Natural Harmonics ( guitar, scientific notation ).


(lettuce) #1

NATURAL HARMONICS, FRETS 0 - 24 ( WITH OCTAVE NUMBER, SHARPS )

_When you use your finger to produce a harmonic,
you are splitting the string into halves, thirds, quarters, and so on.
When you play a natural harmonic at the 12th fret for example,
the string is vibrating on both sides of where you are touching.
The only part of the string that is not vibrating is where you were touching the string.
That point is called the ‘node’.

The string vibrates in 2 divided sections.

The whole length of the string vibrates in two separate sections,
divided at the point where the string was touched to produce the harmonic._

NATURAL HARMONICS, GUITAR ( SHARPS ) :

Fret 4 : G#4 C#5 F#5 B5 D#6 G#6

Fret 5 : E4 A4 D5 G5 B5 E6

Fret 7 : B3 E4 A4 D5 F#5 B5

Fret 9 : G#4 C#5 F#5 B5 D#6 G#6

Fret 12 : E3 A3 D4 G4 B4 E5


Fret 16 : G#4 C#5 F#5 B5 D#6 G#6

Fret 19 : B3 E4 A4 D5 F#5 B5

Fret 24 : E4 A4 D5 G5 B5 E6


4th, 9th, & 16th fret harmonics all sound identical ( on a single string ).
They produce a tone 2 octaves higher than the 4th fret
of that string would normally make.

7th & 19th fret harmonics sound identical on a single string.
They produce a tone one octave higher than the 7th fret of that string would normally make.

5th and 24th fret harmonics produce an identical tone 2 octaves higher than the fundamental ( open string ).

The 12th fret harmonic produces a tone 1 octave higher than the open string ( fundamental ).


Playing the harmonic at the 12th fret splits the string in half.

Playing the harmonics at the 7th or 19th frets splits the string into thirds.
7th and 19th fret natural harmonics produce exactly the same note.

Playing the harmonics at frets 5 and 24 splits the string into quarters.

5th and 24th fret harmonics produce the same note.


SINE WAVE :

Only contains its fundamental frequency with no additional harmonics.

SQUARE WAVE :

A square wave is constructed from only odd harmonics.

SAWTOOTH WAVE :

A sawtooth wave contains both even and odd harmonics of the fundamental frequency.

TRIANGLE WAVE :

A triangle wave contains only odd harmonics.


***Waveforms tend not to contain even harmonics if they are vertically symmetrical.
Square and triangle waves are perfectly mirrored above and below the horizontal center line,
so they don’t have any even harmonics.

The sawtooth, on the other hand, is lopsided and does contain even harmonics.
This rule of thumb applies to other waveforms too.**


(James Britt / Neurogami) #2

Related: “Why you can’t tune a guitar – and how to fix it”

http://blog.discmakers.com/2017/08/why-you-cant-tune-a-guitar/

tl;dr: finger pressure bends notes, plus (almost all) guitars use equal temperament.


(joule) #3

TRIANGLE WAVE :

A triangle wave contains only odd harmonics.

In case there is some math/synth nerd that could give some input…

I always thought that triangles were mostly a visual thing, derived from the need to make a mathematically cheap pseudo-sine… I’m very sceptical that absolute turning points (as in the triangle) have any universal relevance beyond human abstraction. Pretty sure it’s an harmonically arbritrary shape, that just looks visually pleasing to us?

Still, it’s interesting that it would only contain odd harmonics, but that could be a property from constructing something linear with harmonics… Also, the amplitude of the triangle harmonics don’t follow any specific pattern, iirc.


(lettuce) #4

I put the information together from a few different magazine articles that I found online.

As far as I understand it the harmonic series is what ‘just temperament’ is based on and ‘equal temprament’ deviates from the harmonic series.

It was interesting to me though that the harmonics on fret 5 are actually EADGBE, rather than ADGCEA ( as ADGCEA are the notes of fret 5 when they are fretted ).

Also, people dont usually use the harmonics from past fret 12 so it was interesting to see that, for example, fret 24 produces exactly the same harmonic as fret 5.

