Ludger
Hofmann-Engl (for* Musopen*)

*Croydon Family Groups*

**Part
1 - December 2009**

Popular belief
holds that there is a strong connection between musical and
mathematical thinking. Whether this is true or not, there have been
only a limited number of individuals who have applied computational
means to music. However, particularly since the 2^{nd} half
of the 20^{th} century some schools have emerged applying
mathematical methods directly or indirectly to music.

**1. Individuals making the connection between music and mathematics**

This section does not claim to be comprehensive, but highlights simply important developments.

**1. 1. Pythagoras
(ca 570 – 495 BC)**

No
primary source exists on Pythagoras. However, Nicomachus (ca 120 –
60 *BC*) claims that it was Pythagoras who discovered that
consonant and dissonant sounds are the result of number relationships
with the *diapason * (1/2), *diapente * (2/3)and *diatessaron* (3/4)
to be consonant but not so the tone in between the *diapente* and the *diatessaron* (8/9).

Additionally,
the *Pythagorean scale* has been attributed to Pythagoras by placing consecutively *fifths* upon each other. This is: *Fa
– Do – So – Re – La – Mi – Si * resulting
in the scale: *Do, Re, Mi, Fa, So, La, Si, Do.* However,
placing 7 octaves upon each other and 12 fifths, produces the *Pythagorean* *Comma. * This is:

This *comma* amongst other *commas* has led to the introduction
of the equal temperament in later times.

**1.2. Euclid (around 300 BC)**

Euclid
wrote Κατατομη ανόνος (*about the divisions on a the monochord*).
His work on the *elements of music* has been lost. The *sectio canonis* (below after Hawkins)
contained in the *Introductio Harmonica* has been attributed to
Kleoneides with high probability. The sectio canonis contains the
instruction – to our knowledge for the first time in history - on
how to divide the length of the monochord in order to obtain musical
intervals.

Diapason = Octave

Diapente = Fifth

Diatessaron = Fourth

Diezeuctic tone = major Second

Limma ~ minor Second

While *diapason*, *diapente* and *diatessaron*
belong, according to theorem XIX, to the unmovable sounds, the *diezeuctic* tone
and the *lima* belong, according to theorem XX, to the class of moveable sounds.

**1.3. Jing Fang** (Chinese: 京房 78 – 37 BC)

According to a chapter within *Hou Han Shu* of Sima Biao (司馬彪 243 - 306 AD), it appears that Jing Fang was the first person to compute the
*Mercator Comma*. This is, overlaying 53 fifths and 31 octaves produces the near value of which in *cent*
is:
(producing
a beat of about 1 period per 1 to 2 seconds).

**1.4. Ptolomy** (Greek: Κλαύδιος Πτολεμαῖος Latin: Klaúdius Ptolemaîus ca 90 – ca 168 AD)

Ptolomy wrote a three volumes comprising work called Αρμονικά (theory
of music). He divides scales into *tetrachords* (interval of a fourth) and classifies those *tetrachords* according to their
three internal intervals into 8 classes. The three main classes are:

Enharmonic

Chromatic

Diatonic

Chromatic itself is further subdivided into *Soft* and *Intense*, and
Diatonic into *Soft*, *Tonic*, *Diatonic*, *Intense* and *Equable*. The *Diatonic Intense* Tetrachord is:

This is, a minor second (15/16) is followed by a small major second (10/9)
and a large major second (9/8) which results in a fourth (4/3). This tetrachord corresponds to *just tuning*.

What is remarkable about Ptolemy's contribution is not only that he was able to depict intervals in form of fractions but that he understood that adding intervals, requires the fractions to be multiplied. This is:

**1.5.
Francisco de Salinas** **(1513 – 1590)**

In
his *History of the Science and Practice of Music* (1776), John
Hawkins reports that Salinas describes in the fifths chapter of his
third book three different tuning systems. According to Hawkins,
these are:

The Diatonic Genus as:

The Chromatic Genus as:

The Enharmonic Genus as:

