1. Tuning Basics INART 50 Science of Music

2. Three Fundamental Facts Frequency ≠ Pitch (middle A is often 440 Hz, but not necessarily) Any pitch class can be duplicated by multiplying the frequency by 2/1 or 1/2 Musical intervals are associated with ratios: multiply a given frequency by a given ratio, traverse a given interval in “pitch space.” To span the same distance in the opposite direction, flip the ratio.

3. Pythagoras An interval based on 2/1 (or ½) is profound since pitch class is duplicated. The next most profound interval is based on 3/2 It’s consonant It establishes the primacy of the numbers 1, 2, and 3 It sets up a variety of symmetries: 2/1 1/1 diapason (“through all”)

4. Pythagoras An interval based on 2/1 (or ½) is profound since pitch class is duplicated. The next most profound interval is based on 3/2 It’s consonant It establishes the primacy of the numbers 1, 2, and 3 It sets up a variety of symmetries: 1/1 2/1 3/22/3 Go up and down by 3/2

5. Pythagoras An interval based on 2/1 (or ½) is profound since pitch class is duplicated. The next most profound interval is based on 3/2 It’s consonant It establishes the primacy of the numbers 1, 2, and 3 It sets up a variety of symmetries: 1/1 2/1 3/2 2/3 Transpose by 2/1 4/3

6. Pythagoras An interval based on 2/1 (or ½) is profound since pitch class is duplicated. The next most profound interval is based on 3/2 It’s consonant It establishes the primacy of the numbers 1, 2, and 3 It sets up a variety of symmetries: 1/1 2/1 3/2 4/3

7. Pythagoras 1/1 2/1 3/2 4/3 Continue to derive new pitch classes by traversing intervals of 3/2. Transpose all ratios to fall within the range between 1/1 and 2/1. 3/2 x 3/2 = 9/4 transpose by ½: 9/4 x 1/2 = 9/8 9/8

8. Pythagoras 1/1 2/1 3/2 4/3 Continue to derive new pitch classes by traversing intervals of 3/2. Transpose all ratios to fall within the range between 1/1 and 2/1. 9/8 x 3/2 = 27/16 9/8 27/16

9. Pythagoras 1/1 2/1 3/2 4/3 Continue to derive new pitch classes by traversing intervals of 3/2. Transpose all ratios to fall within the range between 1/1 and 2/1. 27/16 x 3/2 = 81/32 9/8 27/16 transpose by ½: 81/32 x 1/2 = 81/64 81/64

10. Pythagoras 1/1 2/1 3/2 4/3 Continue to derive new pitch classes by traversing intervals of 3/2. Transpose all ratios to fall within the range between 1/1 and 2/1. 81/64 x 3/2 = 243/128 9/8 27/16 81/64 243/128

11. Pythagoras 1/12/13/24/3 Continue to derive new pitch classes by traversing intervals of 3/2. Transpose all ratios to fall within the range between 1/1 and 2/1. This is the basis of the major scale: 9/827/1681/64243/128 7 pitch classes 2 step sizes, large and small Problems: The ratios get increasingly awkward, and less consonant. After 12 such successive pitch classes, the result is extremely close to 2/1, but not quite. The symmetry doesn’t hold up.

12. Pythagorean Tuning (based on perfect fifths) 1/19/881/644/33/227/16243/1282/1 9/89/8256/2439/89/89/8256/243

13. Just Intonation As an alternative, consider the pitch classes of the first 6 harmonics With a fundamental f of 100 Hz, the 6 harmonics are: 100 ( f ) 200 ( 2f ) 300 ( 3f ) 400 ( 4f ) 500 ( 5f ) 600 ( 6f )

14. Just Intonation As an alternative, consider the pitch classes of the first 6 harmonics The ratios of these harmonics to the fundamental are: 100 ( f ) 200 ( 2f ) 300 ( 3f ) 400 ( 4f ) 500 ( 5f ) 600 ( 6f ) 1/1 2/1 3/1 4/1 5/1 6/1

15. Just Intonation As an alternative, consider the pitch classes of the first 6 harmonics For simplicity, consider just the ratios: 1/1 2/1 3/1 4/1 5/1 6/1

