If an Octave is defined by this:
doubling of frequency
Why should the way to move from one key to the next be governed by a different rule (i.e. move along a different curve in a X,Y diagram) than moving about 12 keys,
which is nothig else but applying the rule from key-to-key 12 times?
There is a function which dictates how to move from one key ...
I think there are 2 main reasons to be talking about n-TET scales:
You're interested in finding a scale which matches just intonation more closely than 12-TET, thereby preserving the practical advantages of equal temperament while improving upon 12-TET's out-of-tune-ness.*
You're interested in composing music with an unusual or exotic sound, so you're ...
You should probably use a standard notation such as this image from the Wikipedia page on equal temperament and the "N-TET" scales. Take note of the "- " and " +" markers which indicate shifts from the 12-TET Western scale pitches.
This is another answer trying to help understanding also the question to people who can't cope with ratios and other abstract terms:
Imagine you have a tone of 12 Hz frequency (a string waving 12 times/second).
How must the 12 half steps between the octava (24 Hz) be tuned, so that the differences between all half steps are equal?
The question implies: If ...
Since there are 7 pitches per octave, and you have 7 letter-names, I'd say it's OK to do what you're already doing, because anything else would be less convenient.
A temperament with 7 pitches per octave doesn't distinguish between major and minor intervals. For example, between major thirds and minor thirds -- every third is two steps and every step is 1/7 ...
Our note system is a logarithmic scale for frequency. A logarithmic scale turns equal fractions into equal distances. You can define equal temperament as a constant step size of 1/12 on the log_2 scale of frequency.
Going back to the linear scale, this means that a semitone translates into a factor of 2^(1/12) (the twelfth root of two).
The reason for this ...
Start by considering the equal division of octaves into one part. That is, think about changing pitch by octaves only.
If we start with A1=55 Hz, we have the following pitches:
A1 55 Hz
A2 110 Hz
A3 220 Hz
A4 440 Hz
A5 880 Hz
You can see that when ...
Possibly a simple way to look at it is to look at a guitar neck. An octave there is divided into 12 parts - equal as far as each fret is a semitone away from its neighbour. But looking carefully, it's fairly obvious that each fret isn't the same size. In fact, the eleventh fret is very nearly half the size of the first one, from nut to fret 1. Go further, ...
A simple way is to look at ratios as suggested above. One can divide an interval equally arithmetically such that the length (size, or more technically "measure") of each subinterval is identical. Dividing an interval arithmetically in 12 pieces (I can explain the 12 but it takes more math.) yields, 1=12/12, 13/12, 14/12, 15/12, 16/12, 17/12, 18/12, 19/12, ...
The division of notes has to do with human perception and psychoacoustics. One description of human perception is the Weber-Fechner law, where a human will perceive equal changes in some sensory input, such as sound level or sound pitch, not by absolute level or value difference, but by the ratio of the change. e.g. larger values need a proportionately ...
The intervals between notes are "equal" not in the sense that the difference in Hz between them is the same, but the ratio a between them is the same. Let's say g is one semitone higher than f, then g = a f.
Note Hz Ratio a to previous note, rounded to 3 decimal places
A#4 466.16 1.059 (466.16 / 440.0 = 1.059, and so on down the column)...
What happens if you go down by the same steps:
1 step down : 403.33Hz
2 steps down : 366.67Hz
3 steps down : 330.Hz
11 steps down : 36.67Hz
12 steps down : 0Hz
13 steps down : -36.67Hz
So, using your "equally divided" logic, we are at zero Hz after 12 steps, and the next step beyond that is minus 37 Hz! What does that even mean? But ok, let's ...