Teff

Teff is a temperament in which the generator is a small quarter tone of about 47 cents, and the period is half an octave. The period itself functions as 24/17, 17/12, 10/7 and 7/5, tempering out 50/49 and 289/288. One period up and one generator down gives 11/8, which is also 18/13 (hence tempering out the grossma, 144/143). One period and two generators up gives a perfect fifth (3/2), and four of these gives a 5/1, tempering out 81/80 and making this a meantone temperament. The four aforementioned commas, plus 78/77, suffice to define teff in the 17-limit.

See Meantone family #Teff for more technical data.

Teff was named by Mason Green, after the grain from which injera (the bread) is made; this is because teff is closely related to injera temperament. Teff is to injera what mohajira is to meantone; both split the generator in half in order to accommodate the 11-limit. But unlike mohajira, teff also handles the 13- and 17-limits with relative ease and low complexity. This comes at the cost of some 7-limit intervals being rather out of tune as a consequence of tempering out 50:49.

Teff forms 2moses of 24, 26, 50, etc. 26edo and 50edo can both be used to tune teff (the former using the patent val, the latter using 50c). However, 76edo works even better for teff, much as 38edo (which is half of 76edo) is an excellent tuning for injera. Here, the large and small steps of teff[26] are in ratio 3:2. Teff[26] has the structure 2(LLLLLLLLLLLLs).

Using 76edo, the 7th harmonic is quite sharp and the 11th harmonic only a tiny bit so, while the 3rd, 5th, 9th, 13th, and 17th harmonics are all flat by different amounts. One might even consider teff to be a 19-limit temperament if the subminor third (which represents both 7/6 and 13/11) is taken as representing a flat 19/16 as well.

Teff, like injera, is analogous to 12edo in more than one way as it tempers out both 81/80 and 50/49. It necessarily falls on the flat end of the meantone spectrum, which is a change from what 12edo listeners may be accustomed to (although tempering the timbre to reflect this difference should help). But the easy availability of a plethora of higher-limit intervals more than makes up for this. Further, the fact that the period is a half-octave means that straight-fretted teff guitars (necessarily using the augmented-fourths tuning) are a possibility.