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Harmony and Melody
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The tuning of classic music instrumentation by means of objective measurement of pitches
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Appendix 1:

Properties of musical temperaments...(content)

.This text is a translation.
Your comments are welcome..

1    Elementary musical properties.....(content)

The main elementary properties of the main musical temperaments are discussed below.
Following webpage leads to a very extended bibliographic list:

1.1 The Pythagoric temperament.....(see also table 1).....(content)

The Pythagoric temperament is obtained by building pure fifths or fourths consecutively (having pitches proportional with 3/2 respectively 4/3), but is not used in practice, because this temperament leads to differing enharmonic notes (e.g. cis and des), leading to problems with construction and concept of instruments and with the transposition of music.
The Pythagoric temperament has generated the fundamentals of music theory, such as:
  • The sequence in which notes are produced: they are produced by consecutive building of fifths, and this sequence governs the sequence also of notation of the symbols for sharp notes (fa, do, sol, re, la, mi, si) and flat notes (si, mi, la, re, sol, do, fa) in musical scores.
  • The difference between enharmonic notes: according to this temperament a dis (re sharp) has a somewhat higher pitch than an es (mi flat), etc ...
  • The determination of musical intervals in comma’s : a full tone has 9 comma’s, a chromatic semitone 5 comma’s, and a diatonic semitone 4 comma’s.
  • etc…
The relative positioning of the notes and the close vicinity to each other of enharmonic notes in the Pythagoric temperament, have allowed for evolution of music from pentatonic music (5 notes per octave) over diatonic music (7 notes per octave) to chromatic and dodecaphonic music (12 notes per octave).
It is quite probable that classic music never shall exceed systems with 12 notes per octave for key-instruments, if executability of the music by musicians is desired (see note below).
Note:
The subdivision of intervals as described above is not 100 % correct from a pure mathematical point of view, but is a very good approximation.
The Pythagoric comma is defined as the interval between two enharmonic notes, for example dis (re sharp) and es (mi flat).
It can be verified that in the Pythagoric temperament all distances between enharmonic notes are equal, and have the value:
1 Pythagoric comma..=..3^12./.2^19..=..1,013643286....(^ signifies "raised to the power ...")
With the above obtained value of a comma it is easy to verify how may times a comma fits ("aproximately") in a full tone, or in a chromatic or diatonic semi-tone.
Enharmonic notes, and therefore also the Pythagoric comma, come to existence because of the fact that in the Pythagoric temperament it is never possible to come back to the original pitch one departed from, when creating consecutive notes. This has to do directly with the mathematical property:
3^m./.2^n....can never....=....1.......(if....m....and....n....are.....integer....numbers).
Further (mathematical) peculiarities:
  • The Pythagoric comma is created at time a 13-th note is created (this is when 3 is raised to the 12-th power). It can be verified that 12 is the first power of three where the expression
      • .
      3^m / 2^n
      .
    comes very close to a value of 1, for combinations of integer values for m and n (3^12 / 2^19..=..1,013643265...)
  • It can be verified one has 53 commas in one octave (5x9 + 2x4 = 53)

    • It can be verified as well that for m = 53 the expression .....3^m./.2^n..... reaches another better approximation of 1 (3^53./.2^84..=..1,002090314...). This approximation is much better than the one obtained with m = 12.
    Construction of music instruments with 53 notes has been proposed in very early days (a.o. by Mercator).
  • Also m = 41 leads to a good approximation of 1: a little worse than for m = 12, but not at all as good as for m = 53 (3^41./.2^65..=..0,98602548...)
  • Following values of m lead to even better approximations of 1 for the expression ....3^m./.2^n.:
    • m = 306.....3^306./.2^485..=..0,998978283...
    • larger values of m: larger values exist, but could not be calculated at ease with available calculation means (at m = 647, or n = 1024 the system stopped)
    The distance to better approximations grows very rapidly.
    In all cases one obtains values for m that are completely useless for the conception of music instruments that can be played by musicians.
  • The value m = 24 also gives a very good approximation, but not as good as for m = 12:
    • m = 24....3^24./.2^38..=..1,027472668
    Experiments have been made with music and instruments with 24 notes in an octave (with intervals of one quarter tone), apparently without convincing results thus far.
.

