Mathematics, Music, and Mathematics, Music, and the Guitarthe Guitar

Martin FlashmanMartin Flashman

Mathematics, Music, and Mathematics, Music, and the Guitarthe Guitar

Martin FlashmanMartin Flashman

Professor of MathematicsProfessor of MathematicsHumboldt State UniversityHumboldt State University

October 21,2006October 21,2006

Mathematics, Music, and the GuitarMathematics, 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.

The Guitar PartsThe Guitar Parts

• Head– Nut

• Neck

• Body– Bridge and Saddle

The HeadThe 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.

Between the neck and the head is a piece called the nut, which is grooved to accept the strings

The NeckThe 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

The bodyThe bodyThe 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 BridgeThe 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.

Building ScalesBuilding Scales(with Audacity)(with Audacity)

• Choose one tone: – A: frequency = 440 cycles/sec (Hertz)

• Double the frequency: cut the string length by 1/2– A2: frequency = 2* 440 = 880 (Octave)

• Triple the frequency: cut the string length by 1/3Then divide by 2 to bring the frequency between A and A2: double the string length to 2/3– 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

MORE SCALE TONESMORE SCALE TONES

• A=440 D = 586.66 E = 660 A2=880

• Continue to multiply frequencies 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 – Pentatonic Scale:ABDEG#A (Play This)

The round of Perfect Fifth’sThe 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.

A Pythagorean Scale based on 3:2A Pythagorean Scale based on 3:2“Pythagorean Scale”

Frequency ratio F to 1 (1<F<2)

String “Fret” ratio

Factor to obtain next ratio

Do 1:1=1 1 9/8

Re 3/2:2/3=9/4

=9/8

8/9 256/243

Mi 16/9:3/2

=32/27

27/32 9/8

FaPerfect Fourth

2:3/2=4:3

=4/3

3/4 9/8

SolPerfect Fifth

3:2

=3/2

2/3 9/8

La 9/8:2/3

=27/16

16/27 256/243

Ti 4/3:3/2=8/9

=16/9

9/16 9/8

Do 2:1 = 2 1/2

Pythagorean A Major ScalePythagorean A Major Scale“Pythagorean Scale”

Frequency ratio F to 1 (1<F<2)

String “Fret” ratio

Factor to obtain next ratio

A 1 = 440 1 9/8

B 9/8 = 495 8/9 256/243

C# 32/27 = 521.48 27/32 9/8

DPerfect Fourth

4/3 = 586.66 3/4 9/8

EPerfect Fifth

3/2 = 660 2/3 9/8

F# 27/16 = 742.5 16/27 256/243

G# 16/9 = 782.22 9/16 9/8

A 2 = 880 1/2

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

Do 4:4=1 1 0 9/8

Re 3/2:2/3=9/4

=9/8

8/9 1/9 10/9

Mi 5:4=5/4 4/5 1/5 16/15

FaPerfect Fourth

2:3/2=4:3

=4/3

3/4 1/4 9/8

SolPerfect Fifth

6:4=3:2

=3/2

2/3 1/3 10/9

La 2*5/6=10/6=5/3 3/5 2/5 9/8

Ti 3/2*5/4=15/8 8/15 7/15 16/15

Do 2:1 = 2 1/2 1/2

A major Scale with Just Intonation (PtolemyA 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=440 1 0 9/8

B 9/8 8/9 1/9 10/9

C#Major Third

5/4 4/5 1/5 16/15

DPerfect Fourth

4/3 3/4 1/4 9/8

EPerfect Fifth

3/2 2/3 1/3 10/9

F# Major Sixth

5/3 3/5 2/5 9/8

G# 15/8 8/15 7/16 16/15

A Octave 2=880 1/2 1/2

Even Tempered ScaleEven Tempered ScaleBased on Equal “step” RBased on Equal “step” R1.059461.05946

“Even Tempered Scale”

Frequency ratio F to 1 (1<F<2)

String “Fret” ratio

Factor to obtain next ratio

Do 1 1 R

Re R2 0.890899 R

Mi R4 0.793701 R

FaPerfect Fourth

R5 1.335 0.749154 R

SolPerfect Fifth

R7 1.498 0.66742 R

La R9 0.561231 R

Ti R11 0.529732 R

Do R12= 2 0.5

A Major Even Tempered ScaleA Major Even Tempered ScaleBased on Equal “step” RBased on Equal “step” R1.059461.05946

“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 R2

B = 493.88 R2 0.890899 R2

C# = 554.37 R4 0.793701 R

D = 587.33 R5 1.335 0.749154 R2

E = 659.26 R7 1.498 0.66742 R2

F# = 739.99 R9 0.561231 R2

G# = 830.61 R11 0.529732 R

A = 880 R12= 2 = 880 0.5

Comparison Comparison Just vs Even Tempered Just vs Even Tempered

Just F ratio Just Fret Ratio ET F ratio ET Fret Ratio

1 1 1 1

9/8 8/9 1.122462 0.890899

5/4 4/5 1.259921 0.793701

4/3 3/4 1.33484 0.749154

3/2 2/3 1.498307 0.66742

5/3 3/5 1.781797 0.561231

15/8 8/15 1.887749 0.529732

2 1/2 2 0.5

Frets and scalesFrets and scalesNote Fret

Frequency(1st string)

Fret positionfrom saddle on Martin 0-16NY

E open 329.6 25F 1 349.2 23.597

F# 2 370.0 22.272

G 3 392.0 21.022G# 4 415.3 19.843

A 5 440.0 18.729

A# 6 466.1 17.678

B 7 493.8 16.685

C 8 523.2 15.749C# 9 554.3 14.865

D 10 587.3 14.031

D# 11 622.2 13.243

E 12 659.2 12.5

Scales, Frets, and logarithmsScales, Frets, and logarithmsFrequency Fret “cent”

1 1 0

1.059463 0.943874 100

1.122462 0.890899 200

1.189207 0.840896 300

1.259921 0.793701 400

1.33484 0.749154 500

1.414214 0.707107 600

1.498307 0.66742 700

1.587401 0.629961 800

1.681793 0.594604 900

1.781797 0.561231 1000

1.887749 0.529732 1100

2 0.5 1200

2.118926 0.471937 1300

2.244924 0.445449 1400

2.378414 0.420448 1500

2.519842 0.39685 1600

2.66968 0.374577 1700

2.828427 0.353553 1800

2.996614 0.33371 1900

3.174802 0.31498 2000

3.363586 0.297302 2100

3.563595 0.280616 2200

3.775497 0.264866 2300

4 0.25 2400

Multiply with the guitar!Multiply with the guitar!

multiply 749*37410^3 0.749 50010^3 0.374 1700

280126 10^6 0.28 2200

Some Guitar Intonation IssuesSome 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.

ThanksThanksThe End of Lecture!The End of Lecture!

Questions?Questions?