Exploration compiled by Dr. Larry Lesser
1. QUICK GUITAR ANATOMY
nut (where neck meets headstock);
bridge (over body/soundboard)
Scale length = Length of OPEN (unfretted) string (between inner edges of nut & bridge)
for my guitar, what is this length (in mm)?
Stringed instruments
- Fretless: viola, cello, or violin, or playing guitar with a slide
- Fretted (diatonic): most mountain dulcimers
- Fretted (chromatic): guitar, banjo, mandolin, fretted bass guitar, ukulele
[Note: We now focus on 12-tone octaves in equal temperament, but there are guitars fretted for just intonation or for more than 12 (e.g., 19) tones per octave ]
Fretting: Pressing string down against the fingerboard, with finger just behind (on nut side) of a fret, and plucking string on the bridge-side of that fret
What does fretting a string do to the pitch(frequency) of the unfretted string?
What does fretting a string do to the length of the string free to vibrate?
Wait — why don’t all 6 unfretted strings (which have equal length) have the same pitch?
Based on Mersenne’s laws, what happens when a guitarist “bends” a fretted note?
2. LOCATION OF GUITAR FRETS
| FRET#
from the nut |
0(open) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| (major)scale note | DO | RE | MI | FA | SO | LA | TI | DO | |||||
| interval formed with open string | unison | m2 | M2 | m3 | M3 | P4 | aug4;
dim5 |
P5 | m6 | M6 | m7 | M7 | octave |
| Distance to bridge as arithmetic sequence? | a | a-d | a-2d | a-3d | a-4d | a-5d | a-6d | a-7d | a-8d | a-9d | a-10d | a-11d | a-12d |
| Distance to bridge as geometric sequence? | a | ar | ar2 | ar3 | ar4 | ar5 | ar6 | ar7 | ar8 | ar9 | ar10 | ar11 | ar12 |
| Measured fret-to-bridge distance (in mm) |
After filling in bottom row, find successive longer-shorter differences — are they constant?
After filling in bottom row, find successive shorter-to-longer ratios in that row, rounded to 2 decimal places — are they constant?
What fret is exactly half the length of the open string? How could you use this to find r ?
Critique guitar-maker “rule of 18” that says to move 1/18 of the remaining string length to the bridge to place the next fret
Extensions:
- what will be the same and what will be different if you did this with a mandolin instead of guitar? Or with a guitar that has a capo on it?
- how could you find the 12 notes of an octave on an UNFRETTED instrument (mark with Post-it flag, blue painting tape or masking tape)?
3. LOCATION OF GUITAR HARMONICS (specifically, natural harmonics on unfretted strings)
| Popular SONG | “Roundabout” (Yes) | “Barracuda” (Heart) |
| Live band | https://www.youtube.com/watch?v=h_uNHS_TPqA | https://www.youtube.com/watch?v=bFpFBLRQ8Ns |
| Close up | https://www.youtube.com/watch?v=2u5_7yg8O_g | https://www.youtube.com/watch?v=RghvrPCzxq0 |
| Fret location of harmonics | 12, and then: 12, 7, 5 | 12 & 5 |
| Claims to fame | #13 hit; in RockBand3 game | #11 hit; in Guitar Hero III game; voted by VH1 in 2012 as 16th best hard rock song of all time; introduced S. Palin at 2008 RNC |
The mth harmonic has m-1 equally-spaced nodes strictly between nut and bridge.
Node n of harmonic m has a distance from the nut that is n/m times the full string length.
n = 1, 2, 3,…, m-1 https://en.wikipedia.org/wiki/File:Moodswingerscale.svg
| Harmonic
m |
Node
n |
n/m =
(dist. from nut)/(full string) |
Full string (in mm) × column 3 | Fret position | Pitch of note relative to fundamental |
| 1 | 0 | 0/1 | 0 | 0 | fundamental |
| 2 | 1 | octave | |||
| 3 | 1 | 5th above octave | |||
| 3 | 2 | 5th above octave | |||
| 4 | 1 | 2 octaves | |||
| 4 | 2 | not found — why? | n/a | n/a | n/a |
| 4 | 3 | 2 octaves |
Example of higher harmonics: all along G string from frets 3 to 12 in U2 hit “New Year’s Day” (e.g., 2:00-2:20 of https://youtu.be/O0IRY_eWuIw?t=121 )
4. MONOCHORD (with ONE movable fret you can play on both sides of):
Using or making a monochord. A monochord can also have multiple strings all tuned to the same note, as here or here.
| ratio of lengths on two sides of the fret | equation to solve to find fret location(s) | location(s) | musical interval when both sides plucked |
| 1:1 | unison (same) | ||
| 2:1 | octave | ||
| 3:2 | perfect 5th | ||
| √2:1 or 1.414:1 | tritone (aug 4th; dim 5th) | ||
| 21/12:1 or 1.059:1 | minor 2nd |
Do ratios of “nicer” numbers sound nicer to you? What might explain your answer?
Non-monochord alternative demo: play an open (unfretted) guitar string and then that same string at a particular fret to recreate (with some approximation) the 2:1, 3:2, 4:3, and sqrt(2):1 ratios
References
Barger & Haehl’s “Guitars, violins and geometric sequences” in April 2007 Mathematics Teaching in the Middle School, 12(8), 462-466.
Haak’s “Using the monochord: A classroom demonstration on the mathematics of musical scales” in March 1982 Mathematics Teacher, 75(3), 238-244.
Lesser’s Mathemusician webpage
Lesser, L. (2004, September). [math of locating harmonic notes on guitar string]. Acoustic Guitar, 15(3), 12.
Matson & Grigoriadou’s “Don’t fret over exponential functions” in Dec 2012/Jan 2013 Mathematics Teacher, 106(5), 384-388.
NCTM (2008; 2025). Illuminations: Exploring Measurement, Sequences, and Curves with Stringed Instruments (https://illuminations.nctm.org/LessonDetail.aspx?ID=U163 or https://www.nctm.org/illuminations-search/?type=Lesson&wst=deb1ac86877adb30a61feda0ebb1107a)
Robertson, W. & Lesser, L. (July 2013). Scientific skateboarding and mathematical music: Edutainment that actively engages middle school students. European Journal of Science and Mathematics Education, 1(2), 60-68. https://www.scimath.net/download/scientific-skateboarding-and-mathematical-music-edutainment-that-actively-engages-middle-school-9388.pdf