41 equal temperament: Difference between revisions
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In music, '''41 equal temperament''', abbreviated 41- |
In music, '''41 equal temperament''', abbreviated 41-TET, 41-[[equal division of the octave|EDO]], or 41-ET, is the [[Musical temperament|tempered]] scale derived by dividing the octave into 41 equally sized steps (equal frequency ratios). {{audio|41-tet scale on C.mid|Play}} Each step represents a frequency ratio of 2<sup>1/41</sup>, or 29.27 [[cent (music)|cents]] ({{Audio|1 step in 41-et on C.mid|Play}}), an interval close in size to the [[septimal comma]]. 41-ET can be seen as a tuning of the [[schismatic temperament|schismatic]],<ref>[http://x31eq.com/schismic.htm "Schismic Temperaments "], ''Intonation Information''.</ref> [[magic temperament|magic]] and [[miracle temperament|miracle]]<ref>[http://x31eq.com/decimal_lattice.htm "Lattices with Decimal Notation"], ''Intonation Information''.</ref> temperaments. It is the second smallest equal temperament, after [[29 equal temperament|29-ET]], whose [[perfect fifth]] is closer to [[just intonation]] than that of [[12 equal temperament#Twelve-tone equal temperament|12-ET]]. In other words, <math>2^{24/41} \approx 1.50042</math> is a better approximation to the ratio <math>3/2 = 1.5</math> than either <math>2^{17/29} \approx 1.50129</math> or <math>2^{7/12} \approx 1.49831</math>. |
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==History and use== |
==History and use== |
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Although 41-ET has not seen as wide use as other temperaments such as [[19 equal temperament|19-ET]] or [[31 equal temperament|31-ET]] {{Citation needed|date=April 2008}}, pianist and engineer [[Paul von Janko]] built a piano using this tuning, which is on display at the [[Gemeentemuseum Den Haag|Gemeentemuseum]] in [[The Hague]].<ref>[ |
Although 41-ET has not seen as wide use as other temperaments such as [[19 equal temperament|19-ET]] or [[31 equal temperament|31-ET]] {{Citation needed|date=April 2008}}, pianist and engineer [[Paul von Janko]] built a piano using this tuning, which is on display at the [[Gemeentemuseum Den Haag|Gemeentemuseum]] in [[The Hague]].<ref>[https://www.jstor.org/stable/932181] Dirk de Klerk "Equal Temperament", ''Acta Musicologica'', Vol. 51, Fasc. 1 (Jan. - Jun., 1979), pp. 140-150</ref> 41-ET can also be seen as an octave-based approximation of the [[Bohlen–Pierce scale]]. |
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41-ET guitars have been built, notably by [https://www.yossitamim.com/ Yossi Tamim]. The frets on such guitars are very tightly spaced. To make a more playable 41-ET guitar, an approach called [http://tallkite.com/misc_files/The%20Kite%20Tuning.pdf "The Kite Tuning"] omits every-other fret (in other words, 41 frets per two octaves or 20.5 frets per octave) while tuning adjacent strings to an odd number of steps of 41. <ref>[https://en.xen.wiki/w/The_Kite_Guitar "The Kite Guitar "], ''Xenharmonic Wiki''.</ref> Thus, any two adjacent strings together contain all the pitch classes of the full 41-ET system. The Kite Guitar's main tuning uses 13 steps of 41-ET (which approximates a 5/4 ratio) between strings. With that tuning, all simple ratios of [[Limit_(music)#Odd_limit|odd limit]] 9 or less are available at spans at most only 4 frets. |
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41-ET is also a subset of 205-ET, for which the keyboard layout of the |
41-ET is also a subset of 205-ET, for which the keyboard layout of the |
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|align=center|[[ |
|align=center|[[Octave]] |
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|align=center|41 |
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|align=center|1200 |
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|align=center|2:1 |
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|align=center|1200 |
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|align=center|0 |
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|align=center|[[Harmonic seventh]] |
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|align=center|33 |
|align=center|33 |
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|align=center|965.85 |
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|align=center|−2.97 |