**1.6. Johannes Kepler** (December 27, 1571 – November 15, 1630)

Kepler wrote the *Harmonices Mundi*
(The harmonies of the world) published by Gottfried Tamachius Tambach in 1619.
Here, Kepler expresses his fascination with the fact that the
ratio between maximum and minimum angular speed of the planets in
orbit corresponds to musical intervals. For instance, he observes
that the maximum and minimum angular speed of earth around the sun
corresponds to the interval
which in musical terms can be related to the interval of a minor second
(i.e. *mi* to *fa*). Taking into account the introduction of the *cent* calculation, *equal temperament*
and the discovery of *categorical perception*, we might conclude that Kepler's *Harmonices Mundi* appears
to be of historical value only. Still, what is truly surprising is
that Kepler's ideas are absurd from a physical point of view. It is
true that:
but still:
.
It is like saying: *“Yesterday I ran 5 kg."*

**1.7. René Descartes** (31 March 1596 – 11 February 1650)

In 1618, the 22 years old Descartes concludes his *Musicae Compendium* with
the words:

Nec scirent hic, inter ignoraniam militarem ab homine desidioso et [*non*] libero, penitusque diversa cogitnati et agenti, tumultuse tui solius gratia esse compostium.

They don't know here, that amongst ignorant warmongers of lazy and constrained people who think and act completely different [to me], in restlessness this work was written for you.

Descartes shows in this compendium his clear understanding of modern and traditional music theoretical ideas, but he delivers apparently for the first time in history a stringent classification of consonances and dissonances as shown in the figure below taken from his compendium.

Here, Descartes puts the diapasson (octave) in the centre (most consonant), the diapente (fifths) and diatessaron (fourth) both into the next layer (less consonant), the hexachordon minus (minor sixth) and the diatonis (major third) into the next layer (even less consonant) and in the outside layer he places the tertia minor (minor third) and hexachordon majus (major sixth). Tritone and seconds are considered to be dissonant. It is interesting to note, that Hofmann-Engl's virtual pitch model computes for the octave 1 Schouten, the fifth 0.75 Schouten, the fourth 0.679 Schouten, the major third 0.56 Schouten, the minor sixth 0.513 Schouten, the major sixth 0.511 Schouten and for the minor third 0.49 Schouten. The minor second on the other end of the spectrum drops down to 0.182 Schouten. These computations might be seen as supporting Descartes's approach.

**1.8. Andreas Werckmeister** (November 30, 1645 – October 26, 1706)

With Werckmeister, questions on how to tune instruments draws to a close
even if the introduction of the *cent* calculation
still lies away more than 100 years from here. Werckmeister proposes
a number of different tunings called *Wohltemperierte Stimmung* (well-proportioned
tuning) leading towards equal temperament as outlined in his *Musicalische Temperatur* (1691).
Particularly, the tuning with the fifths *D - A, F# - C#, C# - G#* and *F - C* narrowed
down by a ¼ (Pythagorean) comma and the fifth *G# - D#* widened by the same amount we obtain a tuning which deviates from equal
temperament by a mean of 2.6 *cent* only.
Starting on *C* with 0 *cent*, we obtain the interval series:

0 96 204 300 396 504 600 702 792 900 1002 1098 [*cent*]

**1.9. Leonhard Euler** (15 April 1707 – 18 September 1783)

Euler produced, as a 24 years old mathematician, in 1731 the *Tentamen novae theoriae musicae ex certissimis harmoniae principiis dilucide
expositae* (Attempt for a new music theory based on depicting in all clarity the evident
principles of harmony). Not dissimilar to his predecessors, Euler
relates the consonant/dissonant attribute of harmonies to ratios, but
this time in relation to prime number factors. However, perhaps of
more relevance for the future are his thoughts as expressed in
*Conjecture sur la raison de quelques dissonances généralement
reçues dans la musique *(Conjecture about the reason of some dissonances received in music) in 1764.
Here, apparently for the first time in history, Euler suggests that
the ear will tolerate small deviations from the exact ratios. This
idea was finally confirmed in the 20^{th} century through the discovery of
categorical pitch perception pioneered by Houtsma in 1968.

**1.10. Gustav Fechner** (19 April 1801 – 28 November 1887)

Fechner as such was not a mathematician nor a music theorist, but an
experimental psychologist. However, his discovery of the *Fechner
law* is a fundamental prerequisite for both the *cent* and the *sone* calculation.
The Fechner law (sometimes also referred to as the Weber-Fechner law)
states that the intensity of a sensationis proportional to the logarithm of the physical intensityof the stimulus. The law can be written as:

or as
with *c* as a constant

It is interesting to note that Ptolemy's understanding that ratio multiplication correlates to the addition of intervals implies, as shown by Hofmann-Engl (1989), the Fechner law.

