Till Eulenspiegel's funny Series

This wikibook deals with the striking similarity in the ratios of the vibrational frequencies of the Balmer Series with the ratios of the sound frequencies of the opening motif of the symphonic poem Till Eulenspiegels lustige Streiche by Richard Strauss (* 1864; † 1949).

Till Eulenspiegel's funny Series.

There is no doubt that not only is superior musicality a prerequisite for such feats of compositional transfer, but that Richard Strauss also possessed the necessary genius and ingenuity to create such high-level works from basic physical material. It is based on a previous publication in the appropriate German speaking Wikibook, which was widely translated with www.DeepL.com/Translator (free version).

Introduction

Title of the book Ein kurtzweilig lesen von Dyl Ulenspiegel gebore vß dem land zu Brunßwick : wie er sein leben volbracht hatt ; XCVI seiner geschichten, Straßburg, 1515

Till Eulenspiegel is said to have been a roving cunning rogue in the 14th  century, who played dumb and played many tricks on his fellow men.

The Balmer series, named after the Swiss mathematician and physicist Johann Jakob Balmer (* 1825; † 1898), describes a sequence of spectral lines in the spectrum of hydrogen atoms that can be described by specific frequencies or wavelengths of electromagnetic radiation and of which five lines are in the range of visible light. These were first detected in sunlight, since the chemical element hydrogen is the main component of stars and sunlight can be studied very well because of its great brightness. Later, the hydrogen lines were also detected in the light of bright stars, such as the brightest star in the night sky, namely Sirius (α Canis Majoris) in the constellation of the Great Dog.

At the time the composition was written, the composers Richard Strauss and the mathematician and composer Hans Sommer (* 1837; † 1922, actually Hans Zincken called Sommer), who was well versed in optics, became friends in Weimar. It could therefore well be that Richard Strauss learned about current physical discoveries at the time via his fatherly friend Hans Sommer, and thus also about the Balmer optical series. With its help, he could have used the frequencies of the five visible spectral lines transferred to sound frequencies - quasi mischievously - for the five notes c - f - g - gis - a of the opening motif of his symphonic poem Till Eulenspiegels lustige Streiche.

The Balmer series

The atoms or molecules of a gas can be excited by adding energy. In this process, the electrons in the atomic shell are brought to higher discrete energy levels. Through spontaneous emission, the electrons eventually fall back to a lower energy level by chance, whereby the energy released in the process is emitted as electromagnetic radiation in the form of a photon, whose oscillation frequency $\nu$ can be expressed via the speed of light $c$, which has a fixed magnitude of 299792458 metres per second:

$\nu ={\frac {c}{\lambda }}$

Here $\lambda$ is the wavelength of the light particle emitted by the atom, and each wavelength corresponds to a particular saturated colour. White light has a continuous spectrum with practically all visible wavelengths:

.

If atoms are excited by light, the photons with the wavelengths matching the energy levels are destroyed (absorbed) in the process, and a dark line can be seen in the corresponding absorption spectrum at the associated wavelength. If a gas is excited to form a plasma, photons are only produced (emitted) at the wavelengths that match the energy levels, and these can then be seen as bright lines in the emission spectrum.

The position of these lines on the screen of an optical spectrometer depends on the angles of refraction of the glass prism used or on the properties of the diffraction grating used, which depend on the wavelength of the light examined. The angles of refraction achieved are thus determined by the geometric arrangement in the optical devices used.

Obtaining the hydrogen emission spectrum with an optical prism: The light source on the left with hydrogen excited to glow is imaged into a slit via a converging lens. The slit is imaged to infinity by a collimator lens, and these rays are sent through a triangular prism. This prism splits the incident light into different directions depending on the wavelength (dispersion); red light is deflected more weakly than violet light. With another lens, the slit is imaged onto a screen (bottom right) where the spectral lines become visible.

These wavelengths are also known from the absorption spectrum of the light of the stars, first and foremost our sun, as they consist to a very large extent of hydrogen and have a high energy turnover. The English physician, physicist and chemist William Hyde Wollaston (* 1766; † 1828) was the first to describe such dark lines in the solar spectrum in 1802. They were also described independently of him in 1814 by the German optician and physicist Joseph von Fraunhofer (* 1787; † 1826) and systematically investigated.

Johann Jakob Balmer around 1880.

When the excited electron is in a hydrogen atom and passes from a higher energy level $E_{m}$ to the second lowest energy level $E_{2}$, the wavelengths $\lambda _{m,2}$ of the Balmer series result. The spectral lines of this series are named after the Swiss mathematician and physicist Johann Jakob Balmer (* 1825; † 1898), who was able to present the mathematical regularity 1884 by means of his empirically found, generalised Balmer formula for the wavelengths $\lambda _{m,n}$ in the Verhandlungen der Naturforschenden Gesellschaft in Basel:

$\lambda _{m,n}=A\left({\frac {m^{2}}{m^{2}-n^{2}}}\right)$ mit $m>n$ und $n\geq 1$

Here $A=\lambda _{\infty ,n}\approx 364.506820{\text{ nanometres}}$ is a constant wavelength in the ultraviolet. Balmer called it the fundamental number of hydrogen and gave its value as 3645.6 angstroms and 364.56 nanometres respectively, slightly higher due to the measurement accuracies of the time.

However, the existence of energy levels was not yet known at that time and the contemporary researchers were electrified by the lack of explanation. Johann Jakob Balmer wrote about this in his publication:

Es sind besonders die numerischen Verhältnisse der Wellenlängen der ersten vier Wasserstofflinien, welche die Aufmerksamkeit reizen und fesseln. Die Verhältnisse dieser Wellenlängen lassen sich nämlich überraschend genau durch kleine Zahlen ausdrücken.

