Re: Irrational time signature and tuplets

2014-06-13 Thread Hans Aberg 
On 13 Jun 2014, at 01:26, Mark Stephen Mrotek carsonm...@ca.rr.com wrote:

 Thank you for your detailed and informative response. I shall take time to
 study it and the references that you provide.

You are welcome.

In short, one can use any mathematically irrational number and work it back to 
rational approximations using continued fractions. Continued fractions are also 
useful when computing equal temperament approximations of intervals. For 
example, for the perfect 5th rational interval 3/2, one takes the continued 
fractions of log_2(3/2). The convergents, then, are 7/12, 24/41, 31/53, ..., 
where the denominators are the equal temperaments 12, 41, 53, ..., and the 
numerators 7, 24, 31, ..., the approximations of the perfect fifth in these 
temperaments.



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Re: Irrational time signature and tuplets

2014-06-12 Thread David Kastrup 
Hans Aberg haber...@telia.com writes:

 How might one display the time signature 8 + 2 sqrt(5) over 8, and the
 tuplet 1:1 + sqrt(5), where “sqrt is the square root sign? - It is
 for experimenting with a popular meter in 12 [1-2].

 1. https://en.wikipedia.org/wiki/Leventikos
 2. https://www.youtube.com/watch?v=vYR_pvRWO_g

\documentclass{article}
\usepackage[active,textmath,displaymath,tightpage]{preview}
\begin{document}
$8+2\sqrt{5}\over8$
$1\over1+\sqrt{5}$
\end{document}

run through LaTeX, run
dvips -E -i -Pwww

on the resulting DVI file, work with the resulting EPS files.  It might
be cute to integrate something like that into LilyPond like
preview-latex URL:http://www.gnu.org/software/auctex/preview-latex
integrates it into Emacs editing, but that would end up to be a bunch of
work for somebody™.



-- 
David Kastrup

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Re: Irrational time signature and tuplets

2014-06-12 Thread Malte Meyn 

This can be done in LilyPond without using LaTeX:

\version 2.19.6

{
  \override Staff.TimeSignature.stencil = #ly:text-interface::print
  \override Staff.TimeSignature.text =
  \markup \override #'(baseline-skip . 0) \center-column \number {
\concat {
  8+2
  \tiny √
  \hspace #-0.2
  \override #'(offset . -25)
  \override #'(thickness . 1.6)
  \underline 5
}
8
  }
  c'1
  \once \override TupletNumber.text =
  \markup \concat {
1:1+
\tiny √
\hspace #-0.15
\override #'(offset . -16)
\override #'(thickness . 1.6)
\underline 5
  }
  \tuplet 1/1 {
c'4 4 4 4
  }
}

Malte

On 12.06.2014 11:11, Hans Aberg wrote:

How might one display the time signature 8 + 2 sqrt(5) over 8, and the tuplet 1:1 + 
sqrt(5), where “sqrt is the square root sign? - It is for experimenting with a 
popular meter in 12 [1-2].

1. https://en.wikipedia.org/wiki/Leventikos
2. https://www.youtube.com/watch?v=vYR_pvRWO_g



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Re: Irrational time signature and tuplets

2014-06-12 Thread Hans Aberg 
On 12 Jun 2014, at 13:21, Malte Meyn lilyp...@maltemeyn.de wrote:

 This can be done in LilyPond without using LaTeX:

I have made an example incorporating your code; it looks pretty neat. For the 
midi, I wrote out the ornaments in a separate version using the 5/3 ratio, so 
as to be closer to the (1+sqrt 5)/2 = 1.618...; so this is 19 = 5+3+3+5+3, 
instead of the traditional 12 = 3+2+2+3+2, the latter which is used as a base 
for the engraving. It is from a common tune, Egejki Majki, played by the Boys 
from Bouf/Buf in the videos [1-2]. If someone wants to make snippet, please go 
ahead!