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|align=center|[[ |
|align=center|[[Perfect fifth]] |
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|align=center|24 |
|align=center|24 |
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|align=center|702.44 |
|align=center|702.44 |
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|align=center|+0.48 |
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|align=center| |
|align=center|Grave fifth |
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|align=center|23 |
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|align=center|673.17 |
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|align=center|262144:177147 |
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|align=center|678.49 |
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|align=center| |
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|align=center|−5.32 |
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|align=center|[[Septimal tritone]] |
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|align=center|20 |
|align=center|20 |
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|align=center|585.37 |
|align=center|585.37 |
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|align=center| |
|align=center|Eleventh harmonic |
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|align=center|19 |
|align=center|19 |
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|align=center|556.10 |
|align=center|556.10 |
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|align=center bgcolor="#D4D4D4"|15:11 |
|align=center bgcolor="#D4D4D4"|15:11 Wide fourth |
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|align=center bgcolor="#D4D4D4"|18 |
|align=center bgcolor="#D4D4D4"|18 |
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|align=center bgcolor="#D4D4D4"|526.83 |
|align=center bgcolor="#D4D4D4"|526.83 |
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|align=center bgcolor="#D4D4D4"|−10.12 |
|align=center bgcolor="#D4D4D4"|−10.12 |
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|align=center|27:20 |
|align=center|27:20 Wide fourth |
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|align=center|18 |
|align=center|18 |
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|align=center|526.83 |
|align=center|526.83 |
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|align=center|+7.28 |
|align=center|+7.28 |
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|align=center|[[ |
|align=center|[[Perfect fourth]] |
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|align=center|17 |
|align=center|17 |
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|align=center|497.56 |
|align=center|497.56 |
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|align=center|−0.48 |
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|align=center | |
|align=center |Septimal narrow fourth |
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|align=center |16 |
|align=center |16 |
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|align=center |468.29 |
|align=center |468.29 |
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|align=center |−2.48 |
|align=center |−2.48 |
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|align=center|[[ |
|align=center|[[Septimal major third|Septimal (super)major third]] |
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|align=center|15 |
|align=center|15 |
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|align=center|439.02 |
|align=center|439.02 |
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|align=center bgcolor="#D4D4D4"| |
|align=center bgcolor="#D4D4D4"|Undecimal major third |
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|align=center bgcolor="#D4D4D4"|14 |
|align=center bgcolor="#D4D4D4"|14 |
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|align=center bgcolor="#D4D4D4"|409.76 |
|align=center bgcolor="#D4D4D4"|409.76 |
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|align=center| |
|align=center|Classic major third |
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|align=center|13 |
|align=center|13 |
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|align=center|380.49 |
|align=center|380.49 |
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|align=center|−5.83 |
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|align=center bgcolor="#D4D4D4"| |