Der Unterschied zwischen den berechneten und beobachteten Wellenlängen ist so klein, dass die Übereinstimmung im höchsten Grade überraschen muss.

Aus diesen Vergleichungen ergibt sich [...], dass die Formel auch für die fünfte [...] Wasserstofflinie zutrifft.

The coefficients $k_{m,2}$ of the Balmer series, which result from the bracket expression for $n=2$, were also given by Johann Jakob Balmer:

$k_{m,2}={\frac {m^{2}}{m^{2}-4}}$

These coefficients $k_{m,2}$ and the associated wavelengths $\lambda _{m,2}$ read with the additional definition $r_{3}:=1$ for the first five spectral lines of the Balmer series with $3\leq m\leq 7$:

m	Rationaler Wert von $k_{m,2}$	Dezimalwert von $k_{m,2}$	Verhältnis $r_{m}=r_{m-1}\cdot {\frac {k_{m-1,2}}{k_{m,2}}}$	Dezimalwert des Kehrwerts ${\frac {1}{r_{m}}}$	Wellenlänge $\lambda _{m,2}=A\cdot k_{m,2}=A{\frac {k_{3,2}}{r_{m}}}$ in Nanometer
3	${\frac {9}{5}}$	$1.8$	$1$	$1$	656.112
4	${\frac {16}{12}}={\frac {4}{3}}$	$1.{\overline {3}}$	${\frac {27}{20}}$	$0.{\overline {740}}$	486.009
5	${\frac {25}{21}}$	$1.{\overline {190476}}$	${\frac {189}{125}}$	$0.{\overline {661375}}$	433.937
6	${\frac {36}{32}}={\frac {9}{8}}$	$1.125$	${\frac {8}{5}}$	$0.625$	410.070
7	${\frac {49}{45}}$	$1.0{\overline {8}}$	${\frac {81}{49}}$	$0.{\overline {604938271}}$	396.907

The intensity of the five longest wavelength and visible spectral lines of the Balmer series $H_{\alpha }$, $H_{\beta }$, $H_{\gamma }$, $H_{\delta }$ and $H_{\epsilon }$ over the wavelength of light $\lambda$..

In his publication, Balmer gave, among other things, the wavelengths determined by Anders Jonas Ångström (* 1814; † 1874) in his work on the solar spectrum of 1862 experimentally determined wavelengths for the visible hydrogen lines:

m	Wavelength $\lambda _{m,2}$ in Nanometres	Designation	Designation after Fraunhofer	Colour designation	Colour	Frequency of Hydrogen line $\nu _{m,2}$ in Hertz
3	656.2	$H_{\alpha }$	C-Linie	Red		$4.569\cdot 10^{14}$
4	486.1	$H_{\beta }$	F-Linie	Blue-green		$6.168\cdot 10^{14}$
5	434.0	$H_{\gamma }$	vor G	Blue		$6.909\cdot 10^{14}$
6	410.1	$H_{\delta }$	h-Linie	Purple		$7.311\cdot 10^{14}$
7	396.8	$H_{\epsilon }$	nahe vor $H_{I}$	Purple		$7.553\cdot 10^{14}$

Der sichtbare Bereich des Wasserstoffspektrums mit den Linien der Balmer-Serie. The wavelengths of the emitted light become longer and longer from (violett) on the left to (rot) on the right, the frequencies smaller and smaller. Such light spectra can be obtained, for example, by refracting light originating from a light source at a prism. Depending on the dispersion of the optical glass of which the prism is made, different angles of refraction result for different wavelengths.

Alternatively, these series can also be calculated with the Rydberg constant $R_{\infty }={\frac {4}{A}}\approx 1.09737316\cdot 10^{7}\,\mathrm {m^{-1}}$ for the frequencies $\nu _{m,n}$:

$\nu _{m,n}=c\cdot R_{\infty }\left({\frac {1}{n^{2}}}-{\frac {1}{m^{2}}}\right)$ mit $m>n$ und $n\geq 1$

The frequency $\nu _{A,2}$, which results with the fundamental number of hydrogen for $m\rightarrow \infty$ in the Balmer series with $n=2$, can therefore be calculated as follows:

$\nu _{A,2}=\nu _{\infty ,2}=c\cdot R_{\infty }{\frac {1}{4}}\approx 8.224605\cdot 10^{14}\,{\text{Hertz}}$

The composition

Till Eulenspiegel's Merry Pranks

Richard Strauss in 1894, the year in which the symphonic poem Till Eulenspiegel's Merry Pranks was written..

Till Eulenspiegel's Merry Pranks in 2001 played by the United States Navy Band.

Till Eulenspiegel's Merry Pranks was composed nine years after the discovery of the Balmer series by the German composer Richard Strauss (* 1864; † 1949). It is a tone poem for large orchestra with a performance duration of fifteen minutes, written between 1893 and 1894 after Richard Strauss had spent several months in Greece and Egypt recovering from the late effects of pneumonia and after his opera Guntram had been completed.

Originally, Richard Strauss had apparently planned to have the work begin immediately with the well-known and striking six-act Till Eulenspiegel motif. The five-bar prologue was only added by Richard Strauss as the beginning of the composition in the course of the revision. In the course of the composition, the Till-Eulenspiegel motif only reappears at the end.

The prologue was finally titled by the composer with the words "Es war einmal ein Schalknarr..." and "leisurely", the Till Eulenspiegel motif that immediately follows with the words "...Namens Till Eulenspiegel" and "allmählich lebhafter"..

The beginning of the composition is notated in F major, and the five notes of the Till Eulenspiegel motif are c - f - g - gis - a. They are repeated twice:

The six bars directly after the prologue of the symphonic poem Till Eulenspiegels lustige Streiche with the complete Till Eulenspiegel motif to be played by a solo horn.

