26-TET and mixolydian mode - sound ancient with modern method

 Mixolydian mode is often referred to as a medieval church mode and can be traced to ancient Greece. For composing, to bring out the taste of the mode, the melody should accent the flat-seven note in the mode. Thus, the tunning of the note flat-seven is critical. I have discussed the 53-TET is great for perfect 5th. This 26 TET is great for minor 7th.


Harmonic series again, the just intonation of minor 7th is way flatter than the 12-TET that we normally use, -31 cent, about a quarter semitone. Finding an alternative temperament is needed. In the harmonic series, the 7th harmonics is the minor 7th interval to the fundamental. The 4th and 7th are the interval of minor 7th in the same octave. The ratio of 7:4 sounds harmonious in just intonation, as the ratio is from the harmonic series. There are also other ratios of minor 7th in the harmonic series, e.g. 9:5, 16:9, etc.

Harmonic series in C

Borrowing the same method by Pytheogoen, but we stack in 7:4 minor 7th interval instead. After stacking 7/4 26 times and bringing down 21 octaves, the comma is only off by 10.526432 cents. [(7/4)^26]/[2^21]= 0.9939381422= -10.526432 cents. It is smaller than the Pytheogoen comma. After tempering the comma out in 26-TET, the minor 7th interval is only off by 0.405 cents.

Here is an example of the tuning. The calculation is the spreadsheet in C249:F275.



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