Tripod Notation

I'm writing up an expostion of what has now bloated into two different notation schemes. I've still got some things to tweak and add but this tells you what I'm up to:

http://x31eq.com/magic/tripod.pdf

Graham

Dear Graham,

Thank you for sharing this paper. Now I got some idea how your decimal notation examples I saw before are read, although I am still in the process of digesting your approach :)

I assume I am not the most qualified person on this list to comment on your text. However, there was little feedback so far and I assume you would like some. So, here I go...

First of all, what are your plans with this paper? Do you intend to publish it somewhere? What kind of venue would you aim for?

I must say, I do like your somewhat casual way of writing (more precisely, I envy it -- as a non-native English speaker :).

However, I feel your paper contains a few simplifications which can cause misunderstanding. Perhaps you do this on purpose to keep things more easy for the reader. In case you want to publish it, it would depend on the venue whether such simplifications are suitable or not. Anyway, just in case, I point out some to give you an idea what I am talking about. For example, in the very beginning you are talking about the "contrapunctualists of old" and you are saying "steps are the building block of melody and leaps are the building blocks of harmony". There is certainly something to this, but one can argue that such a statement is over-simplified :) What style are you talking about here anyway, when you talk about "harmony": Ars Nova, Renaissance, Baroque? You later go on saying "a newer school of thought...", without specifying who you are referring to.

In the context of this mailing list, a more relevant issue is when you are talking about our common 7 nominals as "meantone names". Again, there is certainly something to this, but you very likely know yourself that this notation can do more than that. For example, these nominals can also mean pitches in Pythagorean tuning. Actually, this is possibly the more common meaning in general. In the microtonal music community there are several notation schemes which complement the notion of Pythagorean tuning with accidentals for various commas, such as Helmholtz / Ellis / Wolf / Monzo notation (HEWM), Extended Helmholtz Ellis JI Pitch Notation by Sabat & v Schweinitz, or JI Sagittal.

I must say, as a traditionally trained musician, I feel very comfortable with the common nominals and the extensions just mentioned feel natural for me. By contrast, it would be relatively hard for me to get used to your tripod notation. But this is primarily due to training, so no principle argument. Indeed, I find the "circulating comma-pump" -- which you point out -- an interesting feature of the tripod notation. Could you perhaps mention other advantages of this notation scheme compared with notations which explicitly notate commas (like HEWM etc.)?

Your notation examples look pretty nice :) Great to see what Lilypond can do. Concerning the Saggital accidentals, you said

> Sagittal is working cleanly now. You have to override the X- and Y-> extent of Accidental, but once you do that you can load Sagittal by > name instead of installing it as a fake system font. I keep the X- > and Y-extents in an alist along with the strings.

When I had a look into your source in tripod-data.ly I found things like

magicStrings = #'(
(2 0 "\xe2\x88\x86" (0 . 1) (-1.1 . 0.5)) ;rightscrolltripleup, 15:14
(1 1 "\xc2\xbf" (0 . 1) (-1.1 . 0.5)) ;doubleleftbarbdoubleup
(1 0 "\xe2\x88\x82" (0 . 1) (-1.1 . 0.5)) ;leftbarbdoubleup, 3/60
(0 1 "\xc3\xba" (0 . 0.6) (-1.1 . 0.5)) ;rightarcup, 7-comma
(0 0 "\xc3\xaa" (0 . 0.7) (-1.1 . 1.1)) ;natural
(0 -1 "\xc3\x91" (0 . 0.6) (-0.5 . 1.1)) ;rightarcdown, 7-comma
(-1 0 "\x6a" (0 . 1) (-0.5 . 1.1)) ;leftbarbdoubledown, 3/60
(-1 -1 "\x60" (0 . 1) (-0.5 . 1.1)) ;doubleleftbarbdoubledown
(-2 0 "\x5a" (0 . 1) (-0.5 . 1.1)) ;rightscrolltripledown, 14:15
)

