Draft:Sing free/Physics of music

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

Harmonic seventh

Circular membrane

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 $f_{0} = c / D$ . Here, $D$ is the diameter and $c$ 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).

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

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, $f_{m n}$ 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]

beware the simple explanation

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

flexbooks.ck12.org: MisleadingTemplate:Cross "Wave speed is the distance a wave travels in a given amount of time."
physicsclassroom.com: Template:TickCorrect "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."
en.wikipedia.org: Template:TickCorrect & complete: "Wave speed is a wave property, which may refer to (...)"
compadre.org: MisleadingTemplate:Cross "The wave speed, v, is how fast the wave travels..."
omnicalculator: MisleadingTemplate:Cross "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."

Air modes in a closed tube (harmonic)

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 Template:Nowrap in the form of the Sturm–Liouville formulation), which gives the equation for the pressure wave: $Δ p (x, t) = p_{max} \cos (\frac{2 π x}{λ}) \cos (ω t)$ . The intuition for this boundary condition Template:Nowrap at Template:Nowrap and Template:Nowrap is that the pressure of the closed ends will follow that of the point next to them. Applying the boundary condition Template:Nowrap at Template:Nowrap gives the wavelengths of the standing waves:

λ = \frac{2 L}{n}; n = 1, 2, 3, . . .

And the resonant frequencies are

f = \frac{v}{λ} = \frac{n v}{2 L} .

Frequency	Order	Name 1	Name 2	Name 3
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

Subpages

/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.

Algebra of beat measurement

File:1.52 versus 1.50 resonance.png

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"

$y = y (t), p = y^{2}, E (t) = \int_{0}^{t} p (s) d s .$

For a passage of duration $τ$ plot:

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

where, $\bar{p} = E (τ) / τ$ is the time-averaged "power", with the average taken over the entire duration, $τ$ , or the interval (or chord). The factor $τ$ in this definition of $f$ ensures that the $f$ 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", $⟨ p ⟩$ , 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 $100 Hz$ , and a badly out of tune pair of notes might create a beat frequency of $10 Hz$ . We wish to compare $f$ with $⟨ p ⟩$ , or more specifically with,

$g (t) = ⟨ p ⟩ - \bar{p},$

since both $f$ and $g$ 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:

$p (t) = \sum_{n} c_{n} s i n (\frac{n π t}{τ})$

$E = - \sum_{n} \frac{τ c_{n}}{n π} c o s (\frac{n π t}{τ}) + C$ $= \sum_{ω_{n}} (- ω_{n} τ + 1) c_{n} \sin (ω_{n} t) \approx - \sum_{ω_{n}} ω_{n} τ c_{n} \sin (ω_{n} t)$

Lilypond

Simplify to show just chords

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

<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>

Template:Cob

Useful Lilypond scripts

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

Template:Image frame This shows how to use \new PianoStaff << \new Staff ... \new Staff ...> to create a piano score. Template:Cot