1. https://www.youtube.com/watch?v=NCDe41TtV-Y
2. https://www.youtube.com/watch?v=OfyZZ8OFvvs


 IrrationalMeter.ly 

\version 2.19.6

\header{
  title = Exampel of irrational meter
  subtitle = Egejki Majki, original meter 12 = 3+2+2+3+2
  composer = Traditional (Macedonia)
  copyright = © Hans Åberg
  tagline = ##f  % Removing Music engraving by LilyPond (version)
}


above = { \once \override Script #'script-priority = #-100 }
below = { \once \override TextScript #'script-priority = #-100 }


irrtuplet = \once \override TupletNumber.text =
 \markup \concat {
   1:1+
   \tiny √
   \hspace #-0.15
   \override #'(offset . -16)
   \override #'(thickness . 1.6)
   \underline 5
 }


music =  \new Staff {
  \tempo 4 = 176
  \key d \minor

  \time 12/8
  \set beatStructure = #'(3 2 2 3 2)

  \override Staff.TimeSignature.stencil = #ly:text-interface::print
  \override Staff.TimeSignature.text =
  \markup \override #'(baseline-skip . 0) \center-column \number {
\concat {
  8+2
  \tiny √
  \hspace #-0.2
  \override #'(offset . -25)
  \override #'(thickness . 1.6)
  \underline 5
}
8
  }


  \override TupletNumber.text = #tuplet-number::calc-fraction-text

  \relative c' {
   \repeat volta 2 {
  \once \override TextScript #'padding = #1.0
  \set Staff.midiInstrument = church organ
  \irrtuplet \tuplet 2/3 {f16 g a d,}  a'8\prall d,  a'8\prall d
\irrtuplet \tuplet 2/3 {c16 bes a g}  a8\prall g |
  \irrtuplet \tuplet 2/3 {f16 g a c,}  a'8\prall c,  a'8\prall d
\irrtuplet \tuplet 2/3 {c16 bes a g}  a8\prall r8 |
  \irrtuplet \tuplet 2/3 {b16 c d d}  d8 d
\above d8\trill^\markup{\flat}~d16 c
\irrtuplet \tuplet 2/3 {c16 d c bes}  a8\prall g |
  \irrtuplet \tuplet 2/3 {f16 g e g}  f8\prall e  \times 2/3 {d8 e c}
\irrtuplet \tuplet 2/3 {d16 e f g}  a8\prall g |
  \irrtuplet \tuplet 2/3 {f16 g e g}  f8\prall e  \times 2/3 {d8 e c}
\irrtuplet \tuplet 2/3 {\above d8\trill^\markup{\flat}~d16 a}  d8-. r8 |
  }
}
  }


\score {
  \music
\layout {}
}


musicmidi =  \new Staff {
  \tempo 4. = 186
  \key d \minor

  \time 19/8
  \set beatStructure = #'(5 3 3 5 3)

  \override TupletNumber.text = #tuplet-number::calc-fraction-text

  \relative c' {
\repeat volta 2 {
  \once \override TextScript #'padding = #1.0
  \set Staff.midiInstrument = church organ
\once \override TupletNumber.text = 
  #(tuplet-number::fraction-with-notes 4 4.)
  \tuplet 2/5 {f16 g a d,}  \tuplet 2/3 {a'32 bes32 a16 d,8}  \tuplet 2/3 
{a'32 bes32 a16 d8}
\once \override TupletNumber.text = 
  #(tuplet-number::fraction-with-notes 4 4.)
\tuplet 2/5 {c16 bes a g}  \tuplet 2/3 {a32 bes32 a16 g8} |
  \tuplet 2/5 {f16 g a c,}  \tuplet 2/3 {a'32 bes32 a16 c,8}  \tuplet 2/3 
{a'32 bes32 a16 d8}
\tuplet 2/5 {c16 bes a g}  \tuplet 2/3 {a32 bes32 a16 r8} |
  \tuplet 2/5 {b16 c d d}  \tuplet 2/3 {d8 d}
\tuplet 2/3 {d32 es d es d16 c}
\tuplet 2/5 {c16 d c bes}  \tuplet 2/3 {a32 bes32 a16  g8} |
  \tuplet 2/5 {f16 g e g}  \tuplet 2/3 {f32 g f16 e8}  {d8 e c}
\tuplet 2/5 {d16 e f g}  \tuplet 2/3 {a32 bes32 a16 g8} |
  \tuplet 2/5 {f16 g e g}  \tuplet 2/3 {f32 g f16 e8}  {d8 e c}
\tuplet 2/5 {d32 es d es d16 a}  \tuplet 2/3 {d8-. r8} |
  }
}
  }