|align=center bgcolor="#D4D4D4"|Tridecimal neutral third, thirteenth subharmonic |
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|align=center bgcolor="#D4D4D4"|12 |
|align=center bgcolor="#D4D4D4"|12 |
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|align=center bgcolor="#D4D4D4"|351.22 |
|align=center bgcolor="#D4D4D4"|351.22 |
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|align=center bgcolor="#D4D4D4"|−8.25 |
|align=center bgcolor="#D4D4D4"|−8.25 |
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|align=center| |
|align=center|Undecimal [[neutral third]] |
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|align=center|12 |
|align=center|12 |
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|align=center|351.22 |
|align=center|351.22 |
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|align=center|+3.81 |
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|align=center|[[minor third]] |
|align=center|Classic [[minor third]] |
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|align=center|11 |
|align=center|11 |
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|align=center|321.95 |
|align=center|321.95 |
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|align=center|−1.45 |
|align=center|−1.45 |
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|align=center| |
|align=center|Tridecimal minor third |
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|align=center|10 |
|align=center|10 |
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|align=center|292.68 |
|align=center|292.68 |
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|align=center|+3.47 |
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|align=center|[[ |
|align=center|[[Septimal minor third|Septimal (sub)minor third]] |
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|align=center|9 |
|align=center|9 |
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|align=center|263.41 |
|align=center|263.41 |
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|align=center| |
|align=center bgcolor="#D4D4D4"|Diminished third |
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|align=center|8 |
|align=center bgcolor="#D4D4D4"|8 |
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|align=center|234.15 |
|align=center bgcolor="#D4D4D4"|234.15 |
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|align=center|{{Audio|8 steps in 41-et on C.mid|Play}} |
|align=center bgcolor="#D4D4D4"|{{Audio|8 steps in 41-et on C.mid|Play}} |
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|align=center|256:225 |
|align=center bgcolor="#D4D4D4"|256:225 |
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|align=center|223.46 |
|align=center bgcolor="#D4D4D4"|223.46 |
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|align=center|{{Audio|Just diminished third on C.mid|Play}} |
|align=center bgcolor="#D4D4D4"|{{Audio|Just diminished third on C.mid|Play}} |
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|align=center|+10.68 |
|align=center bgcolor="#D4D4D4"|+10.68 |
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|align=center|[[ |
|align=center|[[Whole tone]], [[major tone]] |
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|align=center|7 |
|align=center|7 |
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|align=center|204.88 |
|align=center|204.88 |
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|align=center| |
|align=center|Whole tone, [[minor tone]] |
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|align=center|6 |
|align=center|6 |
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|align=center|175.61 |
|align=center|175.61 |
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|align=center|−6.79 |
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|align=center| |
|align=center|Lesser undecimal [[neutral second]] |
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|align=center|5 |
|align=center|5 |
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|align=center|146.34 |
|align=center|146.34 |
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|align=center|−4.30 |
|align=center|−4.30 |
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|align=center|[[ |
|align=center|[[Septimal diatonic semitone]] |
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|align=center|4 |
|align=center|4 |
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|align=center|117.07 |
|align=center|117.07 |
