Draft:Sing free/Physics of music

Sing free/Great Gate of Kyiv (ear training) // Sing free/Physics of music

When you strike a bell it oscillates at a variety of frequencies associated with what are called normal modes. Vibrations of a circular membrane explains how all normal mode frequencies can all be expressed in terms of ${\displaystyle f_{0}=c/D}$. Here, ${\displaystyle D}$ is the diameter and ${\displaystyle c}$ is the speed of surface waves on a drum of infinite extent. The structures of the five modes with the lowest frequencies are shown below:

• A website that magically combines readability and rigor "The Well-Tempered Timpani" by Richard Jone, at https://wtt.pauken.org. In particular, a far superior rendition of the images that follow can be found here.
• See also "Pseudo-Degeneracy in Handbell Modes," by Cha, Sungdo, John Buschert, and Daniel King. arXiv preprint arXiv:1004.0491 (2010).

• 
  ${\displaystyle f_{01}=2.4048f_{0}}$
• 
  ${\displaystyle f_{11}=3.8317f_{0}}$
• 
  Mode ${\displaystyle f_{21}=5.1356f_{0}}$
• 
  ${\displaystyle f_{02}=5.5201f_{0}}$
• 
  Mode ${\displaystyle f_{12}=7.0156}$

Three of the five modes shown above exhibit two-fold degeneracy. All involve the presence of symmetry lines (nodes) that pass through the center of the circular drumheads. The subscripts in the labels of mode frequency, ${\displaystyle f_{mn}}$ identify how many node lines are present. Here a The symmetry lines of only on are marked. The other two degenerate modes can be identified by the presence of a node line that crosses the center.^[1]

Shown below are the top five hits from a Google search using the words: wave and speed

1. flexbooks.ck12.org: MisleadingN "Wave speed is the distance a wave travels in a given amount of time."
2. physicsclassroom.com: YCorrect "In the case of a wave, the speed is the distance traveled by a given point on the wave (such as a crest) in a given interval of time."
3. en.wikipedia.org: YCorrect & complete: "Wave speed is a wave property, which may refer to (...)"
4. compadre.org: MisleadingN "The wave speed, v, is how fast the wave travels..."
5. omnicalculator: MisleadingN "Wave speed is the speed at which the wave propagates. We can also define it as the distance traveled by the wave in a given time interval."

This discussion was lifted from the Wikipedia article Acoustic resonance^[2]. That article also presents the case for a semi-open tube, which can serve a highly simplified model of how a horn works. See phys.unsw.edu.au/jw/brassacoustics models that are more accurate but far more complicated.

The table below shows the displacement waves in a cylinder closed at both ends. Note that the air molecules near the closed ends cannot move, whereas the molecules near the center of the pipe move freely. In the first harmonic, the closed tube contains exactly half of a standing wave (node-w:antinode-node). Considering the pressure wave in this setup, the two closed ends are the antinodes for the change in pressure Δp; Therefore, at both ends, the change in pressure Δp must have the maximal amplitude (or satisfy ∂(Δp)/∂x = 0 in the form of the Sturm–Liouville formulation), which gives the equation for the pressure wave: ${\displaystyle \Delta p(x,t)=p_{\text{max}}\cos \left({2\pi x \over \lambda }\right)\cos(\omega t)}$. The intuition for this boundary condition ∂(Δp)/∂x = 0 at x = 0 and x = L is that the pressure of the closed ends will follow that of the point next to them. Applying the boundary condition ∂(Δp)/∂x = 0 at x = L gives the wavelengths of the standing waves:

${\displaystyle \lambda ={\frac {2L}{n}};n=1,2,3,...}$

And the resonant frequencies are

${\displaystyle f={\frac {v}{\lambda }}={\frac {nv}{2L}}.}$

Frequency	Order	Name 1	Name 2	Name 3	Wave representation	Molecular representation
1 · f =   440 Hz	n = 1	1st partial	fundamental tone	1st harmonic
2 · f =   880 Hz	n = 2	2nd partial	1st overtone	2nd harmonic
3 · f = 1320 Hz	n = 3	3rd partial	2nd overtone	3rd harmonic
4 · f = 1760 Hz	n = 4	4th partial	3rd overtone	4th harmonic

• /LilyPond to Audacity documents an awkward effort to link wikimedia's LilyPond to Audacity. I had to create spreadsheets and manually create the Audacity files. We need a better way to do this because it would be helpful for people learning to sing if they could "see" their voice through Audacity.
• /Piano versus orchestral compares both version of the piece.