\score {
  \unfoldRepeats
\musicmidi
\midi {}
}




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Re: Irrational time signature and tuplets

2014-06-12 Thread Hans Aberg 

On 12 Jun 2014, at 11:54, David Kastrup d...@gnu.org wrote:

 Hans Aberg haber...@telia.com writes:
 
 How might one display the time signature 8 + 2 sqrt(5) over 8, and the
 tuplet 1:1 + sqrt(5), where “sqrt is the square root sign? - It is
 for experimenting with a popular meter in 12 [1-2].
 
 1. https://en.wikipedia.org/wiki/Leventikos
 2. https://www.youtube.com/watch?v=vYR_pvRWO_g
 
 \documentclass{article}
 \usepackage[active,textmath,displaymath,tightpage]{preview}
 \begin{document}
 $8+2\sqrt{5}\over8$
 $1\over1+\sqrt{5}$
 \end{document}
 
 run through LaTeX, run
 dvips -E -i -Pwww
 
 on the resulting DVI file, work with the resulting EPS files.  It might
 be cute to integrate something like that into LilyPond like
 preview-latex URL:http://www.gnu.org/software/auctex/preview-latex
 integrates it into Emacs editing, but that would end up to be a bunch of
 work for somebody™.

One has to wrestle a bit, but that was a part of the reason I got curious about 
it, as you note, an interesting example.

The mathematical background is that I replace the proportion 3:2 of the 
original meter in 12 with the slightly larger golden ratio x = (1 + sqrt 5)/2, 
which has as continued fraction convergents the successive ratios of the 
Fibonacci sequence 1, 1, 2, 3, 5, 8, 13,... , that is, 1/1, 2/1, 3/2, 5/3, 8/5, 
13/8, ...; so here 3/2 occurs naturally.

Tying this back to LilyPond, the graphical representation will look the same, 
only that the meter and the tuplet numbers are changed. There is another 
problem in the MIDI file: how to make a better approximation than 3:2. The next 
is 5:3, but then one ends up with tuplets in the notation for the 2 over 8 
parts in the meter that must be suppressed. Doable, but a lot of typing.



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Re: Irrational time signature and tuplets

2014-06-12 Thread Hans Aberg 
On 12 Jun 2014, at 13:21, Malte Meyn lilyp...@maltemeyn.de wrote:

 This can be done in LilyPond without using LaTeX:

Oops, a typo in the code: the tuplets should be 2:(1+sqrt 5) - I forgot to 
change in your code. So hope it’s OK now.


 IrrationalMeter.ly 

\version 2.19.6

\header{
  title = Exampel of irrational meter
  subtitle = Egejki Majki, original meter 12 = 3+2+2+3+2
  composer = Traditional (Macedonia)
  copyright = © Hans Åberg
  tagline = ##f  % Removing Music engraving by LilyPond (version)
}


above = { \once \override Script #'script-priority = #-100 }
below = { \once \override TextScript #'script-priority = #-100 }


irrtuplet = \once \override TupletNumber.text =
 \markup \concat {
   2:1+
   \tiny √
   \hspace #-0.15
   \override #'(offset . -16)
   \override #'(thickness . 1.6)
   \underline 5
 }


music =  \new Staff {
  \tempo 4 = 176
  \key d \minor

  \time 12/8
  \set beatStructure = #'(3 2 2 3 2)

  \override Staff.TimeSignature.stencil = #ly:text-interface::print
  \override Staff.TimeSignature.text =
  \markup \override #'(baseline-skip . 0) \center-column \number {
\concat {
  8+2
  \tiny √
  \hspace #-0.2
  \override #'(offset . -25)
  \override #'(thickness . 1.6)
  \underline 5
}
8
  }