**1.11. Alexander John Ellis** (14 June 1814 – 28 October 1890)

In 1885, Ellis published his paper called *On the Musical Scales of various Nations* in
the Journal of the Society of Arts. Being aware that, in order to
compare various scales and tuning system against each other, a robust measurement
is needed, he introduced the *cent* calculation.
The logarithmic definition is:

At times, the definition is given in the from that 1 *cent* is
the 1200 ^{th} part of an octave written as:

**1.12. Hermann Ludwig Ferdinand von Helmholtz** (31 August 1821 – 8 September 1894)

Helmhotltz's
general importance as his specific importance within music might have
been exaggerated by historians (not essentially different to the
exaggeration of Albert Einstein's importance). In terms of music
theory, Helmholtz wrote: *Die Lehre von den Tonempfindungen als physiologische Grundlage für die
Theorie der Musik* (translated
by Ellis in 1875 as *On the Sensations of Tones* and
more literally: *The teachings of sound sensations as the basis of music theory*)
which was published in 1863. Here, Helmholtz attaches dashes as pre- and suffixes to the musical letters
(capital ones for the lower and lower case ones for the upper
octaves) in order to differentiate between octaves.

**1.13. Carl Stumpf** (21 April 1848 – 25 December 1936)

Carl Stumpf wrote the book called *Tonpsychologie* (*Sound/Musicpsychology*). Perhaps most
remarkable is his study on Consonance/Dissonance.
Unlike his predecessors who had attempted to explain this phenomenon
via ratios and prime number factors, Stumpf refers to the
*Verschmelzungsgrad* (fusion
degree) of intervals. This is, Stumpf asked participants within two
experiments to relate to him whether they perceived simultaneously
played intervals on an organ as one. The results of two experiments
are recorded as:

Interval |
Fusion Degree (experiment 1) |
Fusion Degree (experiment 2) |

Octave |
76% |
- |

Fifth |
62% |
50% |

Fourth |
36% |
36% |

Major Third |
30% |
27% |

Minor Third |
- |
30% |

Minor Sixth |
19% |
- |

Triton |
15% |
24% |

It appears that for the first time ion history, a music theorist uses statistical tools measuring human responses in order to answer a question of music perceptual nature.

**1.14. Stanley Smith Stevens** (4 November 1906 – 18 January 1973)

If Ellis applied the Fechner Law to pitch perception, this time Stevens
applies the same law to the perception of loudness in 1936 coining
the term *sone*. However,
Stevens exceeds the Fechner law as he computes *sone* by
measuring the actual sound pressure and subtracting from it the
perceptual threshold. The computation can be written as:

or as

where
*L* is the perceived loudness in *sone*, *p* the
actual sound pressure measured in *dB* and *p _{o}* the
perceptual threshold also measured in

**1.15. J. F. Schouten** (around 1940)

At the time of writing, no biographical details about J. F. Schouten
have been available to the author. However, Schouten's paper *The
residue and the mechanism of hearing* in: *Proc. Kon. Akad.
Wetenshap. 43* has been quoted in the literature frequently. Here, Schouten demonstrates that even by omitting some lower partials of the
overtone series the pitch sensation of the first partial will be
evoked. This is, overlaying the sine tones at the consecutive
intervals of a *fourth – major third – minor third – small
minor third – large major second – major second – small major
second* produces the pitch
sensation of a compound fifth below the first sine tone.

**1.16. Wilhelm Fucks** (4 June 1902 – 1 April 1990)

Wihelm Fucks published with
co-author Joesef Lauter the *Exatwissenschaftliche Musikanalyse
* (exact scientific music
analysis) in 1965. The book appears never to have been translated
into English and the German original is no longer available. It is
not within the scope of this article to do justice to the whole book,
but in order to give an insight into this work, the method and
results of the first chapter will be presented here.

The team counted the occurrences of exact pitches within the works of 29
composers ranging from Willaert to Nono. Based on these counts they
computed the probability of occurrence *i* of each pitch within
each of the 29 pieces according to the formula:
.
They further computed sigma
with
and the entropy
.
Below are the results:

What this team attempted to demonstrate with this analysis is to show that over a period of nearly 500 years musical compositions had gained complexity measured as an increase in sigma and entropy values. Within the following chapters they continue to apply similar measures to various other parameters producing similar results.