The first five notes of the Till Eulenspiegel motif sounding three times in succession.

The symphonic poem Till Eulenspiegels lustige Streiche bears the opus number 28, and it was premiered under the direction of the German composer Franz Wüllner (* 1832; † 1902) on 5 November 1895 in Cologne.

Thus Spoke Zarathustra

In this context, it is noteworthy that the symphonic poem by Richard Strauss with the opus number 30, Also sprach Zarathustra, which was composed shortly afterwards, also has very clear references to physical number relationships and uses an opening motif with five tones.

The opening motif used in the introduction Sunrise is also derived from a physical number series, namely the natural tone series, whereby its first, second, third, fourth and fifth tones with the frequency ratios 1:1, 2:1, 3:2, 4:3 and 5:4 play a central role in relation to the fundamental.

The first eight bars of the symphonic poem Also sprach Zarathustra with the voices of the trumpets, trombones, timpani and bass drum.

Orchestral recording of the entire introduction with the title Sunrise.

Strauss again uses C as the basic note, which is held out as an octave on upper C and lower C, initially for four bars in the double basses, organ pedal and contrabassoon, before the four C trumpets join in with their distinctive ascending motif. For his trumpet motif, Richard Strauss uses the next four notes of the natural tone row c', g', c and e, so that finally a radiant C major chord is heard, played almost by the entire orchestra, which is immediately changed to C minor by the alteration of the third tone. After a brief triplet interlude by the timpani with the two fourths G and C, the trumpet motif is heard again with an inverted chord sequence of C minor and C major. As in the symphonic poem Till Eulenspielgel's Merry Pranks, the motif is played a third time and then developed further, whereby the built-up C major chord can be interpreted this time as the dominant to the F major that follows. After four bars, the introduction closes with a renewed and sustained C major in fortissimo, played by all instruments.

In the further course of the orchestral piece, Richard Strauss also plays with the number twelve, which is significant in astronomy, in the section The convalescent by introducing a twelve-tone row.

Twelve-tone motive at the beginning of the section "Der Genesende" of the symphonic poem "Also sprach Zarathustra" by Richard Strauss in the cellos, double basses and trombones. The first three tones e - h - e' correspond to the three naturals of the trumpet motif from the introduction to the composition, going up first a fifth and then another fourth. The twelve notes from the note e' to the last note c' lie within an octave and represent a twelve-tone row, since they all have a different pitch: e' - es' - b - ges - g - h - d' - cis' - gis - f - a - c'. The note e sharp, which is tied over from the penultimate to the last bar, is an enharmonic confusion of the following note f..

Twelve-tone motif at the beginning of the section "The convalescent" of the symphonic poem "Also sprach Zarathustra" by Richard Strauss.

In the last section of the composition, entitled Nachtwandlerlied, he uses twelve midnight chimes. The twelve not only has close astronomical connections to the division of the night into the twelve night hours, but also to the twelve night signs of the ecliptic visible at night, which are traversed by the seven changing stars and in which the planet Jupiter resides for one year as seen from Earth.

This symphonic tone poem was premiered at the end of November 1896 by Richard Strauss in Frankfurt am Main.

There are only a few other physical number series that would be suitable for the pitches of musical motifs. Furthermore, it should be noted that Richard Strauss was neither a pioneer nor a supporter of twelve-tone music, so that the fact that he nevertheless works with it can be seen as symbolic and definitely also a little mischievous.

Three songs after poems by Otto Julius Bierbaum

Incidentally, the three poems Traum durch die Dämmerung, Schlagende Herzen and Nachtgang of the composition Drei Lieder nach Gedichten von Otto Julius Bierbaum from 1895 with the opus number 29, which was created between the two symphonic poems, also deal with the astronomical objects sun, moon and stars and their observation in nature.

• Nummer 1: Traum durch die Dämmerung (Fis-Dur / B-Dur / Fis-Dur, sehr ruhig, 2/4-Takt)
  • ... im Dämmergrau, die Sonne verglomm, die Sterne ziehn,
  • ... in ein blaues, mildes Licht.
  • ... in ein mildes, blaues Licht.
• Nummer 2: Schlagende Herzen (G-Dur, lebhaft und heiter, Allegro giocoso, 2/4-Takt)
  • ... du gold'ne Sonne in Himmelshöhn! (in strahlendem C-Dur)
• Nummer 3: Nachtgesang (c-Moll, mäßig langsam, 3/4-Takt)
  • Der Mond goss silbernes Licht ...
  • ... rein wie die liebe Sonne.

Relationships

Intervals

The light frequencies of the first five hydrogen lines ($3\leq m\leq 7$) of the Balmer series visible to the human eye ($n=2$) correspond quite exactly to the ratios of the tone frequencies of the five tones of the first motif of the symphonic poem Till Eulenspiegel's Merry Pranks repeated twice immediately afterwards:

Tone name	Tone frequency $f_{m}$ in Hertz	Frequency ratio to next tone	m	Hydrogen line frequency $\nu _{m,2}$ in Hertz	Frequency ratio to next line	Wavelength $\lambda _{m,2}$ in Nanometres
c	261.6	1.335	3	$4.569\cdot 10^{14}$	1.350	656.1
f	349.2	1.122	4	$6.168\cdot 10^{14}$	1.120	486.0
g	392.0	1.059	5	$6.909\cdot 10^{14}$	1.058	433.9
gis	415.3	1.059	6	$7.311\cdot 10^{14}$	1.033	410.1
a	440.0		7	$7.553\cdot 10^{14}$		396.9

The tone frequencies given in the second column thus refer to the concert pitch A with 440 hertz. The wavelengths given in the last column correspond almost exactly to the five wavelengths given in Balmer's 1884 publication for the hydrogen lines in the visible range (see above).