1.2    The natural (pure) temperament.....(see also table 1).....(content)

One obtains the natural temperament by building pure thirds (proportion 5/4 and 6/5), at some positions in the octave where this is possible or desirable, at the expense of loss of purity of a number of fourths and fifths.
This temperament is not used in practice, for the same reasons as for Pythagoric temperament.
The rich harmony of this temperament, obtained because of its pure thirds, has been the basis for further development of the meantone temperament and numerous selection temperaments.
Note:
In this temperament it can be observed that enharmonic notes are positioned so that the sharps have a lower pitch than the corresponding enharmonic flats. Verification is easy according to pitch values in table 1 (and this property is contrary to the similar property in the Pythagoric temperament, where sharps have a higher pitch than the corresponding enharmonic flats).

1.3    The equal temperament.....(content)

One obtains equal tempering by dividing an octave in twelve strictly equal parts, according to a geometric series of numbers:
  • The reason of this series therefore is the twelfth radical of 2, this is: 1,059463… .
  • Simple tuning instruments are calibrated according to this scale, which leads to the fact that this temperament is often applied in the case of electronic tuning.
The name of this temperament originates from its property that one always obtains the same ratio between the frequency of the notes and the frequency with which they beat.

The equal temperament has mainly merits as a measuring standard, for the case of calculating, measuring and comparing intervals between notes. In order to raise the resolution, the octave is further divided in 100 cents for a semitone, thus 1200 cents in total.
It is a mathematical casuality that pitches of notes in the equal temperament differ only slightly from pitches one obtains when applying "classic" temperaments.

Musical application of equal temperament has the following most important consequences:

  • There are no more pure thirds, neither sharp nor flat, but the fourths and the fifths remain as good as pure
  • The musical character inherent to the different tonalities is completely lost, because all intervals always have the same absolute interval ratio (because of the equal temperament)
  • The harmony is somewhat less brilliant or coloured, because of the lack of pure thirds

1.4    The meantone temperament....(content)

The meantone temperament has its origins in the natural (pure) tempering.
As many as possible pure thirds are installed in the meantone temperament, at wel defined places (see table 3.2), and the sharp third is devided in two equal full tones, what declares the name of this temperament.
The meantone temperament allows for simple construction of musical instruments (only 12 keys per octave). The meantone furthermore allows for limited transposition of scores. The transposition is mainly limited by the existence of a "wolf"-fifth (on gis or as), a fifth that is very different of the normal pure fifth. On the other hand, the "wolf" allows for special musical effects, mostly mournful or tragic.

The meantone is used for organs and harpsicords and belongs in some sense to a specific culture or time: for some music it is important to use this temperament, if one wants to experience the music reproduction as it was performed at the time of composition.

1.5    Selected temperaments.....(content)

Selected temperaments originate from generalised research to escape from the limitations of meantone. Numerous selected temperaments have been developed, mainly in the period of Bach, sometimes earlier as well, and the comparison of the several different temperaments should have been the issue of numerous salon discussions in those days (cfr. Prof. H. Kelletat).
A brief listing of historical references includes (see also "Zur musikalischen Temperatur", prof. H. Kelletat): Schlick (1511), Silbermann, Mattheson, Werckmeister I, II, III (1691) en IV, Kirnberger I, II en III (1779), Neidhardt I, II en III.
And recently: Kelletat (1966), Kellner (1977) en Billeter (1979) (see also "Zur musikalischen Temperatur", prof. H. Kelletat).

Tuning of music instruments by ear always leads to tuning according to meantone or to one of the listed selected temperaments or other related temperaments (cfr. Prof. H. Kelletat).

