.04 F 348.84 1.0666 16/15

B 490.5 C 523.26 1.0667 16/15

but

C# 279.07 D1 290.7 1.0417

F 348.84 F#1 363.38 1.0417

Making the Interval Equal An octave represents a doubling of the

frequency and we recognize 12 intervals in the octave.

Make the interval 1.059463 212

Using equal intervals makes the cents division more meaningful

The following table uses

Breaking Up One IntervalInterval in Cents Frequency Ratio Frequency Note

0 1.00000 261.63 C4

10 1.00579 263.15  

20 1.01162 264.67  

30 1.01748 266.20  

40 1.02337 267.75  

50 1.02930 269.30  

60 1.03526 270.86  

70 1.04126 272.43  

80 1.04729 274.00  

90 1.05336 275.59  

100 1.05946 277.19 D4

ComparisonFrequency

RatioMusical Interval Cents

(Just)Cents

(Equal-Tempered)

1/1 Unison 000 000

2/1 Octave 1200 1200

3/2 Fifth 702 700

4/3 Fourth 498 500

5/3 Major sixth 884 900

5/4 Major third 386 400

6/5 Minor third 316 300

8/5 Minor sixth 814 800

Pitch Discrepancy Groups When pitch discrepancies exist in a

scale, the cent difference from the equal-tempered interval cluster into three groups

Low Group Middle Group High Group

12 cents lowEqual-tempered

frequency12 cents high

Each group has a range of about 7 cents If a player is asked to sharp/flat a tone, (s)he

invariably goes up/down about 10 cents, moving from one group to another.

Complete Scale ComparisonInterval

Ratio to TonicJust Scale

Ratio to TonicEqual Temperament

Unison 1.0000 1.0000

Minor Second 25/24 = 1.0417 1.05946

Major Second 9/8 = 1.1250 1.12246

Minor Third 6/5 = 1.2000 1.18921

Major Third 5/4 = 1.2500 1.25992

Fourth 4/3 = 1.3333 1.33483

Diminished Fifth 45/32 = 1.4063 1.41421

Fifth 3/2 = 1.5000 1.49831

Minor Sixth 8/5 = 1.6000 1.58740

Major Sixth 5/3 = 1.6667 1.68179

Minor Seventh 9/5 = 1.8000 1.78180

Major Seventh 15/8 = 1.8750 1.88775

Octave 2.0000 2.0000

Indian Music Comparisons

Indian sa re ga ma pa dha ni sa

Western do re mi fa sol la ti do

Letter C D E F G A B C

Indian music uses a generalize seven note scale like the do re mi of Western music.

The Reference Raga

The rag is the most important concept of Indian music. The Hindi/Urdu word "rag" is derived

from the Sanskrit "raga" which means "color, or passion".  It is linked to the Sanskrit word "ranj" which means "to color". 

The Alap

An Indian piece will usually open with an alap, notes going up and down the scale to establish position and relationship. They will play around a tone, the tone

evasion becoming very elaborate. It becomes a game between the player

and the listeners. Jazz has similar variations.

Indian Modes

Play Bilawal

Play Kafi

Pitch Variations

In Western music we have similar pitch wanderings (vibrato, for example) that the Indian musician would find strange.

We almost always make abrupt transitions from one note to the next without the slides of Indian music.