Meantone temperament - I want my minor 7th interval sounds pure on my keyboard

If you are reading the article, you should have the basic knowledge of Pythagorean tuning, finding new notes with pure fifth, ratio 3:2 (The whole article is based on frequency ratio. If calculate the string length, invert all the ratio, e.g. 2/3) and the concept of temperament. Since the isn’t any easy and clear explanation on the Internet, so I made my own.

The main prepose of quarter meantone temperament is making the major third on the keyboard sound pure.

Starting note is C1, using 4 ascending fifths to get a major third:

= C1- up 5th - G1- up 5th - D2 -up 5th- A2- up 5th -E3

The ratio of pure sound fifth is 3/2=1.5. The ratio of between of C1 to two octaves and a major third (E3) w/ ascending fifth:

= (3/2)4 =5.0625


The frequency of the harmonic series sounds harmonious and pure. The major third in harmonic series is the fifth partial C1 x 5. It is also C1 ascend two octaves above major third (E3).

Thus, the ratio is 5. 5 and 5.0625 is close, the difference between them is called a syntonic comma. If we tune the keyboard with pure fifth, we cannot have a pure third.

In quarter-comma meantone temperament, we scarify the pure fifth (3/2). The aim is after stacking 4 fifths get to a pure third two octaves above (C1 x 5). The stacking of 4 pure fifths ((3/2)^4 =5.0625) will outshoot the mark of 5, so the new fifth must be shorter.

The new fifth is r:

r^4 = 5

r stacking for 4 times= 5

r = 5^1/4 = 1.4953487812 (also the “quarter” name comes from)

However, the limitation of meantone is if we the temperament starting on C, distance keys (opposite of the circle of fifth) are more deviated from the origin, out of tune, and unusable.

Let’s be weird. I want my keyboard to be sound pure at minor 7th. (C1 – Bb7). 10 ascending shorter fifths (<3/2) = 7th partial on the harmonic series

The new fifth ratio= 7^1/10 =1.214814044

This should be called 1/10-comma meantone temperament. LOL. But the perfect sounding minor 7th has a good use in Mixolydian mode. This 7-limit temperament is absent in Western music, but very important to middle East scale. Exotic.

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