1.6    Well temperament.....(content)

A number of selected temperaments have the property to be applicable with good harmony to all tonalities, and this property is called well temperament.

The referred books of Prof. H. Kelletat are an attempt to give evidence that the temperaments of Kirnberger were the norm during Bach's life, and that there is a relation between those temperaments and "Das Wohltemperierte Klavier", a musical master work of Bach.
Still according to Prof. H. Kelletat, and despite misunderstandings in the last two centuries, it is not right to put on a par the well temperament and the equal temperament (equal temperament: see 1.3).

Well temperaments are among others: Kirnberger I, II en III (1779), Kelletat (1966), Kellner (1977), Billeter (1979) (see "Zur musikalischen Temperatur", prof. H. Kelletat).

A first and more profound introduction with the temperament of Kellner is possible by browsing to his internet pages: 

1.7    Further Studies.....(content)

The fundamental discussion of the several musical temperaments is part of high level musical education in general.
Very extensive literature on this subject is available and it is part of the educational program of music conservatories. We therefore do not further discuss it here.

2    Musico-technical analysis.....(content)

2.1    Basic data.....(content)

Basic data on scales, according to publications in general, are given below to help in further reading and understanding of this text in general.

Table 1: Intervals of scales (^: "^" means "raised to the power …")
Pythagoric
Natural
 Meantone Kirnberger II Temperament
2/1
2/1
 2/1 2/1
C
3^12/2^18
125/ 64
 . .
his
243/128
15/8
 15/8 - 1/4c 15/8
h
16/9
16/9
 16/9 + 1/2c 16/9
b
3^10/2^15
225/128
 . .
ais
27/16
5/3
 5/3 + 1/4c 161/96
a
128/81
8/5
 . .
as 
3^8/2^12
25/16
 25/16 128/81
gis
3/2
3/2
 3/2 - 1/4c 3/2
g
1024/ 729
64/45
 . .
ges
729/512
45/32
 25/18 + 1/2c 45/32
fis
4/3
4/3
 4/3 + 1/4c 4/3
f
3^11/2^17
125/96
 . .
eis
81/64
5/4
 5/4 5/4
e
32/27
6/5
 6/5 - 1/4c 32/27
es
3^9/2^14
75/64
 . .
dis
9/8
9/8
 9/8 - 1/2c 9/8
d
256/243
16/15
 . .
des
3^7/2^11
25/24
 25/24 + 1/2c 256/243
cis
1/1
1/1
 1/1 1/1
c

It is possible to derive a number of "classic" intervals from the above table.

Together with the standard value of the interval, the deviations corresponding with a semi (musical) comma have been calculated also (=  ± 0,656 %).

Table 2: Pitch ratios of a number of intervals
Ratio + 1/2 comma
1,073664
1,118400
1,132380
1,207872
1,258200
1,192960
1,273928
1,509840
Calculated ratio 
 16/15 =
1,06667
 10/9 =
1,11111
 9/8 =
1,125
 6/5 =
1,2
 5/4 =
1,25
 32/27 =
1,185185
 81/64 =
1,265625
 3/2 =
1,5
Ratio  - 
1/2 comma
1,059669
1,103822
1,117620
1,192128
1,241800
1,177410
1,257323
1,490160
Interval
Semi tone
Small second
Large second
Pure flat third
Pure sharp third
Small Pythagoric third
Large Pythagoric third
Fifth

2.2    Characteristics of a number of intervals: summary.....(content)

The following calculation of a number of intervals, has been done using the data on scales as published in the referred books of prof. H. Kelletat. A summary of said data was already given in paragraphs 2.1, 2.2 of the main text and in paragraph 2.1 of this appendix.
The object of the tables below is to be a graphic aid in overseeing the characteristics of the main intervals, in function of the installed musical temperament.