• Click for 2Hz beat frequency 'emerging' from tones at 220 and 222 Hz. Also, szynalski.com can be used to open two tone generators in adjacent tabs.

https://docs.google.com/spreadsheets/d/14hSwf9v3Yw8u5910LVBVZgX2WLhUjTO50nPl_VKXvJ8/edit#gid=0

Averaging over "power"

${\displaystyle y=y(t),\;p=y^{2},\;E(t)=\int _{0}^{t}p(s)ds.}$

For a passage of duration ${\displaystyle \tau }$ plot:

${\displaystyle f(t)={\frac {E(t)-{\bar {p}}t}{\tau }},}$

where, ${\displaystyle {\bar {p}}=E(\tau )/\tau }$ is the time-averaged "power", with the average taken over the entire duration, ${\displaystyle \tau }$, or the interval (or chord). The factor ${\displaystyle \tau }$ in this definition of ${\displaystyle f}$ ensures that the ${\displaystyle f}$ has the dimensions of "power". If the roughness of a mild or severe dissonance is associated with fluctuations in power, then it is reasonable to define a different kind of "power", ${\displaystyle \langle p\rangle }$, that averages over many cycles of musical note, but is shorter than any amplitude fluctuation (beat) that a person might note. A typical musical note might have a frequency of ${\displaystyle 100\,{\text{Hz}}}$, and a badly out of tune pair of notes might create a beat frequency of ${\displaystyle 10\,{\text{Hz}}}$. We wish to compare ${\displaystyle f}$ with ${\displaystyle \langle p\rangle }$, or more specifically with,

${\displaystyle g(t)=\langle p\rangle -{\bar {p}},}$

since both ${\displaystyle f}$ and ${\displaystyle g}$ have long-term time averages of zero.

Since our interval is finite, we do a discrete Fourier series on "power". For simplicity we inc\lude only sine waves:

${\displaystyle p(t)=\sum _{n}c_{n}sin\left({\frac {n\pi t}{\tau }}\right)}$

${\displaystyle E=-\sum _{n}{\frac {\tau c_{n}}{n\pi }}cos\left({\frac {n\pi t}{\tau }}\right)+{\text{C}}}$ ${\displaystyle =\sum _{\omega _{n}}\left(-\omega _{n}\tau +1\right)c_{n}\sin(\omega _{n}t)\approx -\sum _{\omega _{n}}\omega _{n}\tau c_{n}\sin(\omega _{n}t)}$

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Creating Mussorgsky's score in Audacity is time consuming. Instead just settle for the first 5 chords (as he wrote them.)

<score sound="1"> 
{\language "english" 
\new PianoStaff << \new Staff 
{\set Staff.midiInstrument = #"piano" 
\clef treble \tempo 4 = 120 \time 4/4  \key ef \major
<ef' g' bf' ef''>1            %ef
<f'  bf' d'' f''>             %bf
<g' bf' ef'' g''>           %ef
<ef' g' c'' ef''>          %c:m
<f'  bf' d'' f''>            %bf
} 
\new Staff  { \set Staff.midiInstrument = #"piano" 
\clef bass \key ef \major
<g, bf, ef g>1           %ef
<f, bf, d f>             %bf
<ef, g, bf, ef>         %ef
<c, ef, g, c>            %c:m
<d, f, bf, d>            %bf
}         >>      }</score>

This logs a sequence of Lilipond scores I used to learn the system. This sequences uses absolute notation for pitch, which renders it useful for constructing audicity sound files.

Create Lilipond "Score" extension

This shows how to use \new PianoStaff << \new Staff ... \new Staff ...> to create a piano score.

