Teff: Difference between revisions

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'''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]].

'''Teff''' is a [[temperament]] in which the [[generator]] is a small quarter tone of about 47 [[cent]]s, 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).

Teff forms [[2mos]]es 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 (~~necesarily~~ using the ~~[https://en.wikipedia.org/wiki/~~Augmented-~~fourths_tuning~~ augmented-fourths tuning]) are a possibility.

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 {{w|Augmented-fourths tuning|augmented-fourths tuning}}) are a possibility.

[[Category:Temperaments]]

[[Category:Teff]]

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-'''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]].
+'''Teff''' is a [[temperament]] in which the [[generator]] is a small quarter tone of about 47 [[cent]]s, 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).
+Teff forms [[2mos]]es 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 (necesarily using the [https://en.wikipedia.org/wiki/Augmented-fourths_tuning augmented-fourths tuning]) are a possibility.
+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 {{w|Augmented-fourths tuning|augmented-fourths tuning}}) are a possibility.
 [[Category:Temperaments]]
 [[Category:Teff]]