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Transcript
PH 105 Dr. Cecilia Vogel Lecture 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 Equivalent (equal temp) cents 1.000578 semitones 1.0595 100 cents whole tones 1.1225 2 semitones 200 cents octaves 2 12 semitones 1200 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 1200 I log 2 I log R R inverse log log 2 1200 Example, an octave has a ratio of 1200 I log? log 2 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 *not very good scale are in table 9.2 note C4 C#/Db D D#/Eb E F G C5 freq(Hz) 261.63 277.18 293.66 311.13 329.63 349.23 392.00 523.25 interval — semitone whole minor 3rd major 3rd fourth fifth octave ratio 1 1.06 1.12 1.19* 1.26* 1.335 1.498 2 simple ratio 6/5 = 1.2 5/4 = 1.25 4/3 = 1.333 3/2 = 1.5 2/1 = 2 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 C4 C# Db D Eb E F G C5 freq(Hz) 261.63 272.53 279.07 294.33 313.96 327.04 348.84 392.44 523.25 interval — whole-semi semitone whole minor 3rd major 3rd fourth fifth octave ratio simple ratio 1 9 15 8 16 16 15 9/8 6/5 5/4 4/3 3/2 2 6/5 = 1.2 5/4 = 1.25 4/3 = 1.333 3/2 = 1.5 2/1 = 2 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 C4 C# Db D Eb E F G C5 freq(Hz) 261.63 279.39 279.07 294.33 310.03 331.22 348.84 392.44 523.25 interval — 7 5ths- 4 oct 3 oct – 5 5ths whole minor 3rd major 3rd fourth fifth octave *even worse ratio 1 7 3 1 2 2 simple ratio 4 5 3 2 2 3 9/8 1.185* 1.27* 4/3 3/2 2 6/5=1.2 5/4 = 1.25 4/3 = 1.333 3/2 = 1.5 2/1 = 2 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 Cmajor 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) 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 5ths in any key