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Something Old, Something New, Something Borrowed, and … The Blues

Mathematics, Music, and the Guitar Martin Flashman Visiting Professor of Mathematics Occidental College April 21,2006. Something Old, Something New, Something Borrowed, and … The Blues. Bird Studio Program. Mathematics, Music, and the Guitar General Guitar Overview The Problem of Scales

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Something Old, Something New, Something Borrowed, and … The Blues

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  1. Mathematics, Music, and the GuitarMartin FlashmanVisiting Professor of MathematicsOccidental CollegeApril 21,2006 Something Old, Something New, Something Borrowed, and … The Blues

  2. Bird StudioProgram • Mathematics, Music, and the Guitar • General Guitar Overview • The Problem of Scales • Pythagorean / Ptolemaic Proportional Scales • Even (Well) Tempered Scales • Fretting and Scales on the Guitar • Some Guitar Intonation Problems • Where and how to play a note. • The Bridge and the Saddle.

  3. Head Nut Neck Body Bridge and Saddle The Guitar Parts

  4. The strings pass over the nut and attach to tuning heads, which allow the player to increase or decrease the tension on the strings to tune them. In almost all tuning heads, a tuning knob turns a worm gear that turns a string post. Between the neck and the head is a piece called the nut, which is grooved to accept the strings The Head

  5. The face of the neck, containing the frets, is called the fingerboard. The frets are metal pieces cut into the fingerboard at specific intervals. By pressing a string down onto a fret, you change the length of the string and therefore the tone it produces when it vibrates The Neck

  6. The body of most acoustic guitars has a "waist," or a narrowing. This narrowing happens to make it easy to rest the guitar on your knee. The most important piece of the body is the soundboard. This is the wooden piece mounted on the front of the guitar's body, and its job is to make the guitar's sound loud enough for us to hear. The two widenings are called bouts. The upper bout is where the neck connects, and the lower bout is where the bridge attaches. In the soundboard is a large hole called the sound hole. The body

  7. Attached to the soundboard is a piece called the bridge, which acts as the anchor for one end of the six strings. The bridge has a thin, hard piece embedded in it called the saddle, which is the part that the strings rest against. The Bridge

  8. Building Scales • Choose one tone: • A: frequency = 440 cycles/sec (Hertz) • Double the frequency • A2: frequency = 2* 440 = 880 (Octave) • Triple the primal tone frequency then divide by 2 • E: frequency = 3*440/2 = 1320/2 = 660 • Divide A2 frequency by 3 then double. • D: frequency = 2*880/3 = 4/3* 440 = 586.666

  9. MORE SCALE TONES • A=440 D = 586.66 E = 660 A2=880 • Continue multiply by 3/2, 4/3… • Multiply A by 9/4 then divide by 2 • B: 440*9/4=990… 990/2 = 495 • Multiply A by 16/9 • G#: 440*16/9 = 782.22 • Pythagorean Pentatonic Scale:ABDEG#A (Play This)

  10. The round of Perfect Fifth’s • FCGDAEB F#C#G#D#A# FCGDAEB • This gives a total of 12 distinct “chromatic” tones. • The intervals between these tones in the same octave are roughly the same ratio. • HOWEVER: The scales are not the same if you start with a different tonic.

  11. A Pythagorean Scale based on 3:2

  12. Pythagorean A Major Scale

  13. Just Intonation Scale (Ptolemy)Based on triad 4:5:6

  14. A major Scale with Just Intonation(Ptolemy)

  15. Even Tempered ScaleBased on Equal “step” R1.05946

  16. A Major Even Tempered ScaleBased on Equal “step” R1.05946

  17. Comparison Just vs Even Tempered

  18. Scales, Frets, and logarithms

  19. Frets and scales

  20. Some Guitar Intonation Issues • Where and how to play a note. • At the fret. • Vibrato and Bending. • String qualities- multiple positions. • The Bridge and the Saddle. • Varying string length proportions from bridge to nut. • Added tension: “sharper” on higher frets.

  21. 10 Minute Intermission

  22. Something “Old”Ain’t She SweetJava JiveTeddy Bears’ PicnicSunshine / RailroadThis LandJohnny B Goode Something “New”Tomorrow I’ll be goneWhisper It in My Ear I Wanna’ Be with YouThe Rain SongI gotta’ woman Something “Borrowed”Lulu’s Back in TownS’WonderfulGood Luvin’Be Friends with youDon’t think TwiceThe Story of Love The BluesDown and Out Jesse Fuller Medley The Dink SongYou got me …Trouble in Mind Music ProgramSelections from

  23. ThanksThe End!Refreshments OutsidePlease- No food in Bird Studio

  24. C Major Ptolymaic Scale • 264 Hz - C, do (multiply by 9/8 to get:) • 297 Hz - D, re (multiply by 10/9 to get [5/4]:) • 330 Hz - E, me (multiply by 16/15 to get [4/3]:) • 352 Hz - F, fa (multiply by 9/8 to get [3/2]:) • 396 Hz - G, so (multiply by 10/9 to get [5/3]:) • 440 Hz - A, la (multiply by 9/8 to get [15/8]:) • 495 Hz - B, ti (multiply by 16/15 to get [2]:) • 528 Hz - C, do

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