  \override TupletNumber.text = #tuplet-number::calc-fraction-text

  \relative c' {
   \repeat volta 2 {
  \once \override TextScript #'padding = #1.0
  \set Staff.midiInstrument = church organ
  \irrtuplet \tuplet 2/3 {f16 g a d,}  a'8\prall d,  a'8\prall d
\irrtuplet \tuplet 2/3 {c16 bes a g}  a8\prall g |
  \irrtuplet \tuplet 2/3 {f16 g a c,}  a'8\prall c,  a'8\prall d
\irrtuplet \tuplet 2/3 {c16 bes a g}  a8\prall r8 |
  \irrtuplet \tuplet 2/3 {b16 c d d}  d8 d
\above d8\trill^\markup{\flat}~d16 c
\irrtuplet \tuplet 2/3 {c16 d c bes}  a8\prall g |
  \irrtuplet \tuplet 2/3 {f16 g e g}  f8\prall e  \times 2/3 {d8 e c}
\irrtuplet \tuplet 2/3 {d16 e f g}  a8\prall g |
  \irrtuplet \tuplet 2/3 {f16 g e g}  f8\prall e  \times 2/3 {d8 e c}
\irrtuplet \tuplet 2/3 {\above d8\trill^\markup{\flat}~d16 a}  d8-. r8 |
  }
}
  }


\score {
  \music
\layout {}
}


musicmidi =  \new Staff {
  \tempo 4. = 186
  \key d \minor

  \time 19/8
  \set beatStructure = #'(5 3 3 5 3)

  \override TupletNumber.text = #tuplet-number::calc-fraction-text

  \relative c' {
\repeat volta 2 {
  \once \override TextScript #'padding = #1.0
  \set Staff.midiInstrument = church organ
\once \override TupletNumber.text = 
  #(tuplet-number::fraction-with-notes 4 4.)
  \tuplet 2/5 {f16 g a d,}  \tuplet 2/3 {a'32 bes32 a16 d,8}  \tuplet 2/3 
{a'32 bes32 a16 d8}
\once \override TupletNumber.text = 
  #(tuplet-number::fraction-with-notes 4 4.)
\tuplet 2/5 {c16 bes a g}  \tuplet 2/3 {a32 bes32 a16 g8} |
  \tuplet 2/5 {f16 g a c,}  \tuplet 2/3 {a'32 bes32 a16 c,8}  \tuplet 2/3 
{a'32 bes32 a16 d8}
\tuplet 2/5 {c16 bes a g}  \tuplet 2/3 {a32 bes32 a16 r8} |
  \tuplet 2/5 {b16 c d d}  \tuplet 2/3 {d8 d}
\tuplet 2/3 {d32 es d es d16 c}
\tuplet 2/5 {c16 d c bes}  \tuplet 2/3 {a32 bes32 a16  g8} |
  \tuplet 2/5 {f16 g e g}  \tuplet 2/3 {f32 g f16 e8}  {d8 e c}
\tuplet 2/5 {d16 e f g}  \tuplet 2/3 {a32 bes32 a16 g8} |
  \tuplet 2/5 {f16 g e g}  \tuplet 2/3 {f32 g f16 e8}  {d8 e c}
\tuplet 2/5 {d32 es d es d16 a}  \tuplet 2/3 {d8-. r8} |
  }
}
  }



\score {
  \unfoldRepeats
\musicmidi
\midi {}
}




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Re: Irrational time signature and tuplets

2014-06-12 Thread Malte Meyn 



On 12.06.2014 22:25, Hans Aberg wrote:


Tying this back to LilyPond, the graphical representation will look the same, 
only that the meter and the tuplet numbers are changed. There is another 
problem in the MIDI file: how to make a better approximation than 3:2. The next 
is 5:3, but then one ends up with tuplets in the notation for the 2 over 8 
parts in the meter that must be suppressed. Doable, but a lot of typing.

I’m not sure whether I understood correctly, but instead of writing 
something like


\tuplet 3/2 { c8 d e }

and hiding the tuplet number (and bracket), you can simply write

c8*2/3 d e

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RE: Irrational time signature and tuplets

2014-06-12 Thread Mark Stephen Mrotek 
Hans,

(1+sqrt 5)/2 = 1.618... is the golden ratio, phi. 
http://en.wikipedia.org/wiki/Golden_ratio

Do you know of other instances of this ratio in music?