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|align=center|2187:2048 |
|align=center|2187:2048 |
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|align=center|113.69 |
|align=center|113.69 |
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|align=center| |
|align=center|{{Audio|Pythagorean apotome on C.mid|Play}} |
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|align=center|+3.39 |
|align=center|+3.39 |
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|align=center|[[diatonic semitone]] |
|align=center|Classic [[diatonic semitone]] |
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|align=center|4 |
|align=center|4 |
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|align=center|117.07 |
|align=center|117.07 |
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|align=center|16:15 |
|align=center|16:15 |
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|align=center|111.73 |
|align=center|111.73 |
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|align=center| |
|align=center|{{Audio|Just diatonic semitone on C.mid|Play}} |
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|align=center|+5.34 |
|align=center|+5.34 |
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|align=center|−2.42 |
|align=center|−2.42 |
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|align=center|20:19 |
|align=center|20:19 Wide semitone |
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|align=center|3 |
|align=center|3 |
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|align=center|87.80 |
|align=center|87.80 |
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|align=center|20:19 |
|align=center|20:19 |
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|align=center|88.80 |
|align=center|88.80 |
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|align=center| |
|align=center|{{Audio|Novendecimal augmented unison on C.mid|Play}} |
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|align=center|−1.00 |
|align=center|−1.00 |
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|align=center|[[ |
|align=center|[[Septimal chromatic semitone]] |
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|align=center|3 |
|align=center|3 |
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|align=center|87.80 |
|align=center|87.80 |
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|align=center|+3.34 |
|align=center|+3.34 |
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|align=center bgcolor="#D4D4D4"|[[chromatic semitone]] |
|align=center bgcolor="#D4D4D4"|Classic [[chromatic semitone]] |
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|align=center bgcolor="#D4D4D4"|2 |
|align=center bgcolor="#D4D4D4"|2 |
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|align=center bgcolor="#D4D4D4"|58.54 |
|align=center bgcolor="#D4D4D4"|58.54 |
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|align=center bgcolor="#D4D4D4"|−12.14 |
|align=center bgcolor="#D4D4D4"|−12.14 |
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|- |
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|align=center|28:27 |
|align=center|28:27 Wide semitone |
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|align=center|2 |
|align=center|2 |
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|align=center|58.54 |
|align=center|58.54 |
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|align=center|−4.42 |
|align=center|−4.42 |
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|align=center|[[ |
|align=center|[[Septimal comma]] |
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|align=center|1 |
|align=center|1 |
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|align=center|29.27 |
|align=center|29.27 |
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As the table above shows, the 41-ET both distinguishes between and closely matches all intervals involving the ratios in the [[harmonic series (music)|harmonic series]] up to and including the 10th overtone. This includes the distinction between the [[major tone]] and [[minor tone]] (thus 41-ET is not a [[meantone]] tuning). |
As the table above shows, the 41-ET both distinguishes between and closely matches all intervals involving the ratios in the [[harmonic series (music)|harmonic series]] up to and including the 10th overtone. This includes the distinction between the [[major tone]] and [[minor tone]] (thus 41-ET is not a [[meantone]] tuning). These close fits make 41-ET a good approximation for 5-, 7- and 9-[[limit (music)|limit]] music. |
