1. PH 105 Dr. Cecilia Vogel Lecture 14

2. 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

3. Logarithmic Frequency Measures UnitFactor (equal temp) Equivalent cents 1.000578 semitones 1.0595100 cents whole tones 1.12252 semitones 200 cents octaves 212 semitones 1200 cents

4. 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)

5. Cents Calculation  Interval, I, in cents is related to the  Example, an octave has a ratio of

6. 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

7. 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.

8. Major Diatonic Scale  Western music uses a ____________ instead.  A major diatonic scale has  (the 8 th 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.

9. Example  Key of C (major diatonic scale)  play  CDEFGAB  C to D is a  C # /D b is between  similarly with  E to F is a 

10. Scale on Piano  one octave on keyboard  ignore the gray for now

11. Pitch Standard  Current scales based on standard  A 4 =  historically lower  Handel’s 422.5 is closer to A b  Can base your scale on any frequency,  but current instruments are built to perform well for the standard.

12. 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

13. 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

14. 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

15. 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

16. Tempered Scale  The frequencies of 9 octaves of tempered scale are in table 9.2 notefreq(Hz)intervalratiosimple ratio C4C4 261.63—1 C # /D b 277.18semitone1.06 D293.66whole1.12 D # /E b 311.13minor 3 rd 1.19*6/5 = 1.2 E329.63major 3 rd 1.26*5/4 = 1.25 F349.23fourth1.3354/3 = 1.333 G392.00fifth1.4983/2 = 1.5 C5C5 523.25octave22/1 = 2 * not very good

17. Just Diatonic Scale  Just temperament  based on perfect triads  In triad  major 3 rd is exactly 5/4  minor 3 rd is exactly 6/5  fifth is exactly 3/2

18. 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 D b are not the same  Can only tune triads in a particular key  such as C-major  triads will be mistuned in other scales

19. Just Scale  ratios are perfect in key of C: notefreq(Hz)intervalratiosimple ratio C4C4 261.63—1 C#DbC#Db 272.53 279.07 whole-semi semitone D294.33whole9/8 EbEb 313.96minor 3 rd 6/56/5 = 1.2 E327.04major 3 rd 5/45/4 = 1.25 F348.84fourth4/34/3 = 1.333 G392.44fifth3/23/2 = 1.5 C5C5 523.25octave22/1 = 2

20. 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 D b are not the same

21. Pythagorean Scale  fourths and fifths perfect notefreq(Hz)intervalratiosimple ratio C4C4 261.63—1 C#DbC#Db 279.39 279.07 7 5 ths - 4 oct 3 oct – 5 5 th s D294.33whole9/8 EbEb 310.03minor 3 rd 1.185*6/5=1.2 E331.22major 3 rd 1.27*5/4 = 1.25 F348.84fourth4/34/3 = 1.333 G392.44fifth3/23/2 = 1.5 C5C5 523.25octave22/1 = 2 * even worse

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

23. Summary  equal pitch intervals are equal frequency factors  jnd, cents, semitone, whole tone, octaves  Scales  chromatic, 12 notes, 1 semitone apart  major diatonic, 7 notes, whole & semitone intervals  pentatonic, 5 notes, whole and 1½ tone intervals  Staff  Temperaments of diatonic scale  equal temperament: equal semitones  just temperament: perfect intervals in one key  Pythagorean temperament: perfect 5 th s in any key