Why are there 18 notes listed for chromatic scale tuning

Question

The Wikipedia page for the Chromatic scale opens by saying (emphasis added):

The chromatic scale is a musical scale with twelve pitches, each a semitone above or below another.

However if you scroll down to the sub-section on tuning you will see there are eighteen ratios given.

We can exclude one of the 18, because it is the octave ratio (as noted by hpaulj), however this still leaves 17 ratios.

This discrepancy between the 12 semitones of the chromatic scale and the 17 frequency ratios in the tuning appears exist because a separate tuning is given for the sharp/flat pairs: C# / Db, D# / Eb, etc.

It was my understanding that since these notes in these pairs are the same, so why are different tuning ratios given?

Answer 1

The notes and the sharp/flat pairs you refer to are only the same if you are tuning to 12-tone equal temperament. But they become different pitches if you are tuning to just intonation, Pythagorean intonation or some other temperament. The examples you cite in Wikipedia are there to contrast the distinction between 12-tone equal temperament (where there are only 12 chromatic pitches) on the one hand, and other tuning schemes (where there are more than 12 chromatic pitches, utilizing different intervals) on the other hand.

Modern instruments (since roughly the beginning of the 20th century) almost always use 12-tone equal temperament. But in previous centuries, just intonation and many other schemes were used. There were even keyboard instruments with more than 12 keys/notes per octave in earlier centuries.

Musical temperament and the tuning of scales is a vast topic that comes up in various historical, geographic and cultural contexts. It's a difficult subject that requires doing some elaborate maths and listening very critically to the sonic results in order to be able to appreciate the distinctions.

The short answer to the question is that for a bit more than the last 100 years, almost all musicians in the Western tradition (and almost all contemporary musical instruments and their manufacturers) have "given up" on all methods of tuning and temperament other than 12-tone equal temperament. Other tuning schemes are chiefly used by musicians interested in ancient historical performance practice, or more recent avant-garde experimental music. However, there are still musical traditions outside that of Western civilization (Arabic and Middle-Eastern, Indian, Chinese, African) that continue to use other tuning schemes, scales and temperaments. From the Western standpoint, those other traditions are usually labeled under a large heading called "ethnomusicology".

Wikipedia article on Musical Temperament

Answer 2

Your stated "understanding" is wrong. Those notes are not the same.

Many violinist and singers (etc.) will use one or more of those separate tunings, as the ratio between notes in a given key can be much more harmonically accurate (e.g. not irrational numbers) compared to using just 12-tone equal temperament semitone spaced notes. And to a discerning ear, that sounds better.

The reason is that using 12 equally spaced semitones (using the irrational 12th root of 2.0) in equal temperament was a historic compromise between all the different tunings in all the different keys. A compromise was necessary because big expensive instruments, such as pipe organs, could not be retuned for music in each different key signature, and more than 12 keys on a keyboard per octave to support all those different notes were too many to play (although a few organs with lots of "extra" keys per octave on a huge keyboard were were actually made).

Equal temperament is "close enough" for pianos, etc. But on a non-fretted instrument, such as a violin, many musicians don't have to compromise with "close enough" intervals on some types of music, and prefer a purer (non-irrational) ratio between pitch frequencies.