Why are the frets on a guitar not evenly spaced?

[rokdog49]

Originally Posted by dwasifar

A reason to drink.

[justonwo]

Originally Posted by robj144

I'm quoting my own post, because there was a typo. It's supposed to be:

f2/f1 = L1/L2

Which must equal 2^(1/12):

L1/L2 = 2^(1/12)

So,

L2 = L1/2^(1/12)

To illustrate. If L is the length of the open string, the following represents where fret n should be placed as a percentage length of the string:

n = 1, Ln = L/2^(1/12) = 94.39% of L
n = 2, Ln = L/2^(2/12) = 89.09% of L
n = 3, Ln = L/2^(3/12) = 84.09% of L
n = 4, Ln = L/2^(4/12) = 79.37% of L
n = 5, Ln = L/2^(5/12) = 74.82% of L
...
n = 10, Ln = L/2^(10/12) = 56.12% of L
n = 11, Ln = L/2^(11/12) = 52.97% of L
n = 12, Ln = L/2^(12/12) = 50.00% of L

This is very cool. Thanks for taking the time to explain it. I wish I understood music theory better and why we have 12 notes between octaves.

[SongwriterFan]

Originally Posted by Arthur Blake

Turns out, that when dividing intervals, 12 just works best. Not 10, 9, or 8, but 12. -- for a musical octave or a watch face.

Whenever somebody insists I use metric, I say that I will do that once they stop using minutes, hours, days, weeks, months, and years. Instead, they should only use megaseconds, gigaseconds, terraseconds, etc.

Maybe then they'll see just how inconvenient the metric system is.

[SongwriterFan]

Originally Posted by justonwo

I wish I understood music theory better and why we have 12 notes between octaves.

See my prior post (#22) on why there are twelve notes in an octave.

It's all about the cycle of fifths. Of course, you could then question why perfect fifths? (The ratio of 3/2). It's a simple ratio, and it just sounds darned good.

But there's a reason for that, too. For more details, read "On the Sensation of Tone" by Helmholtz.

[Jengstrom]

Originally Posted by robj144

To be even more precise, in order for music to be easily transposed, the modern tempered scale was invented. In order to match it closely to the natural scale of 7 notes, 5 notes were introduced (sharps and flats) so that the ratio between each successive notes was fixed. Mathematically, to form the octave, the ratio between two successive notes has to be 2^(1/12) (2^12/12 = 2, which is the octave). This is pretty close to the natural scale. For instance, a perfect fifth in the natural scale is a ratio of 3/2 = 1.50, where in the tempered scale, it's 2^(7/12) = 1.498. So, this is the reason why the tempered scale is a compromise, and notes always sound a bit "off' on any instrument which uses the tempered scale.

Now, from physics, the fundamental frequency of a string (for the same string) is a constant divided by the vibrating length of the string:

f = c/L.

Hence, the ratio of the frequency of f1 and the next note, f2, is:

f2/f1 = L2/L1

Which must equal 2^(1/12):

L2/L1 = 2^(1/12)

So,

L2 = L1 2^(1/12)

Hence, every successive note must 2^(1/12) closer to the nut as the previous fret.

To put it slightly differently,

1. In western music, there are 12 half steps in an octave.
2. To increment a half step up, you increase the frequency by a twelfth root. Robj144's math used 2^(1/12) to express the twelfth root because keyboards (most, anyway) do not have the square root symbol. It would have been easier to see if it did. Increasing the frequency shortens the string.
3. To increment the next half step, you increase the new frequency by a twelfth root of THAT frequency. Since the string was already shorter by one fret, the frequency interval to move a half step is less. The result is that the distance to the next fret is slightly less than the first.
4. Rinse and repeat. You will find that there are 12 frets to move up an octave. Each increment uses less distance to the next fret. Since an octave is exactly 2:1 frequency ratio, the string length at the 12th fret is exactly half of the length of the open string.

John

[Pinetreebob]

Originally Posted by DenverSteve

Expanding on what Larry said. Here you go. Pretty simple math.

Calculating fret distances. If you divide any scale length by the constant 17.817, you will get the distance from the front edge of the nut to the first fret. Here is one example: a scale length of 25.5" (650mm) divided by 17.817 gives 1.4312173" (36.482011mm), which can be rounded down to 1.431" (36.48mm).

Well I was going to paste it here but it was too jumbled. You can read it here:

https://educationcloset.com/2017/05/...-nothing-fret/

That link is very helpful. It explains it without getting too complicated. Thanks!

[charles Tauber]

Originally Posted by Jengstrom

...Increasing the frequency shortens the string.

Practically, shortening the vibrating string increases frequency.

Since an octave is exactly 2:1 frequency ratio, the string length at the 12th fret is exactly half of the length of the open string.

Now you've done it, introducing complexity.

The fret positions are calculated as you describe and are based on an assumption of "ideal string" behaviour. In the real world, strings don't behave as assumed and don't play in tune if one does as you describe: they play sharper in pitch than the ideal string and its fret placement predicts.

To counter that sharpness, the actual vibrating string length is altered so that it is not the same as the theoretical vibrating string length on which the fret placements are based. That alteration - to have the instrument play better in tune - is referred to as "compensation". One common form of compensation is to increase the actual vibrating string length by moving the saddle away from the nut, making the string longer than that used to calculate fret positions. The result is that the distance from the nut to the 12th fret is shorter than the distance from the 12th fret to the saddle: the two segments are not of equal length and the 12th fret is not half of the actual vibrating string length. Close, but not exactly 1/2. A consequence of that is that the 12th fret harmonic, which is actually half of the actual vibrating string length, does not occur exactly at the 12th fret: it is very slightly closer to the sound hole than is the 12th fret. It's so close that most don't notice the difference.

To avoid confusion, many people refer to the theoretical (ideal) string length used to calculate fret positions as the "scale length", though there isn't universal agreement on the meaning of the term. Some manufacturers include the compensation in the stated "scale length", which is not the length used to calculate the fret placement.

[AZLiberty]

Originally Posted by SongwriterFan

Whenever somebody insists I use metric, I say that I will do that once they stop using minutes, hours, days, weeks, months, and years. Instead, they should only use megaseconds, gigaseconds, terraseconds, etc.

Maybe then they'll see just how inconvenient the metric system is.

You do know that "metric time" actually was part of the French Revolution that gave us the meter, gram, and beheadings right?

https://en.wikipedia.org/wiki/Decimal_time

The base unit was the day, not second though.