If the light frequencies $\nu _{m,2}$ are multiplied by the mean conversion factor $u_{440}=5.713\cdot 10^{-13}$, the corresponding sound frequencies $f_{m}=u_{440}\cdot \nu _{m,2}$ and the musical intervals of the ascending motif by Richard Strauss.

Tuning pitch

At this point it must still be stated that this conversion factor can be chosen arbitrarily, but must somehow be fixed. The tuning of musical instruments in ensemble music has always had to be carried out so that all instruments can produce harmonious sounds together. For this purpose, tuning tones or a concert pitch were also used, the pitch of which was fixed. The French scholar Joseph Sauveur (* 1653; † 1716) and later the German physicist and astronomer Ernst Chladni (* 1756; † 1827) proposed to use the unit of one second as the fundamental time measure of a physical tuning for the determination of the keynote C. This means that the keynote C is the unit of one second. This means that the fundamental C0 has exactly the frequency 1 hertz and all octaves of this fundamental upwards always have exactly twice the frequency. This leads to the following series:

Tone name	Factor	Tone frequency $f$ in Hertz
C0	$2^{0}$	1
C1	$2^{1}$	2
C2	$2^{2}$	4
C3	$2^{3}$	8
C4	$2^{4}$	16
C5	$2^{5}$	32
C6	$2^{6}$	64
C7	$2^{7}$	128
C8	$2^{8}$	256
C9	$2^{9}$	512

If a reference pitch is sought for the transmission of frequencies of electromagnetic waves to sound waves, it is obvious for physicists to orientate themselves on the keynote C of this physical tuning.

From a musical point of view, the first five tones of this series have only a theoretical meaning and are practically not perceptible to humans with a pitch. The tone A is a major sixth (respectively nine semitones) higher than the C directly below it or a minor third (respectively three semitones) lower than the C directly above it. If the A in the last octave between C8 and C9 in this row is set as a concert pitch, it has the same frequency ratio ${2}^{\frac {1}{12}}$ in equal temperament (all twelve successive semitone intervals have the same frequency ratio ${2}^{\frac {1}{12}}\approx 1.059463$) it has the following frequency:

$f_{A}=f_{C8}\cdot {2}^{\frac {9}{12}}=f_{C9}\cdot {2}^{-{\frac {3}{12}}}\approx 430.539{\text{ Hertz}}\approx 431{\text{ Hertz}}$

With a frequency of 430.539 hertz for the concert pitch A, the following pitches result for the five notes of the Till Eulenspiegel motif:

Tone name	Tone frequency $f_{m}$ in Hertz	m
c	256.000	3
f	341.719	4
g	383.567	5
gis	406.375	6
a	430.539	7

This results in a somewhat smaller mean factor $u_{431}=5.591\cdot 10^{-13}$ as the constant of proportionality between the light and sound frequencies $f_{m}=u_{431}\cdot \nu _{m,2}$ of the ascending motif by Richard Strauss.

These pitches were used, for example, in the early Paris tuning of 1829 in the 19th  century. In the course of time, however, the reference pitch became higher and higher for practical reasons - stringed instruments sound fuller and louder when the strings are stretched a little more, although they acquire a higher natural frequency. Richard Strauss was aware of the problems involved and commented on the increased pitch of the concert pitch a few years before his death as follows:.

Die hohe Stimmung unserer Orchester wird immer unerträglicher. Es ist doch unmöglich, dass eine arme Sängerin A-Dur-Koloraturen, die ich Esel schon an der äußersten Höhengrenze geschrieben habe, in H-Dur herausquetschen soll.

Rückkehr zum Pariser A, bevor sich unsere armen Sänger die Paar letzten noch vorhanden Stimmen verschrien haben!

Coincidence or not coincidence

The question arises as to whether these physical and musical facts are coincidental, or whether it can be a matter of coincidence. Since there is apparently no reliable evidence for coincidence, the question cannot be answered.

At the very least, it must be stated that it would be an extremely remarkable coincidence. There are alone ${12}^{4}=20736$ possibilities of having exactly four tones from a stock of twelve tones follow any initial tone. The probability that exactly the five tones of the Till Eulenspiegel motif will be chosen at random from this supply of tones is therefore just under 0.00005. Other motifs have fewer or more than five tones, have a larger range of tones or have a different starting tone, which further and significantly increases the number of possibilities and reduces the probability of a coincidence accordingly.

The probability that the motif with the five notes c - f - g - gis - a, which unlike the many other motifs actually composed is not even diatonic but has chromatic components, is composed by chance and coincidence only a few years after the discovery of the Balmer series and its discussion in specialist circles, is even smaller.

Finally, it must be taken into account that the motif, notated in the key of F major, begins with the note C, whose pitch in the 18th  and 19th  centuries was often determined purely physically with the help of the definition of the time unit of the second.

In the very extensive Répertoire International des Sources Musicales (RISM), there is no other example in more than one million music documents that begins with a motif consisting only of these five notes. Only two other examples in the key of C major with a date of origin before 1893 can be found:

• Carl Czerny (* 1791; † 1857): Klavierübung in C-Dur (6/8-Takt), opus 599, Nummer 38
• Anonymus: Ländler (3/4-Takt) C-Dur

According to these considerations, the probability that this motif was created at this point in time by pure chance is certainly not zero, but it is quite low.

Further examples

The energy levels and resulting spectral lines of the hydrogen atom of the different series.

Let it be borne in mind that a rational sequence of five consecutive numbers, as in the Balmer series, need by no means result in aesthetically perceived pitch ratios. This is not the case with the higher orders in the Balmer series and, for example, with the Lyman series found in 1906 by the US physicist Theodore Lyman (* 1874; † 1954) or the Paschen series found in 1908 by the German physicist Friedrich Paschen. (* 1865; † 1947), for example, this is not the case.