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41-ET also closely matches a number of other intervals involving higher harmonics. It distinguishes between and closely matches all intervals involving up through the 12th overtones, with the exception of the greater undecimal [[neutral second]] (11:10). Although not as accurate, it can be considered a full [[limit (music)|15-limit]] tuning as well. |
41-ET also closely matches a number of other intervals involving higher harmonics. It distinguishes between and closely matches all intervals involving up through the 12th overtones, with the exception of the greater undecimal [[neutral second]] (11:10). Although not as accurate, it can be considered a full [[limit (music)|15-limit]] tuning as well. |
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==Tempering== |
===Tempering=== |
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Intervals not tempered out by 41-ET include the [[diesis]] (128:125), [[septimal diesis]] (49:48), [[septimal sixth-tone]] (50:49), [[septimal comma]] (64:63), and the [[syntonic comma]] (81:80). |
Intervals not tempered out by 41-ET include the [[lesser diesis]] (128:125), [[septimal diesis]] (49:48), [[septimal sixth-tone]] (50:49), [[septimal comma]] (64:63), and the [[syntonic comma]] (81:80). |
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41-ET tempers out |
41-ET tempers out 100:99, which is the difference between the greater undecimal neutral second and the minor tone, as well as the [[septimal kleisma]] (225:224), 1029:1024 (the difference between three intervals of 8:7 the interval 3:2), and the small diesis (3125:3072). |
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==Notation== |
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Using extended pythagorean notation results in double and even triple sharps and flats. Furthermore, the notes run out of order. The chromatic scale is C, B{{music|sharp}}, A{{music|triplesharp}}/E{{music|tripleflat}}, D{{music|flat}}, C{{music|sharp}}, B{{music|doublesharp}}, E{{music|doubleflat}}, D... These issues can be avoided by using ups and downs notation.<ref>https://en.xen.wiki/w/Ups_and_downs_notation</ref> The up and down arrows are written as a caret or a lower-case "v", usually in a sans-serif font. One arrow equals one step of 41-TET. In note names, the arrows come first, to facilitate chord naming. The many enharmonic equivalences allow great freedom of spelling. |
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* C, ^C, ^^C/vvC{{music|sharp}}/vD{{music|flat}}, vC{{music|sharp}}/D{{music|flat}}, C{{music|sharp}}/^D{{music|flat}}, ^C{{music|sharp}}/^^D{{music|flat}}/vvD, vD, |
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* D, ^D, ^^D/vvD{{music|sharp}}/vE{{music|flat}}, vD{{music|sharp}}/E{{music|flat}}, D{{music|sharp}}/^E{{music|flat}}, ^D{{music|sharp}}/^^E{{music|flat}}/vvE, vE, |
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* E, ^E/vvF, ^^E/vF, |
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* F, ^F, ^^F/vvF{{music|sharp}}/vG{{music|flat}}, vF{{music|sharp}}/G{{music|flat}}, F{{music|sharp}}/^G{{music|flat}}, ^F{{music|sharp}}/^^G{{music|flat}}/vvG, vG, |
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* G, ^G, ^^G/vvG{{music|sharp}}/vA{{music|flat}}, vG{{music|sharp}}/A{{music|flat}}, G{{music|sharp}}/^A{{music|flat}}, ^G{{music|sharp}}/^^A{{music|flat}}/vvA, vA, |
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* A, ^A, ^^A/vvA{{music|sharp}}/vB{{music|flat}}, vA{{music|sharp}}/B{{music|flat}}, A{{music|sharp}}/^B{{music|flat}}, ^A{{music|sharp}}/^^B{{music|flat}}/vvB, vB, |
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* B, ^B/vvC, ^^B/vC, C |
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===Chords of 41 equal temperament=== |
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Because ups and downs notation names the intervals of 41-TET,<ref>https://en.xen.wiki/w/41edo#Intervals</ref> it can provide precise chord names. The pythagorean minor chord with 32/27 on C is still named Cm and still spelled C–E{{music|flat}}–G. But the 5-limit ''up''minor chord uses the upminor 3rd 6/5 and is spelled C–^E{{music|flat}}–G. This chord is named C^m. Compare with ^Cm (^C–^E{{music|flat}}–^G). |