Tabel 3.1: equal temperament
Semi tone
1,059
1,059
1,059
1,059
1,059
1,059
1,059
1,059
1,059
1,059
1,059
1,059
1,059
Small second
1,123
1,123
1,123
1,123
1,123
1,123
1,123
1,123
1,123
1,123
1,123
1,123
1,123
Large second
1,123
1,123
1,123
1,123
1,123
1,123
1,123
1,123
1,123
1,123
1,123
1,123
1,123
Flat third
1,183
1,183
1,183
1,183
1,183
1,183
1,183
1,183
1,183
1,183
1,183
1,183
1,183
Sharp third
1,260
1,260
1,260
1,260
1,260
1,260
1,260
1,260
1,260
1,260
1,260
1,260
1,260
Fifth
1,498
1,498
1,498
1,498
1,498
1,498
1,498
1,498
1,498
1,498
1,498
1,498
1,498
Equal temperament
c
cis
d
es
e
f
fis
g
gis
a
b
h
C

Tabel 3.2: Meantone
Semi tone
 1,045 1,070 1,070 1,045 1,070 1,045 1,070 1,045 1,070 1,070 1,045 1,070 1,045
Small second
1,118
1,145
1,118
1,118
1,118
1,118
1,118
1,118
1,145
1,118
1,118
1,118
1,118
Large second
1,118
1,145
1,118
1,118
1,118
1,118
1,118
1,118
1,145
1,118
1,118
1,118
1,118
Flat third
1,196
1,196
1,196
1,168
1,196
1,168
1,196
1,196
1,196
1,196
1,168
1,196
1,196
Sharp third
1,25
1,28
1,25
1,25
1,25
1,25
1,28
1,25
1,28
1,25
1,25
1,28
1,25
Fifth
1,495
1,495
1,495
1,495
1,495
1,495
1,495
1,495
1,531
1,495
1,495
1,495
1,495
Meantone
c
cis
d
es
e
f
fis
g
gis
a
b
h
C

Tabel 3.3: Kirnberger II
Semi tone
 1,054 1,068 1,054 1,055 1,067 1,055 1,067 ,1054 1,061 1,06 1,055 1,067 1,054
Small second
1,125
1,125
1,111
1,125
 1,125
1,125
1,124
1,118
1.125
1,118
1,125
1,124
1,125
Large second
1,125
1,125
1,111
1,125
 1,125
1,125
1,124
1,118
1.125
1,118
1,125
1,124
1,125
Flat third
1.185
1.187
1,185
1,187
1,2
1,185
1,193
1,185
1,187
1,193
1,185
1,2
1,185
Sharp third
1,25
1,266
1,25
1,266
1,262
1,258
1,262
1,25
1,266
1,253
1,266
1,262
1,25
Fifth
1,5
1,5
1.491
1,5
1,5
1,5
1.498
1,5
1,5
1.491
1,5
1,5
1,5
Kirnberger II
c
cis
d
es
e
f
fis
g
gis
a
b
h
C

 
Color code: pure interval
  pure interval ± 0,5 comma (0,656 %)
  Pythagoric third
  Pythagoric third ± 0,5 komma (0,656 %)
  out of tolerance of  ± 0,5 comma

2.3    Characteristics of a number of intervals: circle of fifths.....(content)

Additonally it is possible to exhibit the characteristics of the main intervals on the circle of fifths, in figure 1.
The color code of the lines in the figure is the same as the color code of the tables above.

Figure 1

Figure 1 has been drawn for the temperament of Kirnberger II.
The fifths are given on arcs of 30 degrees as usual, and the thirds are chords of 90 degrees (flat thirds) or 120 degrees (sharp thirds). Figure 1 shows all the pure flat thirds, but the other flat thirds, the seconds, primes and the semi-tones are not drawn in order not to get lost in details.