16. Just Intonation As an alternative, consider the pitch classes of the first 6 harmonics The pitch class ratio of each harmonic may be found by transposing them down by octaves (multiplications of ½) until the ratio lies between 1/1 and 2/1: 1/1 2/1 3/1 4/1 5/1 6/1 1/1 2/1 3/1 x ½ = 3/2 4/1 x ½ = 4/2 = 2/1 5/1 x ½ = 5/2; 6/1 x ½ = 6/2; 5/2 x ½ = 5/4 6/2 x ½ = 6/4 = 3/2 5/4 is very close to the Pythagorean third at 81/64. (5/4 = 80/64) This sounds more consonant, as it is a naturally-occurring harmonic. Thus, the first 6 harmonics are often said to be: fundamental – octave – fifth – octave – third - fifth

17. Just Intonation If the pitch classes of the first 6 harmonics are also found for the fourth (4/3) and fifth (3/2), two new pitch classes appear that are close to Pythagorean ratios, but are simpler and more consonant. 1/1 2/1 3/1 4/1 5/1 6/1 1/1 2/1 3/2 2/1 5/4 3/2 1/1 ? ? ? ? ? ? 4/3 ? ? ? ? ? ? 3/2 To find the pitch classes, do the same procedure as the previous slide, but start with a ratio of 4/3 instead of 1/1. Repeat, starting with a ratio of 3/2. Multiply the starting ratio by values 1-6, then transpose down by octaves (multiply by ½) until the ratio falls between 1/1 and 2/1.

18. Pythagorean Tuning (based on perfect fifths) 1/19/881/644/33/227/16243/1282/1 9/89/8256/2439/89/89/8256/243 Just Tuning (based on natural harmonics) 1/19/85/44/33/25/315/82/1 9/810/916/159/810/99/816/15 Problem: It’s hard to transpose (change the fundamental pitch). e.g., if in the middle of a piece one decides to modulate, considering the pitch 3/2 as do, then a complete scale is not available. The distance from 1/1 to 9/8 is 9/8. But the tone that is the same distance from 3/2, 3/2 x 9/8 = 27/16, is not in the scale.

19. Pythagorean Tuning (based on perfect fifths) 1/19/881/644/33/227/16243/1282/1 9/89/8256/2439/89/89/8256/243 Just Tuning (based on natural harmonics) 1/19/85/44/33/25/315/82/1 9/810/916/159/810/99/816/15 The above just scale was created by the Greek philosopher Ptolemy. It is sometimes called the Ptolemaic just scale. There are a variety of just scales that musicians explore. What all the scales have in common is that they are all based on simple ratios, taken from the pitch classes of natural harmonics. e.g., a scale that had the pitch class of the 17 th harmonic would include a ratio derived by transposing the 17 th harmonic down by octaves until its pitch class is found to be a ratio falling between 1/1 and 2/1: 17/1 x ½ = 17/2;17/2 x ½ = 17/4; 17/4 x ½ = 17/8;17/8 x 1/2 =17/16

20. Equal Temperament As a compromise, since the early 1700s, Western music has used a scale that divides the octave (diapason) into equal perceptual steps. With 12 equal steps per octave, any note may be used as the first note of a scale, and all notes are available. The intervals are not as consonant as just intonation, but the compromise has been considered with the sacrifice as a change of scale can take place in the middle of a piece, without having to stop to retune the instrument. n = 0, 1, 2, … 12 To derive frequencies in twelve tone equal temperament, start with a frequency f, and multiply it by 2 n/12 for n = 0-12.

21. Pythagorean Tuning (based on perfect fifths) 1/19/881/644/33/227/16243/1282/1 9/89/8256/2439/89/89/8256/243 Just Tuning (based on natural harmonics) 1/19/85/44/33/25/315/82/1 9/810/916/159/810/99/816/15 Twelve Tone Equal Temperament (based on perceptually equal subdivisions of the octave) n = 0, 1, 2, … 12

22. The cent measurement In order to compare tuning systems and intervals, the cent increment was created. A cent is 1/100 of a semitone, or 1/1200 of an octave: n = 0, 1, 2, … 1200 Thus, differences among tuning systems can be quantified. An equal tempered fifth, for example, is 700 cents; a 3/2 just fifth is 701.955 cents, indicating a difference of almost 2 cents. An interval ratio, r, may be converted to cents, c, by the equation

23. The cent measurement In order to compare tuning systems and intervals, the cent increment was created. A cent is 1/100 of a semitone, or 1/1200 of an octave: n = 0, 1, 2, … 1200 Thus, differences among tuning systems can be quantified. An equal tempered fifth, for example, is 700 cents; a 3/2 just fifth is 701.955 cents, indicating a difference of almost 2 cents. An interval ratio, r, may be converted to cents, c, by the equation e.g., the interval in cents for the ratio 3/2 is 1200 x log 10 (3/2)/log 10 (2) = 1200 x 0.1761/0.301 = 701.955