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{| class="wikitable" |
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|+ Various [[7-limit_tuning|7-limit]] chords of 41-TET |
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! Chord name !! Chord !! Notes !! As [[Harmonic_series_(music)|harmonics]]<br>or [[Undertone_series|subharmonics]] !! Homonyms |
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| Sus4 || C4 || C-F-G || 6:8:9 || F sus2 |
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| Sus2 || C2 || C-D-G || 8:9:12 or 9:8:6 || G sus4 |
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| Downmajor or down || Cv || C-vE-G || 4:5:6 || |
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| Upminor || C^m || C-^E{{music|flat}}-G || 6:5:4 || |
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| Downminor || Cvm || C-vE{{music|flat}}-G || 6:7:9 || |
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| Upmajor or up || C^ || C-^E-G || 9:7:6 || |
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| Updim || C^dim || C-^E{{music|flat}}-G{{music|flat}} || 5:6:7 || |
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| Downdim || Cvdim || C-vE{{music|flat}}-G{{music|flat}} || 7:6:5 || |
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| Downmajor7 || CvM7 || C-vE-G-vB || 8:10:12:15 || |
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| Down7 || Cv7 || C-vE-G-vB{{music|flat}} || 4:5:6:7 || |
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| Down add7 || Cv,7 || C-vE-G-B{{music|flat}} || 36:45:54:64 || |
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| Up7 || C^7 || C-^E-G-^B{{music|flat}} || 9:7:6:5 || |
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| Upminor7 || C^m7 || C-^E{{music|flat}}-G-^B{{music|flat}} || 10:12:15:18 || ^E down6 |
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| Downminor7 || Cvm7 || C-vE{{music|flat}}-G-vB{{music|flat}} || 12:14:18:21 || |
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| Downmajor6 or down6 || Cv6 || C-vE-G-vA || 12:15:18:20 || vA upminor7 |
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| Upminor6 || C^m6 || C-^E{{music|flat}}-G-^A || 12:10:8:7 || ^E{{music|flat}} downdim down7 |
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| Downminor6 || Cvm6 || C-vE{{music|flat}}-G-vA || 6:7:9:10 || vA updim up7 |
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| Updim up7 || C^dim^7 || C-^E{{music|flat}}-G{{music|flat}}-^B{{music|flat}} || 5:6:7:9 || ^E{{music|flat}} downminor6 |
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| Downdim down7 || Cvdimv7 || C-vE{{music|flat}}-G{{music|flat}}-vB{{music|flat}} || 7:6:5:4 || vE{{music|flat}} upminor6 |
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| Up9 || C^9 || C-^E-G-^B{{music|flat}}-D || 9:7:6:5:4 || |
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| Down9 || Cv9 || C-vE-G-vB{{music|flat}}-D || 4:5:6:7:9 || |
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== References == |
== References == |
Latest revision as of 15:37, 23 May 2024
In music, 41 equal temperament, abbreviated 41-TET, 41-EDO, or 41-ET, is the tempered scale derived by dividing the octave into 41 equally sized steps (equal frequency ratios). Each step represents a frequency ratio of 21/41, or 29.27 cents ( ), an interval close in size to the septimal comma. 41-ET can be seen as a tuning of the schismatic,[1] magic and miracle[2] temperaments. It is the second smallest equal temperament, after 29-ET, whose perfect fifth is closer to just intonation than that of 12-ET. In other words, is a better approximation to the ratio than either or .
History and use[edit]
Although 41-ET has not seen as wide use as other temperaments such as 19-ET or 31-ET [citation needed], pianist and engineer Paul von Janko built a piano using this tuning, which is on display at the Gemeentemuseum in The Hague.[3] 41-ET can also be seen as an octave-based approximation of the Bohlen–Pierce scale.
41-ET guitars have been built, notably by Yossi Tamim. The frets on such guitars are very tightly spaced. To make a more playable 41-ET guitar, an approach called "The Kite Tuning" omits every-other fret (in other words, 41 frets per two octaves or 20.5 frets per octave) while tuning adjacent strings to an odd number of steps of 41. [4] Thus, any two adjacent strings together contain all the pitch classes of the full 41-ET system. The Kite Guitar's main tuning uses 13 steps of 41-ET (which approximates a 5/4 ratio) between strings. With that tuning, all simple ratios of odd limit 9 or less are available at spans at most only 4 frets.