Therefore, there are only very few known cases in which physical number sequences, especially if they can also still be directly perceived with the human senses, are reflected in musical tone sequences. A significant example is the natural overtone series, which results from whole-number ratios in vibrating strings or in air columns and was processed by Richard Strauss in the symphonic poem Also sprach Zarathustra (Engisch: "Thus Spoke Zarathustra", see above).

Title page of the seven orchestral suites "Pythagorische Schmids=Fuencklein" by Rupert Ignaz Mayr from 1692 with an illustration by the German painter Johann Andreas Wolff (* 1652; † 1716).

Canon with blue markings at the four Pythagorean tones g'-c"-d"-g" on the title page of the seven orchestral suites "Pythagorische Schmids=Fuencklein" by Rupert Ignaz Mayr from 1692.

Four-part canon on the title page of the seven orchestral suites "Pythagorische Schmids=Fuencklein" by Rupert Ignaz Mayr from 1692.

Another example with the four Pythagorean tones c' - f' - g' - c" is handed down in the legend of Pythagoras in the forge. The first three of these notes correspond to the first three notes of the Till Eulenspiegel motif. The integer relationship between the numbers four and three, which plays a role there among other things, describes the musical interval of the pure fourth, which appears both in the first interval of the Till Eulenspiegel motif and in the second interval of the Zarathustra motif.

The sounds caused by hammer blows were invented in 1690 by the French-German organist and composer Georg Muffat (* 1653; † 1704) with the organ composition Nova Cyclopeias Harmonica. This composition is framed by an aria, comprises eight variations on the theme Ad Malleorum Ictus Allusio (To allude to the blows of the hammers) and ends with the chant Summo Deo Gloria.

Two years later, the German violinist, composer and court kapellmeister Rupert Ignaz Mayr (* 1646; † 1712), who like Georg Muffat was a pupil of the Italian composer Jean-Baptiste Lully (* 1632; † 1687), the seven orchestral suites dedicated to the Elector of Bavaria Maximilian II Emanuel:

Pythagorische Schmids=Fuencklein

Bestehend in unterschiedlichen Arien / Sonatinen / Ouverturen / Allemanden / Couranten / Gavotten / Sarabanden / Giquen / Menueten / &c.

Mit 4.Instrumenten und beygefügten General-Baß, Bey Tafel=Musicken / Comœdien / Serenaden / und zu anderen fröhlichen Zusammenkunfften zu gebrauchen.

The main keys of the seven suites for solo violin are F major, D major, G major, D minor, F major, D major and B flat major.

→ See also Wikibook Pythagoras in the Forge.

Even the Pythagoreans were of the opinion that the same numerical regularities were evident in astronomy as in music. Therefore, there were repeated attempts to connect the orbits of the planets with harmonic sounds, as for example by Johannes Kepler (* 1571; † 1630) in his work Harmonices mundi libri V (five books on the harmonies of the world) of 1619, in which he transferred astronomical numerical ratios to musical intervals. The German composer Paul Hindemith (* 1895; † 1963) took up this theme and created the symphony The Harmony of the World in 1951 with the three movements Musica instrumentalis, Musica humana and Musica mundana, as well as the libretto and music for the opera of the same name in five acts in 1957. In the introduction to his music-theoretical work Unterweisung im Tonsatz he writes:

Ich weiß mich mit dieser Einstellung zum Handwerklichen des Tonsatzes einig mit den Anschauungen, die gültig waren lange vor der Zeit der großen klassischen Meister. Wir finden ihre Vertreter im frühen Altertum; weitblickende Künstler des Mittelalters und der Neuzeit bewahren die Lehre und geben sie weiter. Was war ihnen das Tonmaterial? Die Intervalle waren Zeugnisse aus den Urtagen der Weltschöpfung; geheimnisvoll wie die Zahl, gleichen Wesens mit den Grundbegriffen der Fläche und des Raumes, Richtmaß gleicherweise für die hörbare wie die sichtbare Welt; Teile des Universums, das in gleichen Verhältnissen sich ausbreitet wie die Abstände der Obertonreihe, so daß Maß, Musik und Weltall in eins verschmolzen.

Quantum physical background

When the spectral lines had been discovered and even in 1884 when Johann Jakob Balmer had empirically found the mathematical regularities and described them with the Balmer formula, the physical and theoretical causes for the emergence of such discrete spectral lines were still completely unknown. It was not until the first half of the twentieth century that quantum mechanics made it possible to describe the structure of atoms and the interaction between charged matter particles (for example electrons) and light particles (photons) very precisely.

The electromagnetic light field reacts with other particles in such a way that a quantum with the energy $E=h\cdot \nu$ is either captured or emitted. The constant $h$ is the Planck's quantum of action and $\nu$ is the frequency of an electromagnetic wave.

The particle nature of light was concluded in the years from 1899 to 1905 by the German researcher Max Planck (* 1858; † 1947) from the laws of thermal radiation and, in an intensified form, his younger colleague Albert Einstein (* 1879; † 1955) in the publication Ueber einen heuristischen Gesichtspunkt betreffend die Erzeugung und Verwandlung des Lichtes from the photoelectric effect (or photoeffect). In 1918 Max Planck was awarded the Nobel Prize in Physics for the discovery of energy quanta and in 1921 Albert Einstein for the discovery of the law of the photoelectric effect. The buzzword of wave-corpuscle dualism has pervaded the difficult discussions about how to understand the physics of microscopic things ever since. Yes, even the sound waves of music have teeny tiny grains: the phonons with an energy proportional to the frequency according to the Planck-Einstein formula.

