.Nothing escapes..

Harmony and Melody

The tuning of

classic music instrumentation

by means of

objective pitch measurement

"Mathematics swims seductively just below the surface of Music"

Prof. E. Eugene Helm, Univ. of Maryland, U.S.A
.

Nederlandstalige versie.

Version Française.

Home J. Broekaert.
.This text is a translation.
Your comments are welcome.

Abstract:
Tuning of classic music instruments, if not done by the ear, requires availability of measuring instruments capable of measuring the pitch of very complex waveforms. It is also required to know the pitches in function of the desired musical temperament. Both items are discussed in this text.
Original Text: Dutch Language.

1   General Introduction  /  Background.....(content)

This text is an attempt to contribute to more general knowledge of techniques used to obtain good musical temperaments when tuning classic musical instruments by means of instrumentation allowing for objective measurement of the pitches, and this because of following reasons:
  • It is not easy to find objective and reliable information on pitches to be obtained for every single note; a multitude of differing temperaments exists
  • Measuring instruments can still be improved for the case of tuning of very complex sounds; autocorrelation techniques seem to offer promising perspectives according to results of simulations that have been done, but SW packages based on this technology are not readily available
It is the hope that the spreading of this text can contribute and motivate in working out improvements for objective musical tuning techniques.
  • An appendix discusses briefly the reason why some musical temperaments are preferred.
  • Also the pitch measuring techniques are discussed briefly.
The most elementary elements of musical theory, such as for example the note naming conventions, are not discussed in this text.
For the explanation of some terms or points of interest one should consult encyclopaedias, or some of the numerous publications on musical theory that already exist.

2    Pitches.....(content)

2.1    General.....(content)

The pitches in a musical scale depend on the chosen musical temperament.
The most recognised temperaments and their matching musicals scales are:

Table 1: Pitches in Hertz, for some musical temperaments
Pythagoric 260,7 278,4 293,2 309,0 330,0 347,6 371,3 391,1 417,7 440,0 463,5 495,0 521,4
Natural 264,0 275,0 297,1 316,8 330,0 352,0 371,3 396,1 412,5 440,0 469,3 495,0 528,0
Equal 261,6 277,2 293,7 311,1 329,5 349,2 370,0 392,0 415,3 440,0 466,2 493,9 523,2
Meantone 263,2 275,0 294,2 314,8 329,0 352,0 367,8 393,5 411,2 440,0 470,8 491,9 526,4
Selection tuning of Kirnberger II
262,4
276,4
295,2
310,9
328,0
349,8
368,9
393,3
414,6
440,0
466,4
491,9
524,8
Selection tuning of Kellner 262,9 276,9 294,1 311,5 329,1 350,5 369,2 393,2 415,4 440,0 467,3 493,7 525,7
Selection tuning of  Kirnberger III 263,1 277,2 294,5 311,8 328,9 350,8 370,0 393,8 415,8 440,0 467,7 493,3 526,2
Temperament
c
cis
d
es
e
f
fis
g
gis
a
b
h
C

Studies of how the referred pitch values came into existence can be found in professional literature, such as:

2.2    The measurement of musical pitches....(content)

Pitches can be measured in Hertz (or cycles per second), but musical tuning instruments are often calibrated according to the equally tempered scale (see Appendix 1, 1.3), where a semi-tone is further divided in 100 cents or in comma’s (see Appendix 1, 1.1).
The following tables document the required values described above, useful for tuning.
The tables do not include the Pythagoric and natural scales, because these are not commonly used.

Table 2.1:  Equal temperament
Hertz
 261,6 277,2 293,7 311,1 329,5 349,2 370,0 392,0 415,3 440,0 466,2 493,9 523,2
Cents
0
100
200
300
400
500
600
700
800
900
1.000
1.100
1.200
Deviation in Hertz
0
0
0
0
0
0
0
0
0
0
0
0
0
Deviation in Cents
0
0
0
0
0
0
0
0
0
0
0
0
0
Deviation in Comma’s
0
0
0
0
0
0
0
0
0
0
0
0
0
Equal temperament
c
cis
d
es
e
f
fis
g
gis
a
b
h
C