Mark

-Original Message-
From: lilypond-user-bounces+carsonmark=ca.rr@gnu.org 
[mailto:lilypond-user-bounces+carsonmark=ca.rr@gnu.org] On Behalf Of Hans 
Aberg
Sent: Thursday, June 12, 2014 12:33 PM
To: Malte Meyn
Cc: lilypond-user@gnu.org
Subject: Re: Irrational time signature and tuplets

On 12 Jun 2014, at 13:21, Malte Meyn lilyp...@maltemeyn.de wrote:

 This can be done in LilyPond without using LaTeX:

I have made an example incorporating your code; it looks pretty neat. For the 
midi, I wrote out the ornaments in a separate version using the 5/3 ratio, so 
as to be closer to the (1+sqrt 5)/2 = 1.618...; so this is 19 = 5+3+3+5+3, 
instead of the traditional 12 = 3+2+2+3+2, the latter which is used as a base 
for the engraving. It is from a common tune, Egejki Majki, played by the Boys 
from Bouf/Buf in the videos [1-2]. If someone wants to make snippet, please go 
ahead!

1. https://www.youtube.com/watch?v=NCDe41TtV-Y
2. https://www.youtube.com/watch?v=OfyZZ8OFvvs


 IrrationalMeter.ly 

\version 2.19.6

\header{
  title = Exampel of irrational meter
  subtitle = Egejki Majki, original meter 12 = 3+2+2+3+2
  composer = Traditional (Macedonia)
  copyright = © Hans Åberg
  tagline = ##f  % Removing Music engraving by LilyPond (version)
}


above = { \once \override Script #'script-priority = #-100 } below = { \once 
\override TextScript #'script-priority = #-100 }


irrtuplet = \once \override TupletNumber.text =  \markup \concat {
   1:1+
   \tiny √
   \hspace #-0.15
   \override #'(offset . -16)
   \override #'(thickness . 1.6)
   \underline 5
 }


music =  \new Staff {
  \tempo 4 = 176
  \key d \minor

  \time 12/8
  \set beatStructure = #'(3 2 2 3 2)

  \override Staff.TimeSignature.stencil = #ly:text-interface::print
  \override Staff.TimeSignature.text =
  \markup \override #'(baseline-skip . 0) \center-column \number {
\concat {
  8+2
  \tiny √
  \hspace #-0.2
  \override #'(offset . -25)
  \override #'(thickness . 1.6)
  \underline 5
}
8
  }


  \override TupletNumber.text = #tuplet-number::calc-fraction-text

  \relative c' {
   \repeat volta 2 {
  \once \override TextScript #'padding = #1.0
  \set Staff.midiInstrument = church organ
  \irrtuplet \tuplet 2/3 {f16 g a d,}  a'8\prall d,  a'8\prall d
\irrtuplet \tuplet 2/3 {c16 bes a g}  a8\prall g |
  \irrtuplet \tuplet 2/3 {f16 g a c,}  a'8\prall c,  a'8\prall d
\irrtuplet \tuplet 2/3 {c16 bes a g}  a8\prall r8 |
  \irrtuplet \tuplet 2/3 {b16 c d d}  d8 d
\above d8\trill^\markup{\flat}~d16 c
\irrtuplet \tuplet 2/3 {c16 d c bes}  a8\prall g |
  \irrtuplet \tuplet 2/3 {f16 g e g}  f8\prall e  \times 2/3 {d8 e c}
\irrtuplet \tuplet 2/3 {d16 e f g}  a8\prall g |
  \irrtuplet \tuplet 2/3 {f16 g e g}  f8\prall e  \times 2/3 {d8 e c}
\irrtuplet \tuplet 2/3 {\above d8\trill^\markup{\flat}~d16 a}  d8-. r8 |
  }
}
  }


\score {
  \music
\layout {}
}


musicmidi =  \new Staff {
  \tempo 4. = 186
  \key d \minor

  \time 19/8
  \set beatStructure = #'(5 3 3 5 3)