Even a particle with a rest mass, such as an electron, can be understood as a matter wave. The term "matter wave", named after the French physicist Louis-Victor de Broglie (* 1892; † 1987) establish the relations between wavelength $\lambda$ and frequency $\nu$ for matter waves, which apply to particles with energy $E$ and momentum $p$ respectively:

$\lambda ={\frac {h}{p}}$

$\nu ={\frac {E}{h}}$

He was awarded the Nobel Prize in Physics in 1929 for his discovery of the wave nature of electrons.

When Max Planck introduced the quantum of action, he had no idea of the universal significance of the constant. It unambiguously and fundamentally links the scale of values of space and time with that of mass, energy and momentum. In the modern International System of Units, mechanics knows only one arbitrary quantity, the second. Metre and kilogram are derived from the second via the definitively fixed values of the universal speed of light $c$ (since 1983) and Planck's constant $h$ (since 2019). The latter has a 1967 decree nailed down to 9 192 631 770 periods of a quantum transition of cesium atoms.

Hydrogen atom

The formulae given concern the matter waves of free-flying particles. A little later, a brilliant theorist built the more general wave equations with forces that attract or repel particles. Make way for the prime example of this new wave mechanics!

The bond between the electron and the proton of the hydrogen atom takes a relatively time-stable, i.e. stationary, form precisely when there is a spatially concentrated standing wave that satisfies the famous Schrödinger equation, proposed in 1926 by the Austrian physicist Erwin Schrödinger (* 1887; † 1961). As with large mechanical objects, such as strings and organ pipes, the possible frequencies of their various forms of vibration or standing waves are very well related to small integers. Since Pythagoras, we know that the harmonic intervals of tones are nothing other than such frequencies whose ratios are two integers. They readily appear as the natural frequencies of vibrating objects.

State basis of the hydrogen atom with principal quantum numbers 1 to 4 (from top to bottom).

The natural frequencies of the hydrogen atom follow with $n=1,2,3,...$ the formula $\nu _{n}={\frac {\nu _{1}}{n^{2}}}$, where $\nu _{1}$ is the highest occurring frequency. The binding energies of the atom are measured negatively with respect to an unbound pair of electron and proton. The result is $E_{n}=-h\cdot \nu _{n}$. Again, Planck's constant $h$ enters. The lowest energy for the principal quantum number 1 $E_{1}=-13.6{\text{ eV}}$ (eV stands for the energy unit electron volt) belongs to the spherical standing wave in which no nodes occur. The excited energies $E_{n}$ have a symmetrical spatial structure with nodal surfaces that divide the waveform into a number of 'lobes' or 'bellies'. Examples on the right of the picture for the main quantum numbers n from 1 to 4. You can see areas of constant amplitude and colours of constant phase.

Quantum mechanics explains how the hydrogen atom glows. It goes from the state $E_{m}$ to the lower state $E_{n}$ with $m>n$. The energy $\Delta E=E_{m}-E_{n}$ is thereby transferred to a photon whose frequency lies in the spectral series and can be determined with the following Rydberg formula:

$\Delta E=h\cdot \nu _{m,n}=|E_{1}|\left({\frac {1}{n^{2}}}-{\frac {1}{m^{2}}}\right)$

One of the methods of physics is a perturbation calculation, according to which already a classically conceived electric oscillation causes the stimulated quantum transitions of hydrogen, exactly with the experimentally confirmed spectrum of frequencies. But to explain the spontaneous emission of light quanta, a thorough particle-like treatment of the light field was needed: the quantum field theory. With its calculation methods, it then correctly emerges that the excited states of the atom are not entirely stable. Photons can emerge from the vacuum and couple to the appropriate states of electrons.

We owe a profound interpretation of the hydrogen spectrum to the Austrian physicist Wolfgang Pauli (* 1900; † 1958), who worked with the physicists Niels Bohr (* 1885; † 1962) from Denmark, Werner Heisenberg (* 1901; † 1976) from Germany, the above-mentioned Erwin Schrödinger and others contributed significantly to the scientific revolution of quantum mechanics. Niels Bohr was awarded the Nobel Prize in Physics in 1922 for his services to research into the structure of atoms and the radiation they emit. Werner Heisenberg was honoured with it in 1932 for founding quantum mechanics, the application of which led, among other things, to the discovery of the allotropic forms of hydrogen.

Wolfgang Pauli succeeded in 1926 in an algebraic formulation of the hydrogen model thanks to a self-contained group of symmetry operators. He was able to derive the spectrum entirely without the troublesome Schrödinger differential equations. The special form of Coulomb attraction between electron and proton allows a high degree of symmetry.

Since then, symmetry groups have played an increasing role in physics. Often such groups are represented in nature in connection with series of whole quantum numbers. Symmetry and harmony are obviously closely connected. The German theoretical physicist Arnold Sommerfeld (* 1868; † 1951) wrote in September 1919 in Munich in the preface of his book Atombau und Spektrallinien:

Seit der Entdeckung der Spektralanalyse konnte kein Kundiger zweifeln, daß das Problem des Atoms gelöst sein würde, wenn man gelernt hätte, die Sprache der Spektren zu verstehen. Das ungeheure Material, welches 60 Jahre spektroskopischer Praxis aufgehäuft haben, schien allerdings in seiner Mannigfaltigkeit zunächst unentwirrbar. Fast mehr haben die sieben Jahre Röntgenspektroskopie zur Klärung beigetragen, indem hier das Problem des Atoms an seiner Wurzel erfaßt und das Innere des Atoms beleuchtet wird. Was wir heutzutage aus der Sprache der Spektren heraus hören, ist eine wirkliche Sphärenmusik des Atoms, ein Zusammenklingen ganzzahliger Verhältnisse, eine bei aller Mannigfaltigkeit zunehmende Ordnung und Harmonie. Für alle Zeiten wird die Theorie der Spektrallinien den Namen Bohrs tragen. Aber noch ein anderer Name wird dauernd mit ihr verknüpft sein, der Name Plancks. Alle ganzzahligen Gesetze der Spektrallinien und der Atomistik fließen letzten Endes aus der Quantentheorie. Sie ist das geheimnisvolle Organon, auf dem die Natur die Spektralmusik spielt und nach dessen Rhythmus sie den Bau der Atome und Kerne regelt.