Table 2.2: Meantone temperament
Hertz
 263,2 275,0 294,2 314,8 329,0 352,0 367,8 393,5 411,2 440,0 470,8 491,9 526,4
Cents
10
86
203
320
397
514
590
707
783
900
1017
1093
1210
Deviation in Hertz
+ 1,2
- 2,2
+ 0,5
+ 3,7
- 0,5
+ 2,8
- 2,2
+ 1,5
- 4,1
0
+ 4,6
- 2,0
+ 3,2
Deviation in Cents
+ 10
- 14
+ 3
+ 20
- 3
+ 14
- 10
+ 7
- 17
0
+ 17
- 7
+ 10
Deviation in Comma’s
+ 0,5
- 0,6
+ 0,1
+ 0,9
- 0,1
+ 0,6
- 0,5
+ 0,3
- 0,8
0
+ 0,8
- 0,3
+ 0,5
Meantone
c
cis
d
es
e
f
fis
g
gis
a
b
h
C

Tabel 2.3: Well temperament of Kellner
Hertz
 262,9 276,9 294,1 311,5 329,1 350,5 369,2 393,2 415,4 440,0 467,3 493,7 525,7
Cents
8
98
203
302
397
506
596
705
800
900
1004
1099
1208
Deviation in Hertz
+ 1,2
- 0,3
+ 0,5
+ 0,4
- 0,5
+ 1,3
- 0,8
+ 1,2
+ 0,1
0
+ 1,2
- 0,2
+ 2,5
Deviation in Cents
+ 8
- 2
+ 3
+ 2
- 3
+ 6
- 4
+ 5
+ 0
0
+ 4
- 1
+ 8
Deviation in Comma’s
+ 0,4
- 0,1
+ 0,1
+ 0,1
- 0,1
+ 0,3
- 0,2
+ 0,2
+ 0,0
0
+ 0,2
- 0,0
+ 0,4
Well temperament of Kellner
c
cis
d
es
e
f
fis
g
gis
a
b
h
C

Tabel 2.4: Well temperament of Kirnberger III
Hertz
 263,1 277,2 294,5 311,8 328,9 350,8 370,0 393,8 415,8 440,0 467,7 493,3 526,2
Cents
10
100
205
304
396
508
600
708
802
900
1006
1098
1210
Deviation in Hertz
+ 1,5
0
+ 0,8
0,7
- 0,7
+ 1,6
0
+ 1,8
0,5
0
+ 1,5
- 0,6
+ 2,9
Deviation in Cents
+ 10
0
+ 5
+ 4
- 4
+ 8
0
+ 8
2
0
+ 6
- 2
+ 10
Deviation in Comma’s
+ 0,4
0
+ 0,2
+ 0,2
- 0,2
+ 0,3
0
+ 0,4
+ 0,1
0
+ 0,3
- 0,1
+ 0,4
Well temperament of Kirnberger III
c
cis
d
es
e
f
fis
g
gis
a
b
h
C
.
.
.
Recommendations for the choice of temperaments:

  • Classic Baroque Music: meantone for organs en clavichords
  • Classic Music in General: Kirnberger (III)
  • Equal temperament: there is, as a matter of fact, no musical ground to choose for this temperament

    • .
    .
    See further: prof. H. Kelletat, or appendix 1
.
.
 

2.3    Instrumentation.....(content)

It should be possible to achieve good tuning, using the above stated values in combination with instrumentation that is commonly available on the market.
Nowadays it is also possible to tune, by using a PC in combination with a sound-card and suitable SW.
A suitable SW, for example, can be downloaded from:
Website http://www.tunelab-world.com
Often problems are encountered in tuning “difficult” sounds such as: the lowest and highest notes of a piano, sounds with high harmonic content, vibrato’s, sounds of a complex nature such as is the case of percussion instruments.
With thick strings of air columns (thus with low notes) the timbre may deviate from a normal harmonic structure, because the string or air column does not sufficiently well correspond with the theoretical ideal model.
For short strings (thus for high notes) the timbre may deviate form a normal harmonic structure, because of the rigidity of the string; the string behaves partly as if it were a vibrating bar.
Because of the above it has been possible to determine that after tuning by the ear, the base frequencies of the highest and lowest piano notes lay lower, respectively higher than normally expected, the lowest and highest octaves are stretched, as can be seen in the figure below (taken from "The Equal Tempered Scale and Pecularities of Piano Tuning", Jim Campbell, http://www.precisionstrobe.com/apps/pianotemp/temper.html).
.
Note:
In comparison with vibrating strings and air pipes (with only one vibrating dimension), the harmonics of percussion instruments are often quite distant from an integer multiple of the base-harmonic, as a result of their structures built around membranes (two vibrating dimensions) or vibrating corpses (three vibrating dimensions): those sounds often “resound”.
Unless one has access to very complex instrumentation, it will be likely that in the case of complex sounds one will ultimately have to tune by the ear .
 