**1.17. Guerino Mazzola** (born 1947)

Mazzola proposed a method of melodic transformations in 1990. For this purpose he made use of four transformation matrices. These matrices are:

Inversion as:

Retrograde as:

Parameter change as:

Arpeggio as:

He then defines a melody as a vector consisting of a pitch component *x* and
a time component *y* as:. Now, transforming this vector with the retrograde matrix, we obtain:

This is, the time line is now reversed. However as pointed out by Hofmann-Engl (2003), inversion is not a reversal in time, but reversing the order of the pitch sequence. Similar issues are pointed out by Hofmann-Engl (2003) with the three remaining matrices and particularly with the parameter change which is: , which means that Mazzola now treats the time/duration and the frequency/pitch dimension as interchangeable which is absurd from a physical as well as from a cognitive point of view. However, it must be noted that it appears that it was Mazzola who, for the first time in history, used matrices in the context of melodic transformations.

**1.18. Ludger Hofmann-Engl** (born 1964)

Hofmann-Engl applies mathematical tools in three areas. These are 1) statistical music analysis, 2) virtual pitch and 3) melodic similarity.

1.18.1. *Statistical music analysis*

Clearly inspired by the work of Fucks, Hofmann-Engl applies statistical tools
to the analysis of a specific piece: *The Passacaglia ungherese*
by Ligeti 1988. Similar to our approach taken in the context of
Fucks, we will focus on one exemplary aspect only. Hofmann-Engl
firstly counts the occurrence of all pitches classified in pitch
classes for which he gives the formula:
.
He then computes the percentage of each pitch class occurring using the formula:
. In case we have an equal distribution of pitch classes we would expect:
. He further defines the *turbulence* as:
and the significant deviation of occurrence as:
. Applied to Ligeti's *Passacaglia ungherese,* he concludes that the pitch classes
*d, e, g, a* occur significantly more often, *c* and *b* occur regularly, *c#, f *and *f#* are non-significantly underrepresented and *d#, g#* and *a#* are significantly
underrepresented. After some further computation he arrives at the
following graph:

Here, the graph shows that *c-major/a-minor* are
the key(s) which underpin the composition. Supported by a further
harmonic analysis, Hofmann-Engl concludes that this composition is in
*a-minor* leading to the summary conclusion that Ligeti made use of more contemporary harmonies in
order to disguise the fact that the piece is actually in a traditional key.

1.18.2. *Virtual Pitch*

Hofmann-Engl further developed Terhardt's ideas in respect of virtual pitch during his postgraduate studies and presented within his unpublished MA thesis (1991). Additionally, he applied the model in various contexts, such as contemporary composition, music analysis, copy right infringement and Riemann's chord classification system, as presented during three international conferences (1999, 2006 & 2008) as well as in two papers published online (2004 & 2008). Recently, his model has been critically reviewed by Goldbach (2009). At the time of writing this model appears to be the most accurate virtual pitch model. The model is given as:

where *V*(*t*) is the strength of the virtual tone *t*, *w*_{s}(*s _{i}*)
the spectral weight of the

A second part of the model calculates the consonance/dissonance degree, called the
*sonance* to any given chord and is given as:

where *S*(*ch*) is the sonance (with unit *Sh* = Schouten)
of the chord *ch*, *v*_{max} is the
virtuality of the strongest root, *k* = 6 *Hh*/*Sh*, *m*
is the number of virtual pitches produced by the chord *ch*,
*v*_{pmax} is the virtuality of the strongest
root in percent (= *v*_{max} divided by the sum
of all virtual pitches of the chord *ch*), *c* = 0.223 (the
maximal limit the strongest root can fetch), *n* the number of
tones the chord *ch* consists of and *i* the *i*th
tone of the chord *ch*.

1.183. *Melodic similarity*

Here, Hofmann-Engl (*PhD thesis* 2003 and published 2009) represents melodies as composed
of melota (pitch sequences), dynama (dynamic sequences) and chronota
(durational sequences). These parameters are then represented as *n+*1
dimensional vectors. This is:

This vectors are then transformed via transformation matrices of the following two types:

Translation matrix as:

Reflection matrix as:

A thank you to Yan-Tak Lau-Davies for providing the information on Jing Fang and Sima Biao.

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Euler, L. (1764).

Fancher, R. E. (1996).

Fucks, W. & Lauter, W. (1965).

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Hawkins, J. (1875).

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Hofmann-Engl, L. (1999).

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