PH 105

Dr. Cecilia VogelLecture 14

OUTLINE units of pitch intervals

cents, semitones, whole tones, octaves staves

scales chromatic, diatonic, pentatonic

consonant intervals octave, fifth, fourth, major third, minor third

temperament equal, just, Pythagorean

Logarithmic Frequency Measures

Unit Factor(equal temp)

Equivalent

cents 1.000578

semitones 1.0595 100 cents

whole tones 1.1225 2 semitones200 cents

octaves 2 12 semitones1200 cents

Cents One cent interval has a ratio of

1.0006 1 cent above 440Hz is Can you tell the difference between

440 Hz and 440.25 Hz? a jnd is a ratio of 1.005

about 8-9 cents 10 cent above 440Hz is Can you tell the difference between

440 Hz and 442.55 Hz? (10 cents)

Cents Calculation Interval, I, in cents is related to the

1200log?

log 2I

log 2 inverse log

1200

IR

Example, an octave has a ratio of

1200log

log 2I R

Semitone An octave is often

each semitone is a factor of multiply 440 Hz (an A) by

you’ll get about 880 Hz Keys on a piano are separated by 12 semitones in order is a

Musical Staff Musical notes are

the x-axis is the y-axis is Fig 8.9

Only the notes in spaces are written in. Notes on lines are letters between. Short lines indicate where sharp/flat

would be , graphically.

Major Diatonic Scale Western music uses a ____________ instead. A major diatonic scale has

(the 8th would be an The intervals are not all semitones

some are

The intervals in major diatonic scale are

Start with any key on the keyboard.

You’ve played a major diatonic scale.

Example Key of C (major diatonic scale) play

CDEFGAB C to D is a

C#/Db is between similarly with

E to F is a

Scale on Piano one octave on keyboard

ignore the gray for now

Pitch Standard Current scales based on standard

A4 = historically lower

Handel’s 422.5 is closer to Ab

Can base your scale on any frequency, but current instruments are built to

perform well for the standard.

Temperament Temperament means

how you tune intervals within your scale.

Equal temperament means all intervals are each semitone is the

a factor of about 1.06 Keys on a piano are usually tuned to

equal temperament, AKA the tempered scale

Consonance An octave ratio is a particularly close

relationship in our hearing. Other simple ratios also tend to be

consonance= Consonant notes have similar Example 440 Hz and 660 Hz

both have 1320, 2640, etc as harmonics

Consonant Intervals See also Table 9.1 Octave interval is simple ratio Fifth is a simple ratio Fourth is a simple ratio Major third is a simple ratio Minor third is a simple ratio

Temperaments Tempered Scale or equal

temperament all intervals are consonant intervals are

Just Scale consonant intervals are perfect in other keys are

Pythagorean Scale fourths and fifths are perfect in major and minor thirds are

Tempered Scale The frequencies of 9 octaves of tempered scale are in table 9.2

note freq(Hz)

interval ratio simple ratio

C4 261.63 — 1

C#/Db 277.18 semitone

1.06

D 293.66 whole 1.12

D#/Eb 311.13 minor 3rd

1.19*

6/5 = 1.2

E 329.63 major 3rd

1.26*

5/4 = 1.25

F 349.23 fourth 1.335

4/3 = 1.333

G 392.00 fifth 1.498

3/2 = 1.5

C5 523.25 octave 2 2/1 = 2

*not very good

Just Diatonic Scale Just temperament

based on perfect triads In triad

major 3rd is exactly 5/4 minor 3rd is exactly 6/5 fifth is exactly 3/2

Just Diatonic Scale To get perfect triads, must sacrifice:

There are two different size whole tones 9/8 (1.125) and 10/9 (1.111).

All semitones are 16/15 (1.067) but two semitones don’t make whole tone. so, for example, C# and Db are not the same

Can only tune triads in a particular key such as C-major triads will be mistuned in other scales

Just Scale ratios are perfect in key of C:

note freq(Hz)

interval ratio simple ratio

C4 261.63 — 1

C#

Db

272.53279.07

whole-semisemitone

D 294.33 whole 9/8

Eb 313.96 minor 3rd 6/5 6/5 = 1.2

E 327.04 major 3rd 5/4 5/4 = 1.25

F 348.84 fourth 4/3 4/3 = 1.333

G 392.44 fifth 3/2 3/2 = 1.5

C5 523.25 octave 2 2/1 = 2

9 15

8 16

16

15

Pythagorean Scale Pythagorean scale based on

A fifth and a fourth make an octave, (3/2)(4/3) = __,

so if you tune a fifth, you’ve tuned a fourth.

To get perfect fifths and fourths in all scales, must sacrifice: major and minor thirds are not good again, C# and Db are not the same

Pythagorean Scale fourths and fifths perfect

note freq(Hz) interval ratio simple ratio

C4 261.63 — 1

C#

Db

279.39279.07

7 5ths- 4 oct3 oct – 5 5ths

D 294.33 whole 9/8

Eb 310.03 minor 3rd 1.185*

6/5=1.2

E 331.22 major 3rd 1.27* 5/4 = 1.25

F 348.84 fourth 4/3 4/3 = 1.333

G 392.44 fifth 3/2 3/2 = 1.5

C5 523.25 octave 2 2/1 = 2

7 4

53

3 1

2 2

22

3

*even worse

Notes on Pythagorean and Just In C-major scale, both have perfect 4th, 5th

Just has good major thirds in C-major but bad in other scales. for example D:A is 1.69, instead of 1.667

Pythagorean has bad major thirds in C-major to have a perfect fifth in another scale. for example E:C is 1.27 not 1.25, but E:A is exactly 1.5

Table 9.3 (jnd about 8.6 cents)

Summaryequal pitch intervals are equal frequency factors

jnd, cents, semitone, whole tone, octavesScales

chromatic, 12 notes, 1 semitone apart major diatonic, 7 notes, whole & semitone intervalspentatonic, 5 notes, whole and 1½ tone intervals

StaffTemperaments of diatonic scale

equal temperament: equal semitonesjust temperament: perfect intervals in one keyPythagorean temperament: perfect 5ths in any key