<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>

Template:Cob

Add gracenotes

Template:Image frame comments go here Template:Cot

<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>

Template:Cob

Using LilyPond \chord

Template:Image frame comments go here Template:Cot

<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>

Template:Cob

Add melody to chords

Template:Image frame comments go here Template:Cot

<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>

Template:Cob

Alto clef

Template:Image frame comments go here Template:Cot

<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>

Template:Cob

Spreadsheets

These spreadsheets were created on Google Chrome and exported online to wikitext

Large spreadsheet

Copy of original spreadsheet showing intermediary calculations Template:Cot

ratio	ef	just	equal	cent	*	ratio	bf	just	equal	cent	*	ratio	ef	just	equal	cent	*	ratio	c:m	just	equal	cent	*	ratio	bf	just	equal	cent
0.6250	g,	97.2272	97.9989	-14	*	0.5625	f,	87.5045	87.3071	4	*	0.5000	ef,	77.7817	77.7817	0	*	0.4167	c,	64.8181	65.4064	-16	*	0.4688	d,	72.9204	73.4162	-12
0.7500	bf,	116.6726	116.5409	2	*	0.7500	bf,	116.6726	116.5409	2	*	0.6250	g,	97.2272	97.9989	-14	*	0.5000	ef,	77.7817	77.7817	0	*	0.5625	f,	87.5045	87.3071	4
1.0000	ef	155.5635	155.5635	0	*	0.9375	d	145.8408	146.8324	-12	*	0.7500	bf,	116.6726	116.5409	2	*	0.6250	g,	97.2272	97.9989	-14	*	0.7500	bf,	116.6726	116.5409	2
1.2500	g	194.4544	195.9977	-14	*	1.1250	f	175.0089	174.6141	4	*	1.0000	ef	155.5635	155.5635	0	*	0.8333	c	129.6362	130.8128	-16	*	0.9375	d	145.8408	146.8324	-12
					*						*						*						*
2.0000	ef^	311.1270	311.1270	0	*	2.2500	f^	350.0179	349.2282	4	*	2.5000	g^	388.9087	391.9954	-14	*	2.0000	ef^	311.1270	311.1270	0	*	2.2500	f^	350.0179	349.2282	4
2.5000	g^	388.9087	391.9954	-14	*	3.0000	bf^	466.6905	466.1638	2	*	3.0000	bf^	466.6905	466.1638	2	*	2.5000	g^	388.9087	391.9954	-14	*	3.0000	bf^	466.6905	466.1638	2
3.0000	bf^	466.6905	466.1638	2	*	3.7500	d^^	583.3631	587.3295	-12	*	4.0000	ef^^	622.2540	622.2540	0	*	3.3333	c^^	518.5450	523.2511	-16	*	3.7500	d^^	583.3631	587.3295	-12
4.0000	ef^^	622.2540	622.2540	0	*	4.5000	f^^	700.0357	698.4565	4	*	5.0000	g^^	777.8175	783.9909	-14	*	4.0000	ef^^	622.2540	622.2540	0	*	4.5000	f^^	700.0357	698.4565	4

Template:Cob

Equal temperament

116.5409 233.0819 466.1638 are the b-flats in the extra quarter note Template:Cot

ef	equal	*	bf	equal	*	ef	equal	*	c:m	equal	*	bf	equal
g,	97.9989	*	f,	87.3071	*	ef,	77.7817	*	c,	65.4064	*	d,	73.4162
bf,	116.5409	*	bf,	116.5409	*	g,	97.9989	*	ef,	77.7817	*	f,	87.3071
ef	155.5635	*	d	146.8324	*	bf,	116.5409	*	g,	97.9989	*	bf,	116.5409
g	195.9977	*	f	174.6141	*	ef	155.5635	*	c	130.8128	*	d	146.8324
		*			*			*			*
ef^	311.1270	*	f^	349.2282	*	g^	391.9954	*	ef^	311.1270	*	f^	349.2282
g^	391.9954	*	bf^	466.1638	*	bf^	466.1638	*	g^	391.9954	*	bf^	466.1638
bf^	466.1638	*	d^^	587.3295	*	ef^^	622.2540	*	c^^	523.2511	*	d^^	587.3295
ef^^	622.2540	*	f^^	698.4565	*	g^^	783.9909	*	ef^^	622.2540	*	f^^	698.4565

Template:Cob

Just temperament

116.6726 233.3452 466.6905 are the b-flats in that extra quarternote Template:Cot

ef	just	*	bf	just	*	ef	just	*	c:m	just	*	bf	just
g,	97.2272	*	f,	87.5045	*	ef,	77.7817	*	c,	64.8181	*	d,	72.9204
bf,	116.6726	*	bf,	116.6726	*	g,	97.2272	*	ef,	77.7817	*	f,	87.5045
ef	155.5635	*	d	145.8408	*	bf,	116.6726	*	g,	97.2272	*	bf,	116.6726
g	194.4544	*	f	175.0089	*	ef	155.5635	*	c	129.6362	*	d	145.8408
		*			*			*			*
ef^	311.1270	*	f^	350.0179	*	g^	388.9087	*	ef^	311.1270	*	f^	350.0179
g^	388.9087	*	bf^	466.6905	*	bf^	466.6905	*	g^	388.9087	*	bf^	466.6905
bf^	466.6905	*	d^^	583.3631	*	ef^^	622.2540	*	c^^	518.5450	*	d^^	583.3631
ef^^	622.2540	*	f^^	700.0357	*	g^^	777.8175	*	ef^^	622.2540	*	f^^	700.0357

