ROCHESTER PHY 103 - Lecture Notes - Scales

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Scales Physics of Music PHY103 Image from www.guitargrimoire.com/Diatonic scale W W H W W W H Major scaleHow universal is this scale? 9,000 Year Old Chinese Flutes JUZHONG ZHANG et al. Nature 1999 Excavations at the nearly Neolithic site of Jiahu in Henan Province China have found the earliest completely playable tightly dated multinote musical instruments One of these flutes can be played: check the file K-9KChineseFlutes.ramNeanderthal flute • This bear bone flute, found in Slovenia in 1995, is believed to be about 50,000 years old. Differing hole spacing suggest a scale with whole and half note spacingsOvertones of the stringDiatonic scale • Notes that are octaves apart are considered the same note. x2 in frequency • An octave+ 1fifth is the 3rd harmonic of a string, made by plucking at (1/3) of the string • If f is the frequency of the fundamental then the third harmonic is 3f W W H W W W H • Drop this down an octave (divide by 2) and we find that a fifth should have a frequency of 3/2f Fifth Major scalePythagorus and the circle of fifths Image from Berg + Stork Every time you go up a fifth you multiply by 3/2. Every time you go down an octave you divide by 2.Pythagorean scale (continued) third fourth The major third is up a fifth 4 times and down an octave twice The fourth is up an octave and down a fifthPythagorean scale (continued) sixth The sixth is up a fifth 3 times and down an octave once The minor third is down a fifth three times and up and octave twice. minor thirdPythagorean scale (continued) 1 2 octave fifth fourth third sixth The musical scale related to pure fractions! Mathematical beauty is found in music.The circle doesn’t exactly close What happens if you go up by a fifth 12 times? The circle of fifths leads us to expect that we will get back to the same note. (3/2)12=129.746337890625 An octave is 2 times the base frequency 7 octaves above is 27 times the base frequency. 27=128.0000 However 27≠ (3/2)12Getting all the notes One fifth has to be bad. The Wolf fifth. So the subdominant chords can be played the bad fifth is placed between C# and ACircle of fifths Guides harmonic structure and key signatures for baroque, romantic and much folk music. Keeping chords in tune in one key is sufficient for much of baroque and folk music.Equivalency over the octave We associate notes an octave apart as the same note It is difficult to determine the octave of a note. Some people sing an octave off by mistake. Other animals (birds) with not recognize songs played in a different octavePerfect fifths and thirds What’s so perfect about the perfect fifth? • We started with a harmonic 3f and then took it down an octave to 3f/2 Frequency ratio 1:3/2 Is the Pythagorean third perfect? • Fifth harmonic 5f is 2octaves +a major third • Take this down two octaves and we find a major third should be 5f/4 = 1.25f. Frequency ratio 1:5/4 • However the Pythagorean major third is 1:(34/26)=1.2656What do they sound like? I used the generate tones function in the old Audition Sequence: Sum of 2 sines prefect, interval followed by Pythagorean interval Then sum of 2 sets of 5 harmonics, perfect inteval followed by Pythagorean interval Perfect fifth with frequency ratio 1:3/2 Wolf fifth (between C# and A ) 1:218/311=1.4798 Perfect third 1:5/4 Pythagorean third 1:34/26=1.2656What do the waveforms look like? One of these is a perfect third, the other is the Pythagorean third. Which one is which?Perfect intervals and periodic waveforms • A sum of perfect harmonics adds to a periodic wave --- for sum of frequencies that are integer ratios - Fourier series • However 5/4 (prefect third) is not an integer times 1 --- so why does the sum of two frequencies in the ratio 1:5/4 produce a periodic waveform?Major and minor chords Perfect major third and fifth Frequency ratios 1:5/4:3/2 For the triad Minor third Major third Minor chord: set the major third. Divide the G frequency by 5/4 so that the G and E give a perfect third. 3/2*4/5=6/5 Frequency ratios 1:6/5:3/2 for the triad Major MinorTriads • Major triad with perfect major third and fifth 1:5/4:3/2 The minor third in the triad has ratio 3/2 divided by 5/4 = 6/5 and so is perfect or true. • However other thirds and fifths in the scale will not be true.Just temperament More than one tuning system which use as many perfect ratios as possible for one diatonic scale. Image From Berg and StorkThe Just Scale Two different whole tones T1 = 9/8 = 1.125 T2 = 10/9 = 1.111 One semitone S = 16/15 = 1.067Listening example (from Butler) a) An ascending major scale beginning on C4, equal temperament followed by the chord progression I-IV-V7 in Cmajor and in F# major. b) The same scale but with Pythagorean tuning, followed by the same chord progression c) The same scale and chords, quarter comma meantone temperament (errors distributed in a few tones) d) Just intonationMeantone temperament In general, a meantone is constructed the same way as Pythagorean tuning, as a chain of perfect fifths, but in a meantone, each fifth is narrowed in order to make the other intervals like the major third closer to their ideal or perfect just ratios. Quarter-comma meantone is the most well known type of meantone temperament, and the term meantone temperament is often used to refer to it specifically. Uses exact 5:4s for major thirds, but flattens each of the fifths by a quarter of a syntonic comma where a syntonic comma is equal to the frequency ratio 81:80, or around 21.51 cents.Just intonation • Desirable if only a few instruments are playing • You want that ringing baroque quality • You are only going to play in one or two keys.Equal temperament Desirable properties of this scale: • Sounds okay for all triads for all keys. • Allows instruments in different keys to play together more easily Give up perfect fifths and thirds for a scale that is good for any key Makes rapid key changes possible. Led to the development of a different type of music.12 tones To make a tempered scale we need to fit 12 notes within an octave -- between f and 2f How can we do this? 1) Linearly f(1+n/12)? where n are integers C C# D D# E F F# G G# A A# B C 1 13/12 7/6 5/4 4/3 17/12 3/2 19/12 5/3 7/4 11/6 23/12 2 Problems with this: 1) it sounds bad, 2) Makes no


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