We generally agree that an octave is the note with a frequency twice the current note. Breaking the octave up into tones and semi-tones takes more thought.
When a note is played on an instrument, we hear the fundamental tone, which we think of as the pitch, and we hear the overtones or harmonics, which give the instrument it’s timber or characteristic tone.
- A computer generated sine wave tone has no overtones, it is a pure tone.
- A flute has very few overtones, and is thought to have a clear tone.
- A guitar string is rich in overtones, which give it is characteristic sound.
Overtones have a very specific relationship to the fundamental frequency.
1st Harmonic is the lowest pitch we hear e.g. the fundamental
2nd Harmonic is an octave higher
3rd Harmonic is a perfect fifth above the octave
4th Harmonic is a two octaves higher than the fundamental
5th Harmonic is a major third above the second octave
6th Harmonic is a perfect fifth above the second octave
7th Harmonic is between a major sixth and a minor seventh above the second octave
8th Harmonic is three octaves higher than the fundamental
9th Harmonic is a major second above the third octave
Pythagoras spent some time trying to reconcile harmonics with semi-tones. Through his efforts, he noticed a phenomenon called the Comma, where tuning by fifths leads to a octaves whose frequency is not a whole multiple of the fundamental.
Over the years, many tuning systems have been derived to try to address or minimize comma.
Just Tuning tries to by using the overtones directly to define the 12 semi-tones in an octave. This results in a uneven relationship between notes within an octave, so a song written for one key cannot be easily transposed to another key.
The modern system adopted by Western Music is 12 Tone-Equal-Temperment (12TET), which tries to normalize the relationship of notes by defining the distance between any two semi-tones by a logarithmic formula, specifically 2^(1/12). So, if you have a note at 60Hz,
- the octave is 60Hz * 2^(12/12) = 120Hz, and
- the fifth is 60Hz * 2^(5/12) = 80.09Hz.
The drawback to 12TET is that several notes have a very small drift from the overtones. The benefit is that for the first time, songs can be transposed with ease by simply raising or lowering every note in the song by the same number of semi-tones, and it sounds the same as it did in the original key.
Transposing is typically done to move a song into the vocal range of a particular singer, which is done quite commonly these days thanks to equal temperament.
Non-Fretted string instruments like the violin, are still tuned by fifths.
Fretted or fixed pitch instruments are tuned in 12TET.
It may trouble you to find out that there is no absolute correct frequency or pitch to use for the note call an A, although there have been a number of conventional standards used over the years. Here are just 3 of the more widely used reference points.
- A432 Used by Mozart, is suppose to correspond with planetary resonance.
- A440 Widely used standard which some believe is the most dissonant.
- A444 Makes the C above Middle C 528Hz which has been found to repair DNA.
In the real world, it’s difficult to be precise anyway. Tuners and Tuning Forks can be off. They may suffer from poor manufacturing, temperature drift, or dirt and oil affecting resonance, etc.
Record players never had perfectly calibrated rotational speeds, so the playback speed would affect pitch. There was even a thing called Wow and Flutter, where the rotational speed would change over time, causing pitch fluctuations.
With tapes, the lack of a perfectly calibrated rotational speed was still an issue, and over time, the tapes would stretch, thus playing back at a lower pitch.
Even Modern Digital Recorders can vary. I’ve imported audio captured by two different recorders at the same time, and the playback of one was a little longer than the other. I ultimately had to shrink one a tiny bit in order to get them in sync.
So in the end, it’s only important to be in tune with the other instruments in your band. Often, musicians tune to the instrument that is hardest to tune, such as an acoustic piano, which can also drift with temperature.
Join the discussion and let us know your thoughts.