94-edo / garibaldi[53] fret markers for fretless guitar

There was recently a discussion about Garibaldi temperament in 94-edo, so maybe someone is interested in this project idea I intend to put into practice some day (once I'm bored with the 31-edo guitar I still don't have, that is).

The idea is that "fret markers" for garibaldi[53] in 94-edo are painted on the neck of a fretless guitar (or another suited fretless stringed instrument). Here's the fretboard design for the first octave:

http://www.abload.de/img/diamond_frets_94l6xei.png

The "frets" are arranged in a nested structure:

* The centers of the black diamonds mark garibaldi[3] (1:1, 4:3, 3:2), which contains the root and both generators.
* When the red part of the diamond in the middle is ignored, the diamond centers (indicated by numbers) mark garibaldi[12], generated by stacking 5 fifths downwards and 6 fifths upwards.
* The diamond centers and edges, as well as the fret lines between the diamonds mark garibaldi[53].
* The large steps in the former garibaldi[53] MOS are 2 steps in 94-edo. If we add the notes that are approximately in the middle of those large steps, we have a full 94-edo scale. Those large steps occur between a diamond center and its edge, and (sometimes, but not always) between a diamond edge and an adjacent single fret line.

As a rule of thumb, you can associate the diamond centers with "simple" pythagorean intervals. The left edge of a blue diamond and the right edge of a red diamond usually represent simple 5-limit intervals, while the other edges usually represent simple 7-limit intervals. Some of the most important 11- or 13-limit intervals are represented by single fret lines.

For a notation system, I use a garibald[12] interval notation based on numbers 0 to 12 (see picture). Here are the pythagorean intervals associated with these base intervals:
(94-edo steps and simpler ratios in brackets)

0 (0) - 1:1
1 (7) - 256:243 (21:20)
2 (16) - 9:8
3 (23) - 32:27 (13:11)
4 (32) - 81:64 (33:26)
5 (39) - 4:3
6 (48) - 729:512 (10:7)
7 (55) - 3:2
8 (62) - 128:81 (52:33)
9 (71) - 27:16 (22:13)
10 (78) - 16:9
11 (87) - 243:128 (40:21)
12 (94) - 2:1

The form of this 12-tone MOS is sLsLsLssLsLs, with s = 7\94 and L = 9\94, so the chroma is c = L - s = 2\94. This means 2 steps in 94-edo, or the difference between a diamond center and one of its edges.

So we have 12 base intervals (13 when counting the octave), and if we add a pair of accidentals that represent the chroma we have a complete interval notation for Garibaldi, or 94-edo. I decided to use '/' and '\' for a chroma up and down, resp., because from my experience the most common interpretation is a syntonic comma 81:80, though it can also represent the septimal comma 64:63, or the pythagorean comma 531441/524288.

For example, the major and minor scale can be written as following:

major scale:
1:1 9:8 5:4 4:3 3:2 5:3 15:8 2:1
0 2 4\ 5 7 9\ 11\ 12

minor scale:
1:1 9:8 6:5 4:3 3:2 8:5 9:5 2:1
0 2 3/ 5 7 8/ 10/ 12

4\, 9\ and 11\ refer to the left edge of the diamonds labeled 4, 9 and 11, while 3/, 8/ and 10/ refer to the right edges of the corresponding diamonds.

This notation can be used just like standard guitar tab notation, except there are additional microtonal accidentals. For example, if the guitar is tuned in standard tuning using perfect 4ths, a Dm chord can be notated like following:

e|1/
B|3
G|2
D|0
A|x
E|x

(some other chords don't work that well with this string tuning, try a C major chord, for example)

An overtone scale between the 8th and 16th harmonic can be written as following:

1:1 9:8 5:4 11:8 3:2 13:8 7:4 15:8 2:1
0 2 4\ 5// 7 8// 10\ 11\ 12

To avoid that the notation becomes overly messy in the 11-limit or higher, it's a good idea to add a half-sharp '^' and a half-flat 'v' which replace '//' and '\\', resp., so 11:8 can be written as 5^, and 13:8 as 8^.

Note that this is a general Garibaldi notation, so it also works with other tunings that support Garibaldi temperament, like 41- or 53-edo. Similarly, also the fretboard design can be adapted for those tunings: in 41-edo, the 12 small steps in garibaldi[53] vanish (-> between each diamond, remove a fret line), and in 53-edo, all steps in garibaldi[53] become equal (obviously).

Hope I could inspire someone! Also, any suggestions about alternative accidentals? '/' and '\' look like slide notation, and slides are a pretty cool thing on a fretless instrument.

- Gedankenwelt

On Tuesday 09 April 2013 9:25:20 gedankenwelt94 wrote:

> The idea is that "fret markers" for garibaldi[53] in
> 94-edo are painted on the neck of a fretless guitar (or
> another suited fretless stringed instrument). Here's the
> fretboard design for the first octave:
>
> http://www.abload.de/img/diamond_frets_94l6xei.png
>
> The "frets" are arranged in a nested structure:
>
> * The centers of the black diamonds mark garibaldi[3]
> (1:1, 4:3, 3:2), which contains the root and both
> generators. * When the red part of the diamond in the
> middle is ignored, the diamond centers (indicated by
> numbers) mark garibaldi[12], generated by stacking 5
> fifths downwards and 6 fifths upwards. * The diamond
> centers and edges, as well as the fret lines between the
> diamonds mark garibaldi[53]. * The large steps in the
> former garibaldi[53] MOS are 2 steps in 94-edo. If we
> add the notes that are approximately in the middle of
> those large steps, we have a full 94-edo scale. Those
> large steps occur between a diamond center and its edge,
> and (sometimes, but not always) between a diamond edge
> and an adjacent single fret line.