41-ET is also a subset of 205-ET, for which the keyboard layout of the Tonal Plexus is designed.
Interval size[edit]
Here are the sizes of some common intervals (shaded rows mark relatively poor matches):
interval name | size (steps) | size (cents) | midi | just ratio | just (cents) | midi | error |
Octave | 41 | 1200 | 2:1 | 1200 | 0 | ||
Harmonic seventh | 33 | 965.85 | 7:4 | 968.83 | −2.97 | ||
Perfect fifth | 24 | 702.44 | 3:2 | 701.96 | +0.48 | ||
Grave fifth | 23 | 673.17 | 262144:177147 | 678.49 | −5.32 | ||
Septimal tritone | 20 | 585.37 | 7:5 | 582.51 | +2.85 | ||
Eleventh harmonic | 19 | 556.10 | 11:8 | 551.32 | +4.78 | ||
15:11 Wide fourth | 18 | 526.83 | 15:11 | 536.95 | −10.12 | ||
27:20 Wide fourth | 18 | 526.83 | 27:20 | 519.55 | +7.28 | ||
Perfect fourth | 17 | 497.56 | 4:3 | 498.04 | −0.48 | ||
Septimal narrow fourth | 16 | 468.29 | 21:16 | 470.78 | −2.48 | ||
Septimal (super)major third | 15 | 439.02 | 9:7 | 435.08 | +3.94 | ||
Undecimal major third | 14 | 409.76 | 14:11 | 417.51 | −7.75 | ||
Pythagorean major third | 14 | 409.76 | 81:64 | 407.82 | +1.94 | ||
Classic major third | 13 | 380.49 | 5:4 | 386.31 | −5.83 | ||
Tridecimal neutral third, thirteenth subharmonic | 12 | 351.22 | 16:13 | 359.47 | −8.25 | ||
Undecimal neutral third | 12 | 351.22 | 11:9 | 347.41 | +3.81 | ||
Classic minor third | 11 | 321.95 | 6:5 | 315.64 | +6.31 | ||
Pythagorean minor third | 10 | 292.68 | 32:27 | 294.13 | −1.45 | ||
Tridecimal minor third | 10 | 292.68 | 13:11 | 289.21 | +3.47 | ||
Septimal (sub)minor third | 9 | 263.41 | 7:6 | 266.87 | −3.46 | ||
septimal whole tone | 8 | 234.15 | 8:7 | 231.17 | +2.97 | ||
Diminished third | 8 | 234.15 | 256:225 | 223.46 | +10.68 | ||
Whole tone, major tone | 7 | 204.88 | 9:8 | 203.91 | +0.97 | ||
Whole tone, minor tone | 6 | 175.61 | 10:9 | 182.40 | −6.79 | ||
Lesser undecimal neutral second | 5 | 146.34 | 12:11 | 150.64 | −4.30 | ||
Septimal diatonic semitone | 4 | 117.07 | 15:14 | 119.44 | −2.37 | ||
Pythagorean chromatic semitone | 4 | 117.07 | 2187:2048 | 113.69 | +3.39 | ||
Classic diatonic semitone | 4 | 117.07 | 16:15 | 111.73 | +5.34 | ||
Pythagorean diatonic semitone | 3 | 87.80 | 256:243 | 90.22 | −2.42 | ||
20:19 Wide semitone | 3 | 87.80 | 20:19 | 88.80 | −1.00 | ||
Septimal chromatic semitone | 3 | 87.80 | 21:20 | 84.47 | +3.34 | ||
Classic chromatic semitone | 2 | 58.54 | 25:24 | 70.67 | −12.14 | ||
28:27 Wide semitone | 2 | 58.54 | 28:27 | 62.96 | −4.42 | ||
Septimal comma | 1 | 29.27 | 64:63 | 27.26 | +2.00 |
As the table above shows, the 41-ET both distinguishes between and closely matches all intervals involving the ratios in the harmonic series up to and including the 10th overtone. This includes the distinction between the major tone and minor tone (thus 41-ET is not a meantone tuning). These close fits make 41-ET a good approximation for 5-, 7- and 9-limit music.
41-ET also closely matches a number of other intervals involving higher harmonics. It distinguishes between and closely matches all intervals involving up through the 12th overtones, with the exception of the greater undecimal neutral second (11:10). Although not as accurate, it can be considered a full 15-limit tuning as well.
Tempering[edit]
Intervals not tempered out by 41-ET include the lesser diesis (128:125), septimal diesis (49:48), septimal sixth-tone (50:49), septimal comma (64:63), and the syntonic comma (81:80).
41-ET tempers out 100:99, which is the difference between the greater undecimal neutral second and the minor tone, as well as the septimal kleisma (225:224), 1029:1024 (the difference between three intervals of 8:7 the interval 3:2), and the small diesis (3125:3072).
Notation[edit]
Using extended pythagorean notation results in double and even triple sharps and flats. Furthermore, the notes run out of order. The chromatic scale is C, B♯, A♯/E
, D♭, C♯, B
, E
, D... These issues can be avoided by using ups and downs notation.[5] The up and down arrows are written as a caret or a lower-case "v", usually in a sans-serif font. One arrow equals one step of 41-TET. In note names, the arrows come first, to facilitate chord naming. The many enharmonic equivalences allow great freedom of spelling.
