The Japanese Society for Aesthetics

Aesthetics No.28 (2024): 39-53

Conceptual Development and Interaction among Related
Disciplines in Hugo Riemann’s Musik-Lexikon:
Focused on Theory of Harmony *NISHIDA Hiroko
Kyushu University, Fukuoka
Abstract: This study aimed to elucidate how articles related to music theory and acoustics in Hugo Riemann’s(1849-1919) Musik-Lexikon changed during his lifetime from the first (1882) to the eighth edition (1916). In
the recent development of Riemann studies (Rehding 2003, Holtmeier 2011), the significance of Musik-Lexikon
has not been surveyed. This work includes many revised articles and various types of revisions. Specifically,
the publications of his Handbuch der Harmonielehre (1887) and Vereinfachte Harmonielehre (1893) brought
to the fourth edition (1894) a turning point in articles on harmony, whereas the new terminology and symbol
notation permeated gradually only until the seventh edition (1909). Regarding articles on acoustic fields, thecriticism of psychological and physiological methods led his “Ideen zu einer ‘Lehre von den Tonvorstellungen’”in the eighth edition (1916), which could truly contribute to music theory. Regarding harmonic dualism, anambivalence concerning the application of undertones was detected among the editions. Furthermore, theincremental historicization of his own theory and the evaluation of other theorists lost coherence in the text.
Thus, the editions of Musik-Lexikon can be considered as the essence of Riemann’s theoretical development,
revealing fluctuation in his descriptions. It suggests that Musik-Lexikon encouraged Riemann to situate hisfunctional theory within the broader history.
Keywords: Hugo Riemann, Musik-Lexikon, editions, music theory, acoustics

1. Introduction
1.1. Purpose
Recently, the neo-Riemannian theory has become popular in the field of music theory,
particularly in the United States. This theory, based on Hugo Riemann’s (1849–1919) theory of
harmony, has developed multifariously and has been applied to late Romantic and film music. In
addition, Riemann’s own theory has been re-examined. For example, Rehding (2003) historicized
his theory and aesthetics relating to the contemporary academic background, Nishida (2019)
discussed the relationship between his and neo-Riemannian theories, and Holtmeier (2011)
surveyed how his music theory had been received in German-speaking areas in the early twentiethcentury.
However, because Riemann covers a wide variety of writings, it is still difficult to grasp theentire picture and development. Therefore, this study focuses on Musik-Lexikon (1882–), where*

This paper is based on the Japanese version printed in Bigaku 72, No. 2 (2021): 48-59, published by theJapanese Society for Aesthetics.

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NISHIDA Hiroko

the development process was recorded, and elucidates how articles in music theory and acoustics
have changed from the first to the eighth (1916) editions in his lifetime.
1.2. Object and Preceding Studies
First, let us look at the synopsis of Musik-Lexikon. What concepts did Riemann have when
he published this lexicon? On the front cover of the first edition in 1882, the following is printed:“Theory and history of the music, the musicians of old and new periods, including informationon their works, with a complete organology [Theorie und Geschichte der Musik, die Tonkünstler
alter und neuer Zeit mit Angabe ihrer Werke, nebst einer vollständigen Instrumentalkunde]1”. In
its preface, Riemann noted that, because he assumed the “musicians and music lovers” were thereader, the lexicon would evoke the interest of “the highly educated musicians and the man inmusicology,” while, theories of composition, such as the theory of harmony and counterpoint,
were described only in the outline (Riemann 1882, V–VI). Furthermore, he appreciates thecollaborators in Western countries for providing biographical information2. Possibly, he intended
to create a comprehensive lexicon of Western music theory and history.
Subsequently, Musik-Lexikon was expanded, revised many times, and translated into
different languages. This study investigated eight German editions that Riemann himself engaged
in editing (Table 1). From the seventh edition onward, the title became Hugo Riemanns MusikLexikon.
As shown in Table 1, Musik-Lexikon continued to be revised throughout Riemann’s lifetime.
Thus, tracing this shift will also lead to an understanding of the development process of hismusical thoughts. However, previous studies referred to some articles only in a specific edition,and the descriptions in each edition were not compared. Above all, although A. Rehding touchedon several topics, such as tonality, undertones, and cadence (Rehding 2003, 48, 88, 103), he
concentrated on only one edition. This might be because descriptions in the lexicon were viewed
as supplementary data for understanding his main writings. Moreover, at the end of Rehding’sbook, 13 articles were translated into English (Rehding 2003, 186–198) that were all based on