You earlier mentioned

On Jan 6, 2009, at 2:09 PM, Graham Breed wrote:
> The main problem is finding the right character codes. I have a > PDF that gives decimal and hex codes
> for the characters. Neither are consistent with what FontForge > shows. Lilypond goes with the Unicode numbers FontForge gives. And > Guile encodes Unicode using UTF-8. So you have to go into > FontForge, find
> the Unicode numbers, and convert them to UTF-8. I do the last step > in Python like this:
>
>>>> u"\u00fa".encode("utf-8")
> '\xc3\xba'

So, I understand that you look up the Sagittal character code in the available list, translate that with Python to Unicode, and then enter that result into a specification as shown above. Is that still the way you do things? Could you perhaps explain whether I missed something here?

Thank you!

Best
Torsten

PS: You obviously spend quite some time on the typographic layout of your paper. You may want to fine-tune your headings in Fraktur (Gothic script, Blackletter, http://en.wikipedia.org/wiki/Blackletter). Firstly, your kerning is not correct (http://en.wikipedia.org/wiki/Kerning). For example, in the heading "1 Preliminaries" there is clear a gap before the first "e", and in "1.1 Steps and Leaps" there are again clear gaps before the "e"s, in particular in "Leap". Also, an important feature of Fraktur are the ligatures. For example, in "1.2 The Lattice" the "tt" often form a ligature. Finally, some letters in Fraktur have different shapes depending on their position in the word. For example, as far as I know in "Musical" and "Possibilities" the "s" usually has a different shape which more looks like an "f"

As you were typesetting your paper in Latex, you can actually take care of all these things ;-) For example, you might use the package yfonts. I'll send you an example privately.

--
Torsten Anders
Interdisciplinary Centre for Computer Music Research
University of Plymouth
Office: +44-1752-586219
Private: +44-1752-558917
http://strasheela.sourceforge.net
http://www.torsten-anders.de

On Jan 29, 2009, at 11:27 AM, Graham Breed wrote:

> I'm writing up an expostion of what has now bloated into two
> different notation schemes. I've still got some things to
> tweak and add but this tells you what I'm up to:
>
> http://x31eq.com/magic/tripod.pdf
>
> Graham
>
>

Torsten Anders wrote:
> Dear Graham,
> > Thank you for sharing this paper. Now I got some idea how your > decimal notation examples I saw before are read, although I am still > in the process of digesting your approach :)
> > I assume I am not the most qualified person on this list to comment > on your text. However, there was little feedback so far and I assume > you would like some. So, here I go...
> > First of all, what are your plans with this paper? Do you intend to > publish it somewhere? What kind of venue would you aim for?

I plan to keep it on my website, link to it from the tuning page, and from a page listing the other things I hope to do relating to magic temperament.

> I must say, I do like your somewhat casual way of writing (more > precisely, I envy it -- as a non-native English speaker :).

Thank you! I try to follow Orwell, the writer rather than the temperament class:

http://www.mtholyoke.edu/acad/intrel/orwell46.htm

(One day somebody has to write a paper on the temperament class with chapter quotes taken from the writer. There are some good ones in here because style can be applied to music as well.)

I know a lot of academic papers have a pseudo-objective style that I really don't like. And for a native speaker, at least, that isn't even easy to emulate.

> However, I feel your paper contains a few simplifications which can > cause misunderstanding. Perhaps you do this on purpose to keep things > more easy for the reader. In case you want to publish it, it would > depend on the venue whether such simplifications are suitable or not. > Anyway, just in case, I point out some to give you an idea what I am > talking about. For example, in the very beginning you are talking > about the "contrapunctualists of old" and you are saying "steps are > the building block of melody and leaps are the building blocks of > harmony". There is certainly something to this, but one can argue > that such a statement is over-simplified :) What style are you > talking about here anyway, when you talk about "harmony": Ars Nova, > Renaissance, Baroque? You later go on saying "a newer school of > thought...", without specifying who you are referring to.

It's generally aimed at musicians without experience in microtonality. That's a tough audience, but I do my best. So there are no mentions of "just intonation" for example.

