Physics 371 March 14, 2002

• Scales (end)

names of intervals

transposition

the natural scale

the tempered scale

meantone tuning

The Natural Scale

bugle, bach trompet, natural horn etc.

1 2 3 4 5 6 7 8 9 10C C G C E G (Bb) C D E

example: find tones of horn in G

the problem of transposition how many keys do I need to retune? what if I don’t retune? what about adding more keys to the keyboard?

• equal musical intervals - equal frequency ratio• 44on a “multiplicative” number line (=log scale) equal ratios are equidistant• advantage: in graphs below equal intervals have same length

1 1.5 2 3 4 6 8 12

fifthoctave octave octave

JUSTx Triad

x xx

2

Help in visualizing scales:

C D E F G A B C

1

The Tempered Scale

Octave divided into 12 equal intervals

Semitone ratio x: x12 = 2

x = 1.05946….

How to calculate tempered frequency ratios:

Example on blackboard: fifth - 7 semitones major third - 4 semitones

3/27/2001

Tempered Scale - good or bad?

all 12 halftone intervals are identical (100 cent)solves problem of transposition

how good (“exact”) are the tempered intervals? the fifth is good (called “perfect” by musicians) 700 cent but should be 702 cent ratio = 1.498 major third is sharp! 400 cent instead of 386 cent 1.26 instead of 1.25example: 480+605 Hz 480+600 Hzminor third is flat! 300 cent 315 centfourth is good 498 cent instead of 500 cent

Meantone tuning - another interesting compromise

1. Use just major third, divided into two equal intervals

2. This defines a wholetone interval - how many cents?

(answer: 193.2 cents)

3. five whole tones use up 966 cents of the octave

4. Problems remain - see homework # 7

What do musicians not bound to a fixed scale (e.g unfretted strings) actually play?

Measurements on violin soloists (Green 1937)

Minor third mean 296 cents (range 289-301)Major third 406 400-417 (just: 386!!)Fifth 708 692-731 (just: 702!!)

Maybe one likes what one is used to……Evidence: originally, tempered scale sounded bad now just or meantone sounds out of tune

Other cultures - other scales…...

Example: measurements on Burmese Xylophone show thatwholetones are smaller and half tones larger thanWestern scales, tending toward seven almost equalsteps.Similar scales are used in tuning of the Burmese harp.

Demo: scale with seven equal intervals within octave.

Tuning matters mostly in polyphonic music - Only when playing cords is consonance the main issueBecame wide spread only in late Renaissance (after 1600)