Background in music theory

Auditory perception

The human ear is a remarkable measuring instrument that has a logarithmic sensitivity to both the amplitudes and frequencies of sound waves. This means that for us, loudness increases by the same increment when amplitude is multiplied by the same factor - not when the same amount is added. In the same way, the pitch increases for us by the same step when the frequency is increased by a factor. The factor of two is perceived particularly clearly, namely as an octave. At intervals of octaves, the melodies sound so similar that the notes are given the same name, accompanied by dashes or numbers, if you want to indicate the octave position.

Many people are familiar with the logarithmic measure with which we describe the ratios of amplitudes. A factor of ten was defined as 20 decibels. defined. An amplitude factor $q$ can be converted into a level $Q$ in the unit of measurement decibel (dB) with the decadic logarithm $\operatorname {lg}$ or $\log _{10}$ or with a logarithm to any base $\log$:

$Q\,{\text{in dB}}=20\,{\text{dB}}\cdot \log _{10}\;q=20\,{\text{dB}}\cdot {\frac {\log \;q}{\log \;10}}$

The same principle works for the frequencies or the perceived pitches. A factor of two was defined as 1200 cents. Every twelve semitones - or every small second - have a size of 100 cents by definition in music theory for equal temperament. Twelve semitones of the same size directly above each other make an octave, which consequently has a size of 1200 cents. The pitch difference $c$ in the unit of measurement cent (C) at a given frequency ratio $q$ can also be calculated with the following formula with a logarithm to any base $\log$ or with the logarithm dualis $\operatorname {ld}$ or $\log _{2}$ respectively:

$c\,{\text{in Cent}}=1200\,{\text{C}}\cdot \log _{2}\;{q}=1200\,{\text{C}}\cdot {\frac {\log \;{q}}{\log \;{2}}}$

Intervals

Two tones that are in a whole-number vibration ratio to each other as a musical interval are perceived as sounding consonant. This is particularly clear in the case of the prime and the octave, where the frequency ratio is 1:1 and 1:2 respectively. The larger the two integers become, the smaller the effect of perceived consonance and thus of harmonic perception. For trained ears, this effect can still be heard well with the pure fifth (frequency ratio 2:3) and with the pure fourth (frequency ratio 3:4). Instruments without fixed pitches, such as strings or trombones, as well as vocal ensembles, can also intonate major and minor thirds (frequency ratio 4:5 and 5:6) or minor and major sixths (frequency ratio 5:8 and 3:5) purely. Particularly in the case of major and minor triads, which are composed of three notes spaced a major third, minor third or fifth apart, the use of pure intervals produces a particularly harmonious sound.

The following table shows the frequency ratios for musical intervals from the prime to the octave in pure tuning and in relation to the fourth, fifth and octave in Pythagorean tuning:

Interval	Frequency ratio equal temperament	Frequency ratio Pythagorean temperament (decimal value)	Frequency ratio in equal temperament (decimal value)	Deviation between Pythagorean temperament and equal temperament in Cent
Prime	1:1	1.0000	1.0000	0
Minor second	15:16	0.9375	0.9439	12
Big second	8:9	0.8889	0.8909	4
Minor third	5:6	0.8333	0.8409	16
Major third	4:5	0.8000	0.7937	-14
Fourth	3:4	0.7500	0.7492	-2
Tritone	25:36	0.6944	0.7071	31
Fifth	2:3	0.6667	0.6674	2
Minor sixth	5:8	0.6250	0.6300	14
Major sixth	3:5	0.6000	0.5946	-16
Minor seventh	9:16	0.5625	0.5612	-4
Major seventh	8:15	0.5333	0.5297	-12
Octave	1:2	0.5000	0.5000	0

The tritone is therefore the interval between the fourth and the fifth. You can read off from the large numerical ratio 25:36 without having heard the diphthong: a dissonance comes out. In equal temperament, the two tones have the following irrational frequency ratio (also not a gentle sound):

$2^{-{\frac {6}{12}}}=2^{-{\frac {1}{2}}}={\sqrt {\frac {1}{2}}}$.

Positive deviations in the last column with the unit of measurement cent mean that the tone of the pure tuning is higher than the tone of the equal temperament, and negative deviations mean that the tone of the pure tuning is lower than the tone of the equal temperament.

It is noteworthy that in the whole-number ratios given in the table above, among the numbers with only one digit, seven alone does not appear, which underlines that it is a special number from the point of view of some medieval authors.

The next table compares the corresponding musical pure intervals with the frequency ratios in the Balmer series:

Interval	Frequency ratio in Pythagorean temperament	Frequency ratio in Balmer series	Ratio of frequency ratios at Balmer series and at Pythagorean temperament	Deviation between Balmer series and equal temperament in percent	Deviation between Balmer series and equal temperament in Cent
Prime	1:1	1:1	1:1	0	0
Fourth	3:4	20:27	80:81	-1.6	-20
Fifth	2:3	125:189	375:378	-1.3	-16
Minor sixth	5:8	5:8	1:1	0	0
Major sixth	3:5	49:81	245:243	0.8	14

The tones of the Balmer series on the monochord. Above, the spectrum with the five lines of hydrogen and their coefficients of the Balmer formula. Below, the monochord with the five corresponding string lengths, all of which have the same string tension and string thickness and, by shortening the string length, have a shorter wavelength and thus a higher frequency. The constant $A$ comes from the Balmer formula and represents the wavelength for $m=\infty$..

