1. 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

2. Bird Studio Program 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. The Guitar Parts Head –Nut Neck Body –Bridge and Saddle

4. The Head 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. worm gear Between the neck and the head is a piece called the nut, which is grooved to accept the strings

5. The Neck 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

6. The body 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.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.

7. The Bridge 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.

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 “Pythagorean Scale” Frequency ratio F to 1 (1<F<2) String “Fret” ratio Factor to obtain next ratio Do1:1=119/8 Re3/2:2/3=9/4 =9/8 8/9256/243 Mi16/9:3/2 =32/27 27/329/8 Fa Perfect Fourth 2:3/2=4:3 =4/3 3/49/8 Sol Perfect Fifth 3:1=3:2 =3/2 2/39/8 La9/8:2/3 =27/16 16/27256/243 Ti4/3:3/2=8/9 =16/9 9/169/8 Do2:1 = 21/2

12. Pythagorean A Major Scale “Pythagorean Scale” Frequency ratio F to 1 (1<F<2) String “Fret” ratio Factor to obtain next ratio A1= 44019/8 B 8/9256/243 C#32/2727/329/8 D Perfect Fourth 4/33/49/8 E Perfect Fifth 3/22/39/8 F#27/1616/27256/243 G#16/99/169/8 A2=8801/2

13. Just Intonation Scale (Ptolemy) Based on triad 4:5:6 “Ptolemaic Scale” Frequency ratio F to 1 (1<F<2) String “Fret” ratio and complement Factor to obtain next ratio Do4:4=11 09/8 Re3/2:2/3=9/4 =9/8 8/9 1/910/9 Mi5:4=5/44/5 1/516/15 Fa Perfect Fourth 2:3/2=4:3 =4/3 3/4 1/49/8 Sol Perfect Fifth 6:4=3:2 =3/2 2/3 1/310/9 La2*5/6=10/6=5/33/59/8 Ti3/2*5/4=15/88/1516/15 Do2:1 = 21/2

14. A major Scale with Just Intonation(Ptolemy ) “Ptolemaic Scale” Frequency ratio F to 1 (1<F<2) String “Fret” ratio and complement Factor to obtain next ratio A 1=4401 09/8 B 8/9 1/9 C# Major Third 5/44/5 1/516/15 D Perfect Fourth 4/33/4 1/49/8 E Perfect Fifth 3/22/3 1/310/9 F# Major Sixth 5/33/59/8 G# 15/88/1516/15 C Octave 2=8801/2

15. Even Tempered Scale Based on Equal “step” R  1.05946 “Even Tempered Scale” Frequency ratio F to 1 (1<F<2) String “Fret” ratio Factor to obtain next ratio Do 1 1 R Re R2R2 0.890899 R Mi R4R4 0.793701 R Fa Perfect Fourth R 5  1.335 0.749154 R Sol Perfect Fifth R 7  1.498 0.66742 R La R9R9 0.561231 R Ti R 11 0.529732 R DoR 12 = 2 0.5

16. A Major Even Tempered Scale Based on Equal “step” R  1.05946 “Even Tempered Scale” A Frequency ratio F to 1 (1<F<2) String “Fret” ratio Factor to obtain next ratio A = 440 1 = 440 1 R B = 493.88 R2R2 0.890899 R C# = 554.37 R4R4 0.793701 R D = 587.33 R 5  1.335 0.749154 R E = 659.26 R 7  1.498 0.66742 R F# = 739.99 R9R9 0.561231 R G# = 830.61 R 11 0.529732 R A = 880R 12 = 2 = 880 0.5

17. Comparison Just vs Even Tempered Just F ratioJust Fret RatioET F ratioET Fret Ratio 1111 9/88/91.1224620.890899 5/44/51.2599210.793701 4/33/41.334840.749154 3/22/31.4983070.66742 5/33/51.7817970.561231 15/88/151.8877490.529732 21/220.5

18. Scales, Frets, and logarithms FrequencyFret“cent” 110 1.0594630.943874100 1.1224620.890899200 1.1892070.840896300 1.2599210.793701400 1.334840.749154500 1.4142140.707107600 1.4983070.66742700 1.5874010.629961800 1.6817930.594604900 1.7817970.5612311000 1.8877490.5297321100 20.51200 2.1189260.4719371300 2.2449240.4454491400 2.3784140.4204481500 2.5198420.396851600 2.669680.3745771700 2.8284270.3535531800 2.9966140.333711900 3.1748020.314982000 3.3635860.2973022100 3.5635950.2806162200 3.7754970.2648662300 40.252400

19. Frets and scales NoteFret Frequency (1st string) Fret position from saddle on Martin 0-16NY Eopen329.625 F1349.223.597 F#2370.022.272 G3392.021.022 G#4415.319.843 A5440.018.729 A#6466.117.678 B7493.816.685 C8523.215.749 C#9554.314.865 D10587.314.031 D#11622.213.243 E12659.212.5

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 10 Minute Intermission

22. Music Program Selections from Something “Old” Ain’t She Sweet Java Jive Teddy Bears’ Picnic Sunshine / Railroad This Land Johnny B Goode Something “New” Tomorrow I’ll be gone Whisper It in My Ear I Wanna’ Be with You The Rain Song I gotta’ woman Something “Borrowed” Lulu’s Back in Town S’Wonderful Good Luvin’ Be Friends with you Don’t think Twice The Story of Love The Blues Down and Out Jesse Fuller Medley The Dink Song You got me … Trouble in Mind

23. Thanks The End! Refreshments Outside Please- 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