Template:Cob

References and links

Acoustics and Vibration Animations, Daniel A. Russell, Graduate Program in Acoustics, The Pennsylvania State University
https://www.acs.psu.edu/drussell/demos/membranesquare/square.html
Pseudo-Degeneracy in Handbell Modes.^[3]
For animated images of degenerate normal modes on a rectangular drumhead visit [1]
special:Permalink/2383304#Backup_LilyPond_Great_Gate_Kiev

See also User:Guy vandegrift/sandbox/01, w:Consonance and dissonance, w:Interval (music) (discussion), and:J. R. Soc. Interface (2008) 5, 1429–1434 (challenges Hemholtz)

Thomas D. Rossing and Robert Perrin, “Vibrations of bells,” Appl. Acoust. 20, 41-70 (1987).
T. D. Rossing “Acoustics of Bells” American Scientist, 72, 440-447 (1984)
Kristen Menou, Benjamin Audit, Xavier Boutillon, and Holger Vach, “Holographic study of a vibrating bell: An undergraduate laboratory experiment” Am. J. Phys. 66 380 (1998)
N McLachlan, BK Nigjeh, A Hasell, “The design of bells with harmonic overtones,” J. Acoust. Soc. Am. Volume 114, Issue 1, pp. 505-511 (July 2003)
Ralph T. Muehleisen and Anthony A. Atchley, “Fundamental azimuthal modes of a constricted annular resonator: Theory and measurement,” J. Acoust. Soc. Am. 109, 480–487 (2001).
R. Perrin, G. M. Swallowe, T. Charnley and C. Marshall, “On the debossing, Annealing and Mounting of bells,” J. Sound Vib. 227, 409-425 (1999).
Seock-Hyun Kim, Chi-Wook Lee and Jang-Moo Lee, “Beat characteristics and beat maps of the King Seong-deok Divine Bell, ” J. Sound Vib. 281, 21-44 (2005).
https://arxiv.org/ftp/arxiv/papers/1004/1004.0491.pdf
https://www.tablesgenerator.com/mediawiki_tables

↑ Physical and Chemistry Textbook Maps (chem.libretext.org)
↑ w:special:permalink/1109685946#Closed_at_both_ends
↑ Cha, Sungdo & Buschert, John & King, Daniel. (2010). Pseudo-Degeneracy in Handbell Modes. The Journal of the Acoustical Society of America. 117. 10.1121/1.4788550.

[1] Physical and Chemistry Textbook Maps (chem.libretext.org)

[2] w:special:permalink/1109685946#Closed_at_both_ends

[3] Cha, Sungdo & Buschert, John & King, Daniel. (2010). Pseudo-Degeneracy in Handbell Modes. The Journal of the Acoustical Society of America. 117. 10.1121/1.4788550.

[1]

[2]

[3]

Draft:Sing free/Physics of music

Contents

Harmonic seventh

Circular membrane

beware the simple explanation

Air modes in a closed tube (harmonic)

Subpages

Algebra of beat measurement

Lilypond

Simplify to show just chords

Useful Lilypond scripts

Create Lilipond "Score" extension

Add gracenotes

Using LilyPond \chord

Add melody to chords

Alto clef

Spreadsheets

Large spreadsheet

Equal temperament

Just temperament

References and links

Navigation menu

Draft:Sing free/Physics of music

Harmonic seventh

Circular membrane

beware the simple explanation

Air modes in a closed tube (harmonic)

Subpages

Algebra of beat measurement

Lilypond

Simplify to show just chords

Useful Lilypond scripts

Create Lilipond "Score" extension

Add gracenotes

Using LilyPond \chord

Add melody to chords

Alto clef

Spreadsheets

Large spreadsheet

Equal temperament

Just temperament

References and links

Navigation menu

Search