  \override TupletNumber.text = #tuplet-number::calc-fraction-text

  \relative c' {
\repeat volta 2 {
  \once \override TextScript #'padding = #1.0
  \set Staff.midiInstrument = church organ
\once \override TupletNumber.text = 
  #(tuplet-number::fraction-with-notes 4 4.)
  \tuplet 2/5 {f16 g a d,}  \tuplet 2/3 {a'32 bes32 a16 d,8}  \tuplet 2/3 
{a'32 bes32 a16 d8}
\once \override TupletNumber.text = 
  #(tuplet-number::fraction-with-notes 4 4.)
\tuplet 2/5 {c16 bes a g}  \tuplet 2/3 {a32 bes32 a16 g8} |
  \tuplet 2/5 {f16 g a c,}  \tuplet 2/3 {a'32 bes32 a16 c,8}  \tuplet 2/3 
{a'32 bes32 a16 d8}
\tuplet 2/5 {c16 bes a g}  \tuplet 2/3 {a32 bes32 a16 r8} |
  \tuplet 2/5 {b16 c d d}  \tuplet 2/3 {d8 d}
\tuplet 2/3 {d32 es d es d16 c}
\tuplet 2/5 {c16 d c bes}  \tuplet 2/3 {a32 bes32 a16  g8} |
  \tuplet 2/5 {f16 g e g}  \tuplet 2/3 {f32 g f16 e8}  {d8 e c}
\tuplet 2/5 {d16 e f g}  \tuplet 2/3 {a32 bes32 a16 g8} |
  \tuplet 2/5 {f16 g e g}  \tuplet 2/3 {f32 g f16 e8}  {d8 e c}
\tuplet 2/5 {d32 es d es d16 a}  \tuplet 2/3 {d8-. r8} |
  }
}
  }



\score {
  \unfoldRepeats
\musicmidi
\midi {}
}




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Re: Irrational time signature and tuplets

2014-06-12 Thread Hans Aberg 
On 12 Jun 2014, at 22:30, Mark Stephen Mrotek carsonm...@ca.rr.com wrote:

 (1+sqrt 5)/2 = 1.618... is the golden ratio, phi. 
 https://en.wikipedia.org/wiki/Golden_ratio
 
 Do you know of other instances of this ratio in music?

The WP [1] mentions one other case where irrational in music is irrational in 
also the mathematical sense, and it is also a square root. However, I only 
found it after making this example:

The original meter [2] is written 12 = 3+2+2+3+2 subject to interpretation of 
the exact ratios, with duplets or quadruplets on the 3s, and one can also have 
triplets on the 2s, as in the example I posted. Write, as in dance notation, s 
= slow, q = quick; then the meter is s q q s q, with the original, written 
ratio s/q = 3/2.

What I did was setting s/q = x so that also (s + q)/s = x; this gives x + 1 = 
1/x, which is the defining property of the golden ration, as you can see in the 
upper right hand box in your reference [3].

Then, as LilyPond does not handle these irrational time values, the next step 
is to find rational approximations, which can be done via continued fractions 
[4]. To get the denominators, as in this reference, take the integral part of 
the number, invert the fractional part, and repeat. For the golden ratio x the 
formula 1/x = 1 + x will show the it is a sequence of 1s: 1, 1, 1, ... One can 
can see that this leads to the successive quotients of the Fibonacci series 
[5], 1, 1, 2, 3, 5, 8, 13, ..., where the next integer in the series is the sum 
of the two immediate preceding integers. This gives the approximations 1/1, 
2/1, 3/2, 5/3, 8/5, 13/8, ...

But the continued fractions above work with any irrational number. Another idea 
I used was making y = s/q equal to q/(s/2) = 2/y, because of the typical rhythm 
 s/2 s/2 q q s/2 s/2 q. This gives y = sqrt 2, and the continued fractions 
numbers are 1, 2, 2, 2, ..., giving rational approximations 1, 3/2, 7/5, ...

The traditional written value s/q = 3/2, x = (1+sqrt 5)/2 = 1.618..., and y = 
sqrt 2 =
1.414..., but in reality there is a lot of variation in the interpretation.

So one can play around with any mathematically irrational number. But the 
usability in music is another question.


