Sagittal notations for MET-24 tuning
Posted: Mon Jul 31, 2017 5:05 am
Please note that the menu items or buttons for HTML colors or tags are "Harmless Buttons" which I cannot seem to operate in elinks, so I've placed tags around the portion of my post to be formatted in monospaced text using repeated spaces for ASCII lattice diagrams, and also some spaces in Scala files.
MET-24 is a "three-dimensional" or "rank-3" temperament with three generators: the 2/1 octave, a fifth at 703.711 cents (1201 steps of 2048-ed2), and a spacing between chains of fifths at 57.422 cents (98 steps of 2048-ed2).
This post includes two lattice diagrams using a very simple form of mixed sagittal notation equivalent to a kind of keyboard tablature.
In the first lattice, the 1/1 is D on the lower chain of fifths, with notes on the upper chain of fifths shown by a /|\ sign. This sign can stand for 33/32 (e.g. G-G/|\ as 4/3-11/8); 28/27 (e.g. E-E/|\ as 9/8-7/6); and also 91/88 (e.g. F-F/|\ as 13/11-11/9) or 121/117 (e.g. Bb-Bb/|\ as 11/7-13/8).
Since MET-24 tempers out 352/351 (e.g. 32/27 vs. 13/11), 364/363 (e.g. 13/11 vs. 33/28), and also their sum (896/891, e.g. 28/27 vs. 33/32) and minute difference (10648/10647, e.g. 121/117 vs. 91/88), these diverse interpretations of the spacing generator are not so surprising.
In this second lattice, the 1/1 is C on the upper chain of fifths, so that notes on the lower chain are marked with a \|/ sign to show that they are lower by a spacing generator of 57.422 cents. This placement of the 1/1 at C on the upper chain of fifths is one possible choice for an Arab Rast, for example.
MET-24 is a "three-dimensional" or "rank-3" temperament with three generators: the 2/1 octave, a fifth at 703.711 cents (1201 steps of 2048-ed2), and a spacing between chains of fifths at 57.422 cents (98 steps of 2048-ed2).
This post includes two lattice diagrams using a very simple form of mixed sagittal notation equivalent to a kind of keyboard tablature.
In the first lattice, the 1/1 is D on the lower chain of fifths, with notes on the upper chain of fifths shown by a /|\ sign. This sign can stand for 33/32 (e.g. G-G/|\ as 4/3-11/8); 28/27 (e.g. E-E/|\ as 9/8-7/6); and also 91/88 (e.g. F-F/|\ as 13/11-11/9) or 121/117 (e.g. Bb-Bb/|\ as 11/7-13/8).
Since MET-24 tempers out 352/351 (e.g. 32/27 vs. 13/11), 364/363 (e.g. 13/11 vs. 33/28), and also their sum (896/891, e.g. 28/27 vs. 33/32) and minute difference (10648/10647, e.g. 121/117 vs. 91/88), these diverse interpretations of the spacing generator are not so surprising.
Lattice for MET-24, 1/1 = D (on lower 12-note chain of fifths) +0.3 +2.1 -1.1 +0.6 +2.4 -0.6 -3.8 -2.0 -0.3 +1.5 +3.2 +0.1 138.9 842.6 346.3 1050.0 553.7 57.4 761.1 264.8 968.6 472.3 1176.0 679.7 Eb/|\ Bb/|\ F/|\ C/|\ G/|\ D/|\ A/|\ E/|\ B/|\ F#/|\ C#/|\ G#/|\ 13/12--13/8---11/9---11/6---11/8---91/88--14/9---7/6---7/4---21/16--63/32--77/52 / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / 22/21--11/7---13/11--16/9----4/3---1/1---3/2----9/8---22/13--14/11--21/11--63/44 Eb Bb F C G D A E B F# C# G# 81.4 785.2 288.9 992.6 496.3 0 703.7 207.4 911.1 414.8 1118.6 622.3 +0.9 +2.7 -0.3 -3.5 -1.8 +1.8 +3.5 +0.3 -2.7 -0.9 +0.8 Horizontal lines = tempered 3:2 (703.711 cents) Vertical lines = tempered 7:6 (264.844 cents) Quadrangles from lower left = tempered 12:14:18:21 Two adjacent quadrangles = tempered 1-3-7-9 hexany Generators of MET-24 temperament: (2/1, 703.711, 57.422) ! met24c_d.scl ! Canonical version of MET-24 in 2048-ed2, 1/1=D (lower chain) 24 ! 57.42187 81.44531 138.86718 207.42187 264.84375 288.86718 346.28906 414.84375 472.26562 496.28906 553.71093 622.26562 679.68750 703.71093 761.13281 785.15624 842.57812 911.13281 968.55468 992.57812 1050.00000 1118.55468 1175.97656 2/1
In this second lattice, the 1/1 is C on the upper chain of fifths, so that notes on the lower chain are marked with a \|/ sign to show that they are lower by a spacing generator of 57.422 cents. This placement of the 1/1 at C on the upper chain of fifths is one possible choice for an Arab Rast, for example.
Lattice for MET-24, 1/1 = C (on upper 12-note chain of fifths) Eb Bb F C G D A E B F# C# G# +0.9 +2.7 -0.3 -3.5 -1.8 +3.5 -0.3 -2.7 +0.3 +0.8 -2.3 =0.6 288.9 992.6 496.3 0 703.7 207.4 911.1 414.8 1118.6 622.3 126.0 829.7 13/11--16/9--4/3---1/1---3/2----9/8--22/13-14/11--21/11--63/44--14/13--21/13 / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / 8/7--12/7---9/7--176/91-16/11-12/11--18/11--16/13--24/13--18/13--27/26--264/169 Eb\|/ Bb\|/ F\|/ C\|/ G\|/ D\|/ A\|/ E\|/ B\|/ F#\|/ C#\|/ G#\|/ 231.4 935.2 438.9 1142.6 646.3 150.0 853.7 357.4 1061.1 564.8 68.6 772.3 +0.3 +2.0 +3.8 +0.6 -2.4 -0.6 +1.1 -2.1 -0.3 +1.5 +3.2 +0.05 ! met24c_cup.scl ! MET-24 (2048-ed2), (2/1, 703.711, 57.422), 1/1=C on upper chain 24 ! 68.55468 125.97656 150.00000 207.42187 231.44531 288.86718 357.42187 414.84375 438.86718 496.28906 564.84375 622.26562 646.28906 703.71093 772.26562 829.68750 853.71093 911.13281 935.15624 992.57812 1061.13281 1118.55468 1142.57812 2/1Dialogue on other options is most welcome: for example, making a more thorough use of Sagittal symbols, or indicating some notes as comma variations of others.