Comparison of the frequency ratios for pure and equal temperament and for the Balmer series.

Finally, a table comparing the corresponding musical intervals from the Balmer series with the frequency ratios at equal temperament:

Interval	Frequency ratio Balmer series	Frequency ratio in equal temperament	Deviation between Balmer series and equal temperament in percent	Deviation between Balmer series and equal temperament in Cent
Prime	1:1	1.0000	0.0	0
Fourth	20:27	0.7492	-1.1	20
Fifth	125:189	0.6674	0.9	16
Minor sixth	5:8	0.6300	0.8	14
Major sixth	49:81	0.5946	-1.7	-30

The notes of the Balmer series on the fretboard of a guitar. Above, the spectrum with the five lines of hydrogen with the corresponding wavelength ratios. Below, the fingerboard of the guitar with the frets at the semitones in equal temperament. The Till Eulenspiegel motif can be played on all six strings in different pitches.

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Hans Sommer

Hans Sommer around 1890.

The composer and natural scientist Hans Sommer may have conveyed the facts to Richard Strauss via the acoustic-musical variant of the optical-physical Balmer series. It is easy to understand that he was well informed about discoveries in optics and physics.

Hans Sommer was named as the son of Otto Gustav Zincken (* 1809; † 1940) was actually Hans Friedrich August Zincken called Sommer. The father of Otto Gustav Zincken was the Duchy of Brunswick Court Medical Officer Julius Leopold Theodor Friedrich Zincken (* 1770; † 1856), who in turn was the son of the judicial officer Carl Friedrich Wilhelm Zincken (* 1729; † 1806) and his wife Sophie Schläger. This marriage was divorced and both spouses remarried. Sophie Schläger married in 1782 Johann Christoph Sommer (* 1741; † 1802), who was a court councillor and professor of anatomy at the Anatomical-Surgical Institute in Brunswick and whose surname was henceforth used by her Zincken family descendants.

Hans Sommer's father died when he was two and a half years old. His widowed mother, Nanny Langenheim (* 1813; † 1902), married the entrepreneur, optician and pioneer of photography Peter Wilhelm Friedrich von Voigtländer in 1845. (* 1812; † 1878). His father was the optician Johann Friedrich Voigtländer (* 1779; † 1859), who had run the company J. F. Voigtländer, Werkstätte für optische und feinmechanische Instrumente since 1808 and was the descendant of the optician and inventor Johann Christoph Voigtländer (* 1732; † 1797).

Hans and Antonie Sommer had two sons Otto and Richard.

The family tree of Hans Friedrich August Zincken, called Hans Sommer

Hans Sommer was trained as a mathematician, and he was also taught in Göttingen by the physicist Wilhelm Eduard Weber (* 1804; † 1891). Already in 1866 he became professor of mathematics at the polytechnic Collegium Carolinum in Braunschweig. Twelve years later he was appointed rector, and he was involved in the transfer of the Collegium into the Herzogliche Technische Hochschule Carolo-Wilhelmina, which is now the Technische Universität Braunschweig. He conducted research in Braunschweig until 1884, particularly in the field of applied optics.

As one of the pioneers in applied optics, he also helped his stepfather Peter Wilhelm Friedrich Ritter von Voigtländer, who, together with his father Johann Friedrich Voigtländer, ran the optical works Voigtländer & Sohn in Braunschweig as an entrepreneur from 1849..

In 1858, Hans Sommer published his inaugural dissertation in Göttingen to obtain a doctorate in philosophy with the title Zur Bestimmung der Brechungsverhältnisse. In it, he also devotes himself in detail to refraction at the prism, with the help of which white light can be split spectrally in order to be able to recognise and measure spectral lines, for example..

In 1870, Hans Sommer published a book in Braunschweig entitled On the Dioptics of Lens Systems, in which he also discusses the refractive effect.

Hans Sommer and Richard Strauss jointly founded the Anstalt für musikalische Aufführungsrechte (AFMA) in 1903, which is considered the first predecessor organisation of the later Gesellschaft für musikalische Aufführungs- und mechanische Vervielfältigungsrechte (GEMA).

In 2019, the correspondence between Richard Strauss and Hans Sommer was published in book form.

Epilogue

Without the manifold research results and remarkable discoveries from the 19th  century, it would have been very difficult to develop quantum theory. And without the deeper understanding of quantum mechanics, there would probably be no semiconductors, which are present in almost every household today. In explicit reference to light emission, it should be noted that through knowledge of the many different discrete energy levels of semiconductors, light-emitting diodes can be produced today that emit monochromatic light at practically any wavelength in a wide spectral range from the infrared to the ultraviolet. Today, light-emitting diodes are used as energy-efficient light sources in lighting technology and in almost all screens.

So far, no written sources are known for the interdisciplinary hypothesis put forward here that the discovery of the integer ratios in the optical-physical Balmer series might have inspired the transformation into an acoustic-musical motif. However, this story was passed on orally by generations of physicists. The main author had heard it in the mid-1980s in a course at the Technical University of Berlin taught by Professor Gerd Koppelmann (* September 5, 1929; † September 21, 1992). The main author thanks his doctoral supervisor Professor Heinz Niedrig for the nice and informative obituary of his colleague Gerd Koppelmann.

Summary of the project

• Target audience: Musicians and natural scientists
• Learning objectives: Number ratios at light and sound frequencies.
• Book sponsorship/contact person: User:Bautsch
• Are co-authors currently desired? Yes, very much so. Corrections of obvious errors directly in the text; content please via discussion.
• Guidelines for co-authors: Wikimedia-like.