# U-M ENGR 100 - Music Signal Processing (7 pages)

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**View the full content.**## Music Signal Processing

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## Music Signal Processing

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- Pages:
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- School:
- University of Michigan
- Course:
- Engr 100 - Intro Engineering

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Eng 100 Music Signal Processing F14 Project 1 Tone synthesizer transcriber 1 J Fessler Abstract This project teaches you the Matlab Graphical User Interface GUI tools that you will use in subsequent projects It also demonstrates some issues you will face during the projects The goals of this project are 1 to write a Matlab program for a tone synthesizer with an on screen mouse activated keypad 2 to write a Matlab program for a tone transcriber that produces a Matlab stem plot similar to musical staff notation from a pure tonal signal The second project will do the same thing for telephone touch tones the third project will do the same thing for synthesized musical instruments This project will help you do both 2 Background In Lab 2 you learned the frequencies used in a pure tonal version of The Victors In this project you will apply that knowledge to build a tone synthesizer and transcriber You need to learn some musical notation and nomenclature and some graphical Matlab commands These topics are covered next 3 3 1 Basics of musical staff notation Results from Lab 2 In Lab 2 you learned that the musical tones used in The Victors had the following frequencies 392 440 494 523 587 659 Hz rounded to the nearest integer You also inferred that there were missing frequencies 415 466 554 622 Hz You also deduced that these notes are related to each other not by common differences but by a common ratio of 1 06 actually 1 0595 In fact 1 0595 is the 12th root of 2 i e 21 12 That factor suggests that there is a basic set of 12 notes that repeat with their frequencies multiplied by integer powers positive and negative of two This is indeed the case This is the modern 12 tone or chromatic musical scale most Western music is based on it but several non Western musical genres use other scales The repetitions of the 12 frequencies are called octaves because for any 8 notes that make up a major scale the highest note has twice the frequency of the lowest note An 8 note major scale consists of two half steps and 5 whole steps so the ratio of the highest to lower note frequency is 21 12 2 22 12 5 212 12 2 So 880 Hz is one octave above 440 Hz and 220 Hz is one octave below 440 Hz We will focus on the octave spanned by the frequencies in The Victors because many musical compositions use this octave The 88 keys of a full sized piano keyboard span more than 7 octaves 21 12 88 1 27 25 Rightmost piano key 4186 Hz is 3 octaves above 523 Hz 4186 23 523 Leftmost piano key 27 5 Hz is 4 octaves below 440 Hz 440 24 27 5 3 2 MIDI frequency notation The information above is enough to create one kind of notation for most Western music The Musical Instrument Digital Interface MIDI notation represents musical frequencies using integers from 0 to 127 based the formula MIDI 69 12 log 2 F 440 1 where F denotes frequency in Hertz In MIDI notation frequencies of The Victors are represented as the following sequence of integers 71 67 69 71 67 69 71 72 69 71 72 69 71 72 74 76 71 71 72 67 69 71 74 71 69 67 MIDI data also includes information about amplitude duration and some other things This notation is well suited for communicating music between digital devices but is not very visually intuitive for humans 3 3 Musical staff notation Musical staff notation the musical notation you usually see is more complicated The reason is that Western music uses the frequencies used in The Victors more commonly than the missing frequencies from Lab 2 that s why they were missing The 7 tones in The Victors are called naturals and are designated with letters A B C D E F G The 5 missing tones in a single octave are called accidentals or sharps and flats and they are designated by and respectively Some sharps and flats are equivalent A B C D D E F G G A These are pronounced A sharp and B flat etc B is below B between A and B and C is above C between C and D A is the same as B etc A 12 note chromatic scale can be written as either of the following lists A A B C C D D E F F G G or A B B C D D E E F G G A These chromatic scales include the naturals and all the accidentals On a standard piano keyboard the naturals are the white keys and the accidentals sharps and flats are the black keys Certain combinations of notes occur together more frequently in music traditionally Why Because they sound more harmonious because the ratios between their frequencies are very close to ratios of small integers For example A is 440 Hz and E is 659 Hz almost exactly a 3 2 ratio Indeed early Western music was based on using these ratios of small integers rather than the equally spaced logarithms of frequencies used today Proposals to use the 21 12 ratio date back to the 16th century but the more widespread change occurred around when J S Bach composed The Well Tempered Clavier in 1722 These two books had compositions in all 12 keys so an equal tempered scale like we use today would sound equally good but perhaps not perfect for all the pieces In contrast other tuning methods based on small integer ratios might sound good for some keys but not for others The Ill Tempered Clavier The following table summarizes the similarities of the integer ratios and the powers of 21 12 Notes A A B C C D D E F F G G FREQ 440 Hz 20 12 21 12 22 12 23 12 24 12 25 12 26 12 27 12 28 12 29 12 210 12 211 12 Hertz Ratio 440 1 1 440 466 2 none 493 9 9 8 495 523 3 6 5 528 554 4 5 4 550 587 3 4 3 586 6 622 3 none 659 3 3 2 660 698 5 8 5 704 740 0 5 3 733 3 784 0 16 9 782 2 830 6 15 8 825 One need not always start with A Starting with G C and A and looking mostly at naturals we have G 1 1 DO A 9 8 RE B 5 4 MI C 4 3 FA D 3 2 SO E 5 3 LA F 15 8 TI G 2 1 DO Natural Notes Only C Major Interval A Minor C D E F G A B C 1 1 21 1 1 1 21 A B C D E F G A DO RE MI FA SO LA TI DO will be familiar to anyone who has seen the movie The Sound of Music 2 A musical staff consists of 5 parallel horizontal lines with …

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