## Relationship between fundamental and harmonics, in semitones

Chords, scales, harmony, melody, etc.
Hunter
KVRAF
2936 posts since 14 Dec, 2003 from Edinburgh
I'm trying to work out the relationship between the harmonics of a note and the number of semitones that equates to.

Eg
Fundamental = 440Hz
first harmonic = 880Hz = 12 semitones
second harmonic = 1320Hz = just higher than 19 semitones (19 semitones = 1318.51022765148Hz)
third harmonic = 1760Hz = 24 semitones
etc

Is there a calculable relationship here? Basically I want to be able to work out a semitone value that a partial x needs to be detuned to reach either the fundamental or nth harmonic. I'm working in excel so ideally want to develop a formula.

Any input very much appreciated!

BazzaTron
KVRist
401 posts since 29 Mar, 2006 from The Grim North (well, Yorkshire)

Ogg Vorbis
KVRian
1084 posts since 12 Sep, 2008 from Your basement
Hunter wrote:I'm trying to work out the relationship between the harmonics of a note and the number of semitones that equates to.

Eg
Fundamental = 440Hz
first harmonic = 880Hz = 12 semitones
second harmonic = 1320Hz = just higher than 19 semitones (19 semitones = 1318.51022765148Hz)
third harmonic = 1760Hz = 24 semitones
etc

Is there a calculable relationship here? Basically I want to be able to work out a semitone value that a partial x needs to be detuned to reach either the fundamental or nth harmonic. I'm working in excel so ideally want to develop a formula.

Any input very much appreciated!
The overtones do not follow rationed semitones. They get "flatter and flatter" as they ascend. So you're going to have to deal with that unless you're talking about a theoretical vibrating body which is infinitely thin.

LauraMichelJarre
KVRist
437 posts since 1 Sep, 2007 from Oz

PaulSC
KVRian
1185 posts since 13 May, 2004 from SF Bay Area, California
To calculate a pitch difference D in semitones from the frequency ratio R of a given interval, you have to calculate
D = 12*log2(R)
where log2(R) means the base-2 logarithm of R.

So for example the interval from fundamental to fifth harmonic is a ratio of R = 5.
Then D = 12*log2(5) = 27.86313714 semitones
If you like 80s retro sounds, check out my latest tune…

Hunter
KVRAF
2936 posts since 14 Dec, 2003 from Edinburgh
Thanks

PaulSC wrote:To calculate a pitch difference D in semitones from the frequency ratio R of a given interval, you have to calculate
D = 12*log2(R)
where log2(R) means the base-2 logarithm of R.

So for example the interval from fundamental to fifth harmonic is a ratio of R = 5.
Then D = 12*log2(5) = 27.86313714 semitones
I think these are the droids I'm looking for. I knew you'd have an answer Paul! Now to put it into action Taika-Kim
KVRian
1121 posts since 15 May, 2002 from Finland
I'll have to necro this, just to say thanks! I am doing an additive synth, and needed these since I'm using semitone values to define each partial.