Trick for tuning low notes:
In the lowest octaves of a piano or various other instruments, it might be preferable in case pf problems, to tune on the third harmonic (the quinte) rather than the fundamental: reason is that the third harmonic of a note in a low octave usually is very strong and might even surpass the fundamental, as can be seen in figure 1 of appendum 2.
For very fine tuning based on this trick, it is necessary to expand the tables 2.2 to 2.4 with data required for the tuning of the third harmonic. The third harmonic does indeed not always coincide with the quinte.
Extended tables:

Appendix 2 discusses pitch measuring techniques based on autocorrelation, which should allow for very precise pitch measurements, also for very complex sounds.

  • Link to :     Appendix 1     Properties of musical temperaments
  • Link to :     Appendix 2     Pitch measurement techniques

3    Conclusions.....(content)

I hope that in time this text might contribute to the objective set here:

"The tuning of classic music instruments by means of objective pitch measurements"

Also if very complex sounds have to be tuned.

A good quality and user-oriented solution fully compliant with said expectation is not yet available.
It therefore makes sense to work on the SW development that can satisfy the herewith expressed expectations.
Any contribution that can lead to the elaboration and spreading of SW that can be made publically available, and that is meant for measuring pitches of musical sounds by means of autocorrelation techniques, or other techniques that allow for tuning of very complex sounds, will be accepted with great appreciation.

Comments on this publication are always welcome.
Correspondence about this subject is possible with:
 

Ir. Johan Broekaert
bewi, richting elektronica, KULeuven 1967
Nieuwelei, 52
B 2640 Mortsel
Belgium

tel 32 - 3 - 455.09.85


Following persons contributed in the past years in the elaboration of this publication:

My wife Rosette Devriendt, and our children, with whom the exchange of ideas about this subject always lead to interesting discussions.
Mrs. J. Jacobs-Waayeret, professional pianist and singer, teacher of solfege and piano at the musical academy of Mortsel, who gave me the first introductions to more specialised literature
F. Cuypers, orchestral director, director of the musical academy of Mortsel.
Mevr. C. Vandervelden, teacher of solfege and piano at the musical academy of Mortsel
Ing. W. Palmans, Ir. M. Boets and other colleagues in my working environment Agfa-Gevaert N.V., whereby I fear the risk of mentioning more because of the possibility I would forget some.
Mr. J. Mestdagh, composer, orchestral director and teacher of contrapunt en fugue at the musical conservatory of Gent and Brussel, and ir. M. Van Cauwenberghe who introduced me to him.
Mevr. M. Dejonghe-Roberts and Ir. J.J. Caufriez. They assisted in translating this text in English and French, their respective mother tongues.


Note:
The subject has greatly interested me since 1983.
The subject of this internet publication has already been discussed in a paper of mine, with very limited distribution :

"Het Muzikaal Stemmen (het temperen) van Klassieke Muziekinstrumenten op basis van Elektronische Toonhoogtemetingen"

Written by J. Broekaert, on November 26-th 1990 in Mortsel.

The referred text already contains thoughts about possible application of autocorrelation techniques, then just as it is now, by publication of results of a simulation on a spreadsheet.


Content

The tuning of classic music instrumentation by means of objective pitch measurement

1    Introduction..
2    Pitches
    2.1    General
    2.2    The measurement of musical pitches
    2.3    Instrumentation
3    Conclusions


Appendix 1:    Properties of musical temperaments...
    1    Elementary musical properties
        1.1    The Pythagoric temperament
        1.2    The natural (pure) temperament
        1.3    The equal temperament
        1.4    The meantone temperament
        1.5    Selected temperaments
        1.6    Well temperament
        1.7    To probe further
    2    Musico-technical analysis
        2.1    Basic data
        2.2    Characteristics of a number of intervals: overview
        2.3    Characteristics of a number of intervals: circle of fifths
        2.4    Characteristics of intervals: Graphic comparison

Appendix 2:    Pitch measurement techniques...
    1    Application of existing instrumentation
    2    Possibilities for further developments
    3    Practical implementation of autocorrelation techniques

Version 2002-07-26