What I know about counterpoint is the Palestrina style, or Fux. But this isn't a paper on counterpoint. The newer school of thought is the way jazz/rock musicians seem to think. I took that paragraph out a while back. One result is that I go more quickly into explaining the relevance to tripod notation, which is the point of it all.

> In the context of this mailing list, a more relevant issue is when > you are talking about our common 7 nominals as "meantone names". > Again, there is certainly something to this, but you very likely know > yourself that this notation can do more than that. For example, these > nominals can also mean pitches in Pythagorean tuning. Actually, this > is possibly the more common meaning in general. In the microtonal > music community there are several notation schemes which complement > the notion of Pythagorean tuning with accidentals for various commas, > such as Helmholtz / Ellis / Wolf / Monzo notation (HEWM), Extended > Helmholtz Ellis JI Pitch Notation by Sabat & v Schweinitz, or JI > Sagittal.

The names I use are meantone. I assume approximations that are only valid in meantone. The lattices would have different spellings for these other notation systems.

Incidentally, I now have Trojan Sagittal working for 60 note equal temperament. It's in the code bundle. The implications are similar to a JI notation because of the way fifths are approximated.

> I must say, as a traditionally trained musician, I feel very > comfortable with the common nominals and the extensions just > mentioned feel natural for me. By contrast, it would be relatively > hard for me to get used to your tripod notation. But this is > primarily due to training, so no principle argument. Indeed, I find > the "circulating comma-pump" -- which you point out -- an interesting > feature of the tripod notation. Could you perhaps mention other > advantages of this notation scheme compared with notations which > explicitly notate commas (like HEWM etc.)?

That's the point of tricycle notation. I have a keyboard mapping for the 19 note magic scale (Pengcheng) based on tripod notation which I've been playing with for the past year. First, I added stickers to show the tripod degrees. But then I added stickers to show the meantone names for the 19 note scale as well. I found they're very helpful, especially when it comes to moving traditional harmony into magic temperament. All you need to do is get the right meantone names and the right feet and it all works out. And that's all there is, in principle, to tricycle notation.

The magic comma pump in tripod notation was wrong until last night :-P I spotted the error when I was checking the Trojan notation. That's one advantage of having different notations based on the same code.

There are two tripod-related notations. The tripod staff is good where you're using magic temperament, or marvel temperament with an eye to magic. It makes the pitch structure clearer in that unequal notes are marked by inequalities in the staff. It's also a bit larger so notes don't get squashed up as much. Having alternative ways to spell a pitch, while keeping magic temperament, helps here. If the notation fits the music you need less accidentals and the music generally looks cleaner. It does notate the usual commas as in the "Inch Shifts" section.

Tripod pitches are a good fit for numerical notation (jianpu) if you happen to be familiar with that. And the short names can be used for microtonal solfege.

Tricycle notation is firstly an easier way of getting used to magic temperament. It means you keep more that's familiar. It also works out simpler than the notations you mention for music with a lot of comma drift because the commas don't accumulate. I don't know if performers would be comfortable with it or not but I can't think of any critical flaws.

It's also an interesting approach to the meantone vs. Pythagorean question. Because it doesn't specify shifts relative to a fixed scale, you don't have to worry about how the fifths are tuned in that scale. And, as it happens, magic temperament tends to have very good fifths. But you can still think in terms of meantone approximations for the 5-limit harmony.

Maybe the nearest relative to tricycle notation is Vicentino's enharmonic notation as applied to his second archicembalo tuning. Both can specify JI chords but avoid long term pitch drift. Both are torsional as well, in mathematical terms.