That looks really good. So the commas are the larger steps
here? It doesn't look scarily complicated because you can
read the diamonds as decoration.

I count 55 pitches :-S

> This notation can be used just like standard guitar tab
> notation, except there are additional microtonal
> accidentals. For example, if the guitar is tuned in
> standard tuning using perfect 4ths, a Dm chord can be
> notated like following:
>
> e|1/
> B|3
> G|2
> D|0
> A|x
> E|x

Oh, yeah, alterted guitar tab's the way to go.

Do you expect to play accurate chords on a fretless?

> Hope I could inspire someone! Also, any suggestions about
> alternative accidentals? '/' and '\' look like slide
> notation, and slides are a pretty cool thing on a
> fretless instrument.

Sagittal uses half-arrows for syntonic commas. They might
get confused with ^ and v, though.

Graham

--- In MakeMicroMusic@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> That looks really good. So the commas are the larger steps
> here? It doesn't look scarily complicated because you can
> read the diamonds as decoration.

Thanks, I'm glad you like it! :)

Yes, the large steps are {pythagorean, syntonic, septimal} commas, and two large steps make up a quartertone that represents 128:125, 36:35, 33:32 or 1053:1024 (the latter being the difference between 128:81 and 13:8). Interestingly, those are literal quartertones in 94-edo, dividing the whole tone 9:8 (= 16\94) into 4 equal steps.

> I count 55 pitches :-S

Probably my wording, or the dual diamond in the middle were a little confusing. If it helps, you can ignore the red diamond in the middle, and think of its left edge as a single fret line; its center and right edge overlap with the left edge and center of the blue diamond. Here's a scala file of the 53 pitches, with annotations:

! 53of94.scl
!
garibaldi[53] MOS in 94-edo
53
!
25.53191 ! 0/ (black, right edge)
51.06383 ! 0^
63.82979 ! 1\ (red, left edge)
89.36170 ! 1 (red, center)
114.89362 ! 1/ (red, right edge)
140.42553 ! 1^
153.19149 ! 2v
178.72340 ! 2\ (blue, left edge)
204.25532 ! 2 (blue, center)
229.78723 ! 2/ (blue, right edge)
255.31915 ! 2^
268.08511 ! 3\ (red, left edge)
293.61702 ! 3 (red, center)
319.14894 ! 3/ (red, right edge)
344.68085 ! 3^
357.44681 ! 4v
382.97872 ! 4\ (blue, left edge)
408.51064 ! 4 (blue, center)
434.04255 ! 4/ (blue, right edge)
446.80851 ! 5v
472.34043 ! 5\ (black, left edge)
497.87234 ! 5 (black, center)
523.40426 ! 5/ (black, right edge)
548.93617 ! 5^
561.70213 ! 6v (red, left edge)
587.23404 ! 6\ (red, center / blue, left edge)
612.76596 ! 6 (blue, center / red, right edge)
638.29787 ! 6/ (blue, right edge)
651.06383 ! 7v
676.59574 ! 7\ (black, left edge)
702.12766 ! 7 (black, center)
727.65957 ! 7/ (black, right edge)
753.19149 ! 7^
765.95745 ! 8\ (red, left edge)
791.48936 ! 8 (red, center)
817.02128 ! 8/ (red, right edge)
842.55319 ! 8^
855.31915 ! 9v
880.85106 ! 9\ (blue, left edge)
906.38298 ! 9 (blue, center)
931.91489 ! 9/ (blue, right edge)
944.68085 ! 10v
970.21277 ! 10\ (red, left edge)
995.74468 ! 10 (red, center)
1021.27660 ! 10/ (red, right edge)
1046.80851 ! 10^
1059.57447 ! 11v
1085.10638 ! 11\ (blue, left edge)
1110.63830 ! 11 (blue, center)
1136.17021 ! 11/ (blue, right edge)
1148.93617 ! 12v
1174.46809 ! 12\ (black, left edge)
2/1 ! 12 (black, center)

> Do you expect to play accurate chords on a fretless?

Well, that depends on the chord and the string tuning. I know there will be problems, be it correct intonation or sustaining notes on light strings, but I'm happy if I can play single note lines, or some simple basic chords.

I'm more concerned about correct intonation in general, since straight fret lines will mean there's at least some error if I just go by the fret markers, instead of adjusting finger positions. But I suppose with some experience one learns on which strings and in which ranges the finger positions have to be adapted in which way.

> > Hope I could inspire someone! Also, any suggestions about
> > alternative accidentals? '/' and '\' look like slide
> > notation, and slides are a pretty cool thing on a
> > fretless instrument.
>
> Sagittal uses half-arrows for syntonic commas. They might
> get confused with ^ and v, though.

Thanks, that helped! :)

I guess I'll use / and ^ for non-guitar tab software, /! and ^ for ascii tabs, and half and full arrows for physical guitar tabs ... or instead of full arrows, a # with only one vertical line, and a struck through b, as used in maqam notation. I intended to use # and b for two quartertones anyway (representing 17:16 or 18:17 in 94-edo), so that would fit well.

- Gedankenwelt