- C, ^C, ^^C/vvC♯/vD♭, vC♯/D♭, C♯/^D♭, ^C♯/^^D♭/vvD, vD,
- D, ^D, ^^D/vvD♯/vE♭, vD♯/E♭, D♯/^E♭, ^D♯/^^E♭/vvE, vE,
- E, ^E/vvF, ^^E/vF,
- F, ^F, ^^F/vvF♯/vG♭, vF♯/G♭, F♯/^G♭, ^F♯/^^G♭/vvG, vG,
- G, ^G, ^^G/vvG♯/vA♭, vG♯/A♭, G♯/^A♭, ^G♯/^^A♭/vvA, vA,
- A, ^A, ^^A/vvA♯/vB♭, vA♯/B♭, A♯/^B♭, ^A♯/^^B♭/vvB, vB,
- B, ^B/vvC, ^^B/vC, C
Chords of 41 equal temperament[edit]
Because ups and downs notation names the intervals of 41-TET,[6] it can provide precise chord names. The pythagorean minor chord with 32/27 on C is still named Cm and still spelled C–E♭–G. But the 5-limit upminor chord uses the upminor 3rd 6/5 and is spelled C–^E♭–G. This chord is named C^m. Compare with ^Cm (^C–^E♭–^G).
Chord name | Chord | Notes | As harmonics or subharmonics |
Homonyms |
---|---|---|---|---|
Sus4 | C4 | C-F-G | 6:8:9 | F sus2 |
Sus2 | C2 | C-D-G | 8:9:12 or 9:8:6 | G sus4 |
Downmajor or down | Cv | C-vE-G | 4:5:6 | |
Upminor | C^m | C-^E♭-G | 6:5:4 | |
Downminor | Cvm | C-vE♭-G | 6:7:9 | |
Upmajor or up | C^ | C-^E-G | 9:7:6 | |
Updim | C^dim | C-^E♭-G♭ | 5:6:7 | |
Downdim | Cvdim | C-vE♭-G♭ | 7:6:5 | |
Downmajor7 | CvM7 | C-vE-G-vB | 8:10:12:15 | |
Down7 | Cv7 | C-vE-G-vB♭ | 4:5:6:7 | |
Down add7 | Cv,7 | C-vE-G-B♭ | 36:45:54:64 | |
Up7 | C^7 | C-^E-G-^B♭ | 9:7:6:5 | |
Upminor7 | C^m7 | C-^E♭-G-^B♭ | 10:12:15:18 | ^E down6 |
Downminor7 | Cvm7 | C-vE♭-G-vB♭ | 12:14:18:21 | |
Downmajor6 or down6 | Cv6 | C-vE-G-vA | 12:15:18:20 | vA upminor7 |
Upminor6 | C^m6 | C-^E♭-G-^A | 12:10:8:7 | ^E♭ downdim down7 |
Downminor6 | Cvm6 | C-vE♭-G-vA | 6:7:9:10 | vA updim up7 |
Updim up7 | C^dim^7 | C-^E♭-G♭-^B♭ | 5:6:7:9 | ^E♭ downminor6 |
Downdim down7 | Cvdimv7 | C-vE♭-G♭-vB♭ | 7:6:5:4 | vE♭ upminor6 |
Up9 | C^9 | C-^E-G-^B♭-D | 9:7:6:5:4 | |
Down9 | Cv9 | C-vE-G-vB♭-D | 4:5:6:7:9 |
References[edit]
- ^ "Schismic Temperaments ", Intonation Information.
- ^ "Lattices with Decimal Notation", Intonation Information.
- ^ [1] Dirk de Klerk "Equal Temperament", Acta Musicologica, Vol. 51, Fasc. 1 (Jan. - Jun., 1979), pp. 140-150
- ^ "The Kite Guitar ", Xenharmonic Wiki.
- ^ https://en.xen.wiki/w/Ups_and_downs_notation
- ^ https://en.xen.wiki/w/41edo#Intervals