I’m trying to put together a small chart about the harmonics of each guitar string. I want to emulate a guitar a bit in renoise and do some fast arp phrases that run through the harmonics of each string ( or the note that is highlighted by those harmonics anyway ). Might sound pretty cool. E pentatonic minor scale can be played with only the harmonics.

I’m not sure if the information below is correct. Please correct me if you can see some mistakes.

HARMONICS OF EACH STRING :

From here : http://www.michaelnorris.info/theory/harmonicseriescalculator

( From left to right : Partial no, Note name, Frequency ( Hz ), deviance from 12TET in cents )

HARMONICS OF E2 STRING :

Partial 1 : E2, 82.407Hz, 0
Partial 2 : E3 164.814Hz, 0
Partial 3 : B3, 247.221Hz, 2
Partial 4 : E4, 329.628Hz, 0
Partial 5 : Ab4, 412.034Hz, -14
Partial 6 : B4, 494.441Hz, 2
Partial 7 : D5, 576.848Hz, -31
Partial 8 : E5, 659.255Hz, 0
Partial 9 : F#5, 741.662Hz, 4
Partial 10 : Ab5, 824.069Hz, -14
Partial 11 : Bb5, 906.476Hz, -49
Partial 12 : B5, 988.883Hz, 2
Partial 13 : C6, 1071.29Hz, 41
Partial 14 : D6, 1153.696Hz, 0
Partial 15 : Eb6, 1236.103Hz, 0
Partial 16 : E6, 1318.51Hz, 0

HARMONICS OF A2 STRING :

Partial 1 : A2, 110Hz, 0
Partial 2 : A3, 220Hz, 0
Partial 3 : E4, 330Hz, 2
Partial 4 : A4, 440Hz, 0
Partial 5 : C#5, 550Hz, -14
Partial 6 : E5, 660Hz, 2
Partial 7 : G5, 770Hz, -31
Partial 8 : A5, 880Hz, 0
Partial 9 : B5, 990Hz, 4
Partial 10 : C#6, 1100Hz, -14
Partial 11 : Eb6, 1210Hz, -49
Partial 12 : E6, 1320Hz, 2
Partial 13 : F6, 1430Hz, 41
Partial 14 : G6, 1540Hz, -31
Partial 15 : Ab6, 1650Hz, -12
Partial 16 : A6, 1760Hz, 0

HARMONICS OF D3 STRING :

Partial 1 : D3, 146.832Hz, 0
Partial 2 : D4, 293.665Hz, 0
Partial 3 : A4, 440.497Hz, 2
Partial 4 : D5, 587.33Hz, 0
Partial 5 : F#5, 734.162Hz, -14
Partial 6 : A5, 880.994Hz, 2
Partial 7 : C6, 1027.827Hz, -31
Partial 8 : D6, 1174.659Hz, 0
Partial 9 : E6, 1321.491Hz, 4
Partial 10 : F#6, 1468.324Hz, -14
Partial 11 : Ab6, 1615.156Hz, -49
Partial 12 : A6, 1761.989Hz, 2
Partial 13 : Bb6, 1908.821Hz, 41
Partial 14 : C7, 2055.653Hz, -31
Partial 15 : C#7, 2202.486Hz, -12
Partial 16 : D7, 2349.318Hz, 0

HARMONICS OF G3 STRING :

Partial 1 : G3, 195.998Hz, 0
Partial 2 : G4, 391.995Hz, 0
Partial 3 : D5, 587.993Hz, 2
Partial 4 : G5, 783.991Hz, 0
Partial 5 : B5, 979.989Hz, -14
Partial 6 : D6, 1175.986Hz, 2
Partial 7 : F6, 1371.984Hz, -31
Partial 8 : G6, 1567.982Hz, 0
Partial 9 : A6, 1763.979Hz, 4
Partial 10 : B6, 1959.977Hz, -14
Partial 11 : C#7, 2155.975Hz, -49
Partial 12 : D7, 2351.973Hz, 2
Partial 13 : Eb7, 2547.97Hz, 41
Partial 14 : F7, 2743.968Hz, -31
Partial 15 : F#7, 2939.966Hz, -12
Partial 16 : G7, 3135.963Hz, 0