<score sound="1"> 
{\language "english" 
\new PianoStaff << \new Staff 
{\set Staff.midiInstrument = #"piano" 
\clef treble \tempo 4 = 120 \time 4/4  \key ef \major
<ef' g' bf' ef''>1            %ef
<f'  bf' d'' f''>             %bf
<g' bf' ef'' g''>2            %ef
<ef' g' c'' ef''>4 (<g' g''>) %c:m
<f'  bf' d'' f''>2            %bf
<bf bf'>                      %"bf*
} 
\new Staff  { \set Staff.midiInstrument = #"piano" 
\clef bass \key ef \major
<g, bf, ef g>1           %ef
<f, bf, d f>             %bf
<ef, g, bf, ef>2         %ef
<c, ef, g, c>            %c:m
<d, f, bf, d>            %bf
<bf,>                    %bf*
}      >> }</score>

Simplify using LilyPond \chord

comments go here

<score sound="1"> 
{\language "english" 
\new PianoStaff << \new Staff 
{\set Staff.midiInstrument = #"piano" 
\clef treble \tempo 4 = 120 \time 4/4  \key ef \major \partial 8
r8
<ef' g' bf' ef''>1            %ef
<f'  bf' d'' f''>             %bf
<g' bf' ef'' g''>2            %ef
<ef' g' c'' ef''>4 (<g' g''>) %c:m
<f'  bf' d'' f''>2            %bf
<bf bf'>                      %"bf*
} 
\new Staff  { \set Staff.midiInstrument = #"piano" \partial 8
\clef bass \key ef \major
r8 \grace{g,8}
<g, bf, ef g>1 \grace{f,8}          %ef
<f, bf, d f>1  \grace{ef,8}            %bf
<ef, g, bf, ef>2 \grace{c,8}        %ef
<c, ef, g, c>2  \grace{d,8}          %c:m
<d, f, bf, d>2            %bf
<bf,>2                    %bf*
}      
>> }</score>

Simplify using LilyPond \chord

comments go here

<score sound="1"> 
{\language "english" 
\new PianoStaff << \new Staff\chordmode 
{\set Staff.midiInstrument = #"piano" 
\clef treble \tempo 4 = 120 \time 4/4  \key ef \major 
ef1            %ef
bf1           %bf
ef2            %ef
c2:m           %c:m
bf2           %bf
r2                  %"bf*
} 
\new Staff \chordmode  { \set Staff.midiInstrument = #"piano" 
\clef bass \key ef \major
ef,,1          %ef
bf,,1            %bf
ef,,2         %ef
c,,2:m           %c:m
bf,,2          %bf
r2                   %bf*
}      >> }</score>

Add melody to 2

comments go here

<score sound="1"> 
{\language "english" 
\new PianoStaff << 
\new Staff{\set Staff.midiInstrument = #"piano" 
\clef treble\tempo 4 = 120 \time 4/4  \key ef \major 
% Top notes to complete the melody
ef''1     
r1
g''2
ef''4
g''4
r2 
bf'2
}
\new Staff\chordmode 
{\set Staff.midiInstrument = #"piano" 
\clef treble \tempo 4 = 120 \time 4/4  \key ef \major 
ef1            %ef
bf1           %bf
ef2            %ef
c2:m           %c:m
bf2           %bf
r2                  %"bf*
} 

\new Staff \chordmode  { \set Staff.midiInstrument = #"piano" 
\clef bass \key ef \major
ef,,1          %ef
bf,,1            %bf
ef,,2         %ef
c,,2:m           %c:m
bf,,2          %bf
r2                   %bf*
}      >> }</score>

Use alto clef on 1

comments go here

<score sound="1"> 
{\language "english" 
\set Staff.midiInstrument = #"piano" 
\clef alto \tempo 4 = 120 \time 4/4  \key ef \major 
<g, bf, ef g ef' g' bf' ef''>1_"Eb"
<f, bf, d f f'  bf' d'' f''>1_"Bb"
<ef, g, bf, ef g' bf' ef'' g''>2_"Eb"
<c, ef, g, c ef' g' c'' ef''>2_"Cm"
<d, f, bf, d f'  bf' d'' f''>2_"Bb"}</score>