1. https://en.wikipedia.org/wiki/Time_signature#Irrational_meters
2. https://en.wikipedia.org/wiki/Leventikos
3. https://en.wikipedia.org/wiki/Golden_ratio
4. https://en.wikipedia.org/wiki/Continued_fraction
5. https://en.wikipedia.org/wiki/Fibonacci_number



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Re: Irrational time signature and tuplets

2014-06-12 Thread Hans Aberg 
On 12 Jun 2014, at 22:28, Malte Meyn lilyp...@maltemeyn.de wrote:

 I’m not sure whether I understood correctly, but instead of writing something 
 like
 
 \tuplet 3/2 { c8 d e }
 
 and hiding the tuplet number (and bracket), you can simply write
 
 c8*2/3 d e

I haven’t used that latter notation, simply. Though LilyPond admits just a 
sequence of time value fractions, tuples come with beaming structure, so not 
really fractions 3/2 but proportions 3:2, different from 6:4, which might be a 
reason I didn’t use it. Also, I find these irregular meter tricky to get right 
in timing, so I copy in place the right \tuplet with brackets and then edit in 
the notes within the brackets. And, the use of \times is because it is some old 
code I was editing when getting this irrational meter idea.


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RE: Irrational time signature and tuplets

2014-06-12 Thread Mark Stephen Mrotek 
Hans,

Thank you for your detailed and informative response. I shall take time to
study it and the references that you provide.

Mark

-Original Message-
From: Hans Aberg [mailto:haber...@telia.com] 
Sent: Thursday, June 12, 2014 2:20 PM
To: Mark Stephen Mrotek
Cc: Malte Meyn; lilypond-user@gnu.org
Subject: Re: Irrational time signature and tuplets

On 12 Jun 2014, at 22:30, Mark Stephen Mrotek carsonm...@ca.rr.com wrote:

 (1+sqrt 5)/2 = 1.618... is the golden ratio, phi. 
 https://en.wikipedia.org/wiki/Golden_ratio
 
 Do you know of other instances of this ratio in music?

The WP [1] mentions one other case where irrational in music is irrational
in also the mathematical sense, and it is also a square root. However, I
only found it after making this example:

The original meter [2] is written 12 = 3+2+2+3+2 subject to interpretation
of the exact ratios, with duplets or quadruplets on the 3s, and one can also
have triplets on the 2s, as in the example I posted. Write, as in dance
notation, s = slow, q = quick; then the meter is s q q s q, with the
original, written ratio s/q = 3/2.

What I did was setting s/q = x so that also (s + q)/s = x; this gives x + 1
= 1/x, which is the defining property of the golden ration, as you can see
in the upper right hand box in your reference [3].

Then, as LilyPond does not handle these irrational time values, the next
step is to find rational approximations, which can be done via continued
fractions [4]. To get the denominators, as in this reference, take the
integral part of the number, invert the fractional part, and repeat. For the
golden ratio x the formula 1/x = 1 + x will show the it is a sequence of 1s:
1, 1, 1, ... One can can see that this leads to the successive quotients of
the Fibonacci series [5], 1, 1, 2, 3, 5, 8, 13, ..., where the next integer
in the series is the sum of the two immediate preceding integers. This gives
the approximations 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, ...

But the continued fractions above work with any irrational number. Another
idea I used was making y = s/q equal to q/(s/2) = 2/y, because of the
typical rhythm  s/2 s/2 q q s/2 s/2 q. This gives y = sqrt 2, and the
continued fractions numbers are 1, 2, 2, 2, ..., giving rational
approximations 1, 3/2, 7/5, ...

The traditional written value s/q = 3/2, x = (1+sqrt 5)/2 = 1.618..., and y
= sqrt 2 = 1.414..., but in reality there is a lot of variation in the
interpretation.

So one can play around with any mathematically irrational number. But the
usability in music is another question.


1. https://en.wikipedia.org/wiki/Time_signature#Irrational_meters
2. https://en.wikipedia.org/wiki/Leventikos
3. https://en.wikipedia.org/wiki/Golden_ratio
4. https://en.wikipedia.org/wiki/Continued_fraction
5. https://en.wikipedia.org/wiki/Fibonacci_number



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