HARMONICS OF B3 STRING :

Partial 1 : B3, 246.942Hz, 0
Partial 2 : B4, 493.883Hz, 0
Partial 3 : F#5, 740.825Hz, 2
Partial 4 : B5, 987.767Hz, 0
Partial 5 : Eb6, 1234.708Hz, -14
Partial 6 : F#6, 1481.65Hz, 2
Partial 7 : A6, 1728.592Hz, -31
Partial 8 : B6, 1975.533Hz, 0
Partial 9 : C#7, 2222.475Hz, 4
Partial 10 : Eb7, 2469.417Hz, -14
Partial 11 : F7, 2716.358Hz, -49
Partial 12 : F#7, 2963.3Hz, 2
Partial 13 : G7, 3210.241Hz, 41
Partial 14 : A7, 3457.183Hz, -31
Partial 15 : Bb7, 3704.125Hz, -12
Partial 16 : B7, 3951.066Hz, 0

HARMONICS OF E4 STRING :

Partial 1 : E4, 329.628Hz, 0
Partial 2 : E5, 659.255Hz, 0
Partial 3 : B5, 988.883Hz, 2
Partial 4 : E6, 1318.51Hz, 0
Partial 5 : Ab6, 1648.138Hz, -14
Partial 6 : B6, 1977.765Hz, 2
Partial 7 : D7, 2307.393Hz, -31
Partial 8 : E7, 2637.02Hz, 0
Partial 9 : F#7, 2966.648Hz, 4
Partial 10 : Ab7, 3296.276Hz, -14
Partial 11 : Bb7, 3625.903Hz, -49
Partial 12 : B7, 3955.531Hz, 2
Partial 13 : C8, 4285.158Hz, 41
Partial 14 : D8, 4614.786Hz, -31
Partial 15 : Eb8, 4944.413Hz, -12
Partial 16 : E8, 5274.041Hz, 0

The reason I’m not too sure about the above infromation is because i have seen different ratios are used and reduced, like the image below :

math-sound.gif


(joule) #5

The image refers to just intonation or the harmonic series.

Your table of guitar fret frequencies refer to the usual equallly tempered scale? Or did I misunderstand it.


(lettuce) #6

I think maybe the ‘ratios of the physics of sound’ image may contain incorrect information because it lists A440 as the ‘Major second’ of A55.

I believe that the ‘harmonics of each string’ tables frequencies have been adjusted to equal temperament as you said as ‘deviance from 12TET in cents’ was included ( is 12TET just? ). Also because the partials of the A2 string follow a pattern to a point, but then 10Hz is added out of nowhere.:

HARMONICS OF A2 STRING :

Partial 1 : A2, 110Hz, 0
Partial 2 : A3, 220Hz, 0
Partial 3 : E4, 330Hz, 2
Partial 4 : A4, 440Hz, 0
Partial 5 : C#5, 550Hz, -14
Partial 6 : E5, 660Hz, 2
Partial 7 : G5, 770Hz, -31
Partial 8 : A5, 880Hz, 0
Partial 9 : B5, 990Hz, 4
Partial 10 : C#6, 1100Hz, -14
Partial 11 : Eb6, 1210Hz, -49
Partial 12 : E6, 1320Hz, 2
Partial 13 : F6, 1430Hz, 41
Partial 14 : G6, 1540Hz, -31
Partial 15 : Ab6, 1650Hz, -12
Partial 16 : A6, 1760Hz, 0


(Zer0 Fly) #7

@joule take a wavecycle that has even harmonics, like saw or pulse, and duplicate it, loop the whole and invert the copy. The wavecycle is now halved frequency, and has purely odd harmonics.

I think tri is nice, because driven by filters the turning points make nice action ringing at the right spot, and for bass the harmonic spectrum is kind of close to sweet spot for a dull tone spread over multiple harmonics.