It is remarkable to see in the picture that ALL notes are connected by green (= pure) elements, except for the a ("la"). In other words: the relative positioning of ALL notes (except for the a or "la") is very precisely defined, based on simple pitch ratios (pure intervals). The structure of the figure therefore is very tight and stiff: it is not possible to distort the figure without changing the characteristics of the pitch ratios.
It is very remarkable that it is precisely the a ("la") that forms an exception in the above given characteristic, certainly if one considers that the a ("la") is used as reference pitch in classic music with almost no exception!
One has to go back to historical and traditional context, in order to understand or accept that the "la" is thé musical reference pitch by excellence. This subject can be the topic of countless investigations.
Based on pure scientific technical considerations of the above finding on the figure of circle of fifths, it should have been better to choose for a reference note that has a good central positioning and has as many pure green connections (intervals) as possible, such as the g (sol) that is connected by four pure intervals with other notes, and that is called the "dominant" note of the C ("do") sharp scale.
Habits and traditions in classical music are so strong that any proposal to choose for another reference tone will not have the slightest chance of attention or acceptance. Thought has also to be given to the fact that if small deviations from integral purity are accepted, the a ("la") has even more good intervals: in figure 1 the a ("la") has SIX yellow connections with other notes.

Possible practical implementation of the diagram of figure 1:
The black semi circle covers all the diatonic notes of one octave, and is drawn in a position corresponding with C (sharp) and a (flat).
The fundamental notes of the sharp and flat scales in this figure are marked by a black dot including a "c" and a white dot including an "a" on the black semi circle.
Adequate rotation of the black semi circle allows for fast and easy exhibit of the characteristics of the chords in the other tonalities. The typical characteristics of the different tonalities are a consequence of the different characteristics of the intervals contained within the tonality, as explained in depth by prof. H. Kelletat.
It appears clearly that the tonalities "C" and "a" have many pure chords, when using the Kirnberger II temperament.
 

2.4 Characteristics of intervals: Graphic comparison.....(content)

Separate graphic diagrams are displayed in the book of Kelletat, for the fifths and for the sharp and flat thirds of the most important temperaments.
Combination of those diagrams leads to the figures that can be opened below, whereby the Pythagoric scale for these figures was based on an alternate definition of the scale (*).
Remarkable properties of the "Bach-temperaments":

  • There is a very tight fit between the "Bach"-temperaments, and as a group they differ significantly from other temperaments.
  • The deviations of the fifths are very limited (always less than aproximately 5 cents)
  • The (sharp and flat) thirds are very pure for the main tonalities, and they gradually convert to (sharp and flat) Pythagoric thirds for more remote tones (+ 21.5 cents deviation for the sharp Pythagoric thirds, en - 21.5 cents deviation for the flat Pythagoric thirds)
Figure showing the
Deviation of Purity of
Fifhts
Figure showing the
Deviation of Purity of
Sharp Thirds
Figure showing the
Deviation of Purity of
Flat Thirds
Figure showing the
Deviation of Purity of
Fifhts for "Bach-temperaments"
Figure showing the
Deviation of Purity of
Sharp Thirds for "Bach-temp"
Figure showing the
Deviation of Purity of
Flat Thirds for "Bach-temp"
.
 (*) Alternate definition of the Pythagoric scale:

Normally the Pythagoric scale is defined by:

  1. From a upwarts build pure fifths leading to d, g, c, f, bes, es, as, des, ges, ces, fes, beses, eses, …
  2. From a downwards build pure fifths leading to e, b, fis, cis, gis, dis, ais, eis, bis, fisis, cisis, gisis, …
For the figures above the Pythagoric scale was build an alternate way, so that the four most pure thirds of the scale should fall on the notes d, g, c and f.
.
This alternate construction of the scale is obtained by building pure fifths upwards, starting from a, whereby at some points the name of the note should be replaced by the name of an enharmonic note. This leads to the series:
.
   a,  e,  b,  fis,  cis,  gis, (dis ->) es, (ais ->) bes,
  (eis ->)  f,  (bis ->)  c,  (fisis ->)  g,  (cisis ->)  d
.
 
  • Link to     Appendix 2:     Pitch Measuring Techniques
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

Revision 2002-07-25