In melodic music, tuning is the selection of notes added to the musical pallette that the musician uses to make music.
Every instrument has to be tuned somehow. Simple percussion instruments like cymbals do not have to be tuned as carefully as keyboard percussive instruments like the piano. Many instruments allow the player to choose from continuous set of notes, like slide trombone and violin; however, many modern instruments have discreet tones that they can produce, like the keys of a piano and the frets of a guitar.
With discreet tone instruments, the tones have to be selected in such a way that makes sense. Even the number of tones must be decided.
Western music has developed into a modern standard of twelve equally divided tones per octave. That statement deserves careful clarification: an octave is the musical interval between a reference tone and the tone with exactly (in theory, anyway) double the frequency. For example, if a root tone has a fundamental frequency of 110 Hz, the octave of that is 220 Hz. On a relative scale, an octave is divided into 1200 cents. Cents do not translate directly into Hz, but there is a multiplicative relationship between the two, meaning that an increase in 1200 Hz means exactly 2x the Hz. The western octaves are cut up into notes based on cents, not based on Hz.
The newest international standard is that the tone A4 is defined to be 440 Hz. Because Hz are used to measure the absolute frequency, 440 Hz gives the absolute reference tone, then all other tones can easily be related back to A4 using cents. If A4 is 440 Hz, then +1200 cents of that is 880 Hz. +600 cents is not exactly in the middle (660 Hz), due to the purely relative nature of cents and the absolute nature of Hz. There is a mathematical equation to convert from cents to Hz, but it is not necessary to understand tuning. Despite that, here is the equation to convert from Hz to cents:
cents = 1200 x logarithm base 2 of (frequency 2 (in Hz) divided by frequency 1 (in Hz))
or more compactly
cents = 1200 log2 ( f2 / f1 )
So, when tuning, it is easy to cut the octave into twelve pieces of equal cents, because 1200 divided by 12 is 100. Thus, in almost all western music, these notes are considered "in tune" compared to the correct reference frequency:
A - 0 cents
A# or Bb - 100 cents
B - 200 cents
C - 300 cents
C# or Db - 400 cents
D - 500 cents
D# or Eb - 600 cents
E - 700 cents
F - 800 cents
F# or Gb - 900 cents
G - 1000 cents
G# or Ab - 1100 cents
A (again) - 1200 cents
This might sound very easy. The truth is, though, that the entire above scenario is completely artificial.
The truth is, that there is no A B C, do re mi, sa re ga, or anthing like that dictated by nature. The tones of the musical scales are not universally accepted. Scientists will analyze musical tones as simple ratios of frequencies, and although the simplest ratios are pretty well agreed to sound musical, the ratios needed to perform even the simplest melodies are not universal. Tuning by ear has been shown to vary geographically, with some Persian and Arabic cultures using tones that are totally unfamiliar to European ears, African cultures using tones foreign to both European and Asian cultures, and some Indonesian gamelan players using tunings that are totally different from anything else known.
In the West, using major and minor scales, the most common tones are:
A - 0 cents
Bb - 111.73 cents ("minor second")
B - 203.91 cents ("major second")
C - 315.64 cents ("minor third")
C# - 386.31 cents ("major third")
D - 498.05 cents ("perfect fourth")
E - 701.96 cents ("perfect fifth")
F - 813.69 cents ("minor sixth")
F# - 884.36 cents ("major sixth")
G - 1017.60 cents ("minor seventh")
G# - 1088.27 cents ("major seventh")
These intervals are easy for most European-based cultures to reach a majority consensus, but Indian cultures agree on all but the last few intervals listed.
Notice that the major and minor intervals give only eleven tones, instead of twelve.
Applying classical modal theory, there are two more intervals that may be added: the D#, also called the "augmented fourth; and the Eb, also called the "diminished fifth." Modal theory is well established in western music, yet these two intervals are not, oddly. There are different ways to represent the ratios between a root and either of these two intervals, and you may notice that they are identical in the equally divided octave tuning I mentioned earlier. Based on my own ears, and the typical consensus of information I found available, as well as my own logical reasoning, which I will attack later, I support the following:
D# - 590.22 cents ("augmented fourth")
Eb - 609.78 cents ("diminished fifth")
Now there are thirteen tones. I find it interesteing that just tuning yields either eleven or thirteen intervals, depending on depth of modal thinking, but never twelve.
Chords, however, offer a different way to look at things altogether. Whereas melodic playing relies on modal theory and thus construction of the notes based on the scale, chordal playing relies more on the "harmonic series." Basically, it is just a different way of coming up with the intervals and notes, but many of the notes are the same.
The first portion of the harmonic series generates the octave. The second portion generates the perfect fifth at the same interval as the major and minor scales. The third portion generates the major third and the harmonic seventh (which is different from the major and minor sevenths, perhaps it could be called the "dominant seventh"). The fourth portion of the harmonic series generates the major second, major seventh, and the harmonic 11th and 13th. The reciporicals of these intervals yield the perfect fourth, minor sixth, minor second, and some other less understood intervals.
Generated from the overtones and undertones of the harmonic series:
A - 0 cents
Bb - 111.73 cents
B - 203.91 cents
B# (ut)*- 231.17 cents
Ct (ut)* - 359.47 cents
C# - 386.31 cents
D - 498.05 cents
D# (harm)* - 551.32 cents
Eb (ut)* - 648.68 cents
E - 701.96 cents
F - 813.69 cents
Ft (harm)* - 840.53 cents
Gb(harm)* - 968.83 cents
G (ut)* - 996.09 cents
G# - 1088.27 cents
These should look pretty weird and different for most western musicians, compared to their typical tuning, but the tones from the harmonic series are closely related to the tones used in ancient Babylonian and Persian music, and some of that has carried into modern Arabic and Iranian music.
From this background of three different ways to look at tuning, we should be able to understand alternate tunings better, a=but more importantly, we should understand the limitations and strengths of standard tuning.
From here, please read about:
17-EDO
31-EDO
Scales