Sagittal notations for MET-24 tuning

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mschulter
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Joined: Thu Jul 27, 2017 10:05 am

Sagittal notations for MET-24 tuning

Post by mschulter » 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.
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
 2/1
Dialogue 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.

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Dave Keenan
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Re: Sagittal notations for MET-24 tuning

Post by Dave Keenan » Mon Jul 31, 2017 12:24 pm

Thanks for those, Margo. My apologies for the delay in the appearance of your posts. The first post by every new member must be approved by a moderator (me) as an anti-spam measure. But at least in this forum, once that first post is approved you have full access and full capabilities, unlike the Xenharmonic Discourse test forum.

Apart from the time-zone difference (me being in Australia) you won't believe the additional cause of the delay :) : As I was about to approve your post, I lost internet access. Some investigation showed that I had lost my telephone landline and hence my ADSL connection to the internet. A cell-phone call to my telephone company('s computer) told me a cable had been damaged and it was likely to take 4 days to fix. So it took me some time to make other arrangements to get back online.

If you looked at your posts immediately after I approved them you may have seen collapsed lattice diagrams. If so, you should reload the pages now. I simply had to edit your post to change your <pre>...</pre> to [pre]...[/pre] (which you should use in future). The phpBB forum software that this forum is based on, does not allow HTML tags directly, but uses a form of markup with tags in square brackets known as BBCode. You can read about it here:
https://www.phpbb.com/community/help/bbcode

This will give you all the capabilities of those "harmless buttons".

In addition to the tags listed, there is the [pre]...[/pre] tag which is very similar to the [code]...[/code] tag, however [pre] does not perform any syntax hiliting and it allows you to nest other BBCodes within it (for e.g. bold, italic, color). It also allows you to nest smilies within it, including those that map to Sagittal glyphs (consisting of the long-ASCII form of the Sagittal between colons). You can also use tabs (width 8 spaces) to line up columns within [pre]...[/pre] however you may have to type the tabs in another editor and paste them into your forum post, as tabs are normally interpreted as a command to move between text fields.

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Dave Keenan
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Re: Sagittal notations for MET-24 tuning

Post by Dave Keenan » Mon Jul 31, 2017 2:25 pm

Margo, I believe your notation is the best possible Sagittal notation for those tunings. I note that since (|) is the apotome-complement of /|\, you can avail yourself of alternative spellings such as Bb/|\ (B flat vai) = B(!) (B wao). Also, you may find it more convenient to use the shorthand way of writing these symbols in plain text, using ^ and w.
Bb/|\ = Bb^ = B:b::/|\:
= B(!) = Bw = B:(!):

Similarly, in your second example (using v and m):
F#\!/ = F#v = F:#::\!/:
= F(|) = Fm = F:(|):

You might also consider alternative spellings when ratios of 7 are involved, e.g.
E/|\ = E^ = E:/|\:
= F!) = Ft = F:!):

F\!/ = Fv = F:\!/:
= E|) = Ef = E:|):

But of course as soon as you introduce alternative spellings, you lose the tablature property.

mschulter
Posts: 3
Joined: Thu Jul 27, 2017 10:05 am

Re: Sagittal notations for MET-24 tuning

Post by mschulter » Mon Jul 31, 2017 3:04 pm

Thank you, Dave, for some great alternatives which I will study carefully. I'm fascinated by the apotome complements, and also the ASCII notations for notes a 64/63 comma up or down. As I had guessed, some of our discussion might also fit under ASCII notations.

It occurs to me that while the tablature is useful for many things, a comma style of notation could be helpful if I wished to write out a realization of a version of the Archytas Diatonic, for example. Indeed, comparing some version of the tablature notation with a version of comma notation might help to clarify certain patterns.

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