> Your notation examples look pretty nice :) Great to see what Lilypond > can do. Concerning the Saggital accidentals, you said
> >> Sagittal is working cleanly now. You have to override the X- and Y- >> extent of Accidental, but once you do that you can load Sagittal by >> name instead of installing it as a fake system font. I keep the X- >> and Y-extents in an alist along with the strings.
> > > When I had a look into your source in tripod-data.ly I found things like
> > magicStrings = #'(
> (2 0 "\xe2\x88\x86" (0 . 1) (-1.1 . 0.5)) ;rightscrolltripleup, > 15:14
> (1 1 "\xc2\xbf" (0 . 1) (-1.1 . 0.5)) ;doubleleftbarbdoubleup
> (1 0 "\xe2\x88\x82" (0 . 1) (-1.1 . 0.5)) ;leftbarbdoubleup, 3/60
> (0 1 "\xc3\xba" (0 . 0.6) (-1.1 . 0.5)) ;rightarcup, 7-comma
> (0 0 "\xc3\xaa" (0 . 0.7) (-1.1 . 1.1)) ;natural
> (0 -1 "\xc3\x91" (0 . 0.6) (-0.5 . 1.1)) ;rightarcdown, 7-comma
> (-1 0 "\x6a" (0 . 1) (-0.5 . 1.1)) ;leftbarbdoubledown, 3/60
> (-1 -1 "\x60" (0 . 1) (-0.5 . 1.1)) ;doubleleftbarbdoubledown
> (-2 0 "\x5a" (0 . 1) (-0.5 . > 1.1)) ;rightscrolltripledown, 14:15
> )

Right. The first two numbers describe the pitch in terms of magic intervals. Then there's the string that gets printed (I can do mixed Sagittal now so it isn't always a single glyph). Then there are two pairs for the X- and Y-extents as Lilypond expects them.

> You earlier mentioned
> > On Jan 6, 2009, at 2:09 PM, Graham Breed wrote:
>> The main problem is finding the right character codes. I have a >> PDF that gives decimal and hex codes
>> for the characters. Neither are consistent with what FontForge >> shows. Lilypond goes with the Unicode numbers FontForge gives. And >> Guile encodes Unicode using UTF-8. So you have to go into >> FontForge, find
>> the Unicode numbers, and convert them to UTF-8. I do the last step >> in Python like this:
>>
>>>>> u"\u00fa".encode("utf-8")
>> '\xc3\xba'
> > > So, I understand that you look up the Sagittal character code in the > available list, translate that with Python to Unicode, and then enter > that result into a specification as shown above. Is that still the > way you do things? Could you perhaps explain whether I missed > something here?

That's it. Lilypond does define a Scheme function for the UTF-8 conversion (ly:wide-char->utf-8) but I stick with the raw strings anyway.

> PS: You obviously spend quite some time on the typographic layout of > your paper. You may want to fine-tune your headings in Fraktur > (Gothic script, Blackletter, http://en.wikipedia.org/wiki/ > Blackletter). Firstly, your kerning is not correct (http:// > en.wikipedia.org/wiki/Kerning). For example, in the heading "1 > Preliminaries" there is clear a gap before the first "e", and in "1.1 > Steps and Leaps" there are again clear gaps before the "e"s, in > particular in "Leap". Also, an important feature of Fraktur are the > ligatures. For example, in "1.2 The Lattice" the "tt" often form a > ligature. Finally, some letters in Fraktur have different shapes > depending on their position in the word. For example, as far as I > know in "Musical" and "Possibilities" the "s" usually has a different > shape which more looks like an "f"

I use Gothic Textura Prescius from the Bookhands collection. It's simpler than a lot of other black fonts, and the kind of thing Caxton used. Yes, the kerning looks strange, but I'm not sure it's actually wrong. If you look at some real Caxton here:

http://upload.wikimedia.org/wikipedia/commons/4/4d/Godefrey_of_Boloyne_-_Facsimile_page_1_-_Project_Gutenberg_eText_12369.png

(I'm sure I found a better example before but never mind.)

It's generally messier than the digital font, of course, but it also has some strange kerning. And I don't know if they even had ligatures in those days. The font I used does have the long "s", for example in "just". I think it's good that it isn't used that much because it makes things easier for modern readers, and it happens not to be used in any of my headings, although I didn't deliberately avoid it.

> As you were typesetting your paper in Latex, you can actually take > care of all these things ;-) For example, you might use the package > yfonts. I'll send you an example privately.

Yes, yfonts is good, and I'd use "Gotik" rather than "Fraktur". But it still doesn't suit my purposes as well. If you can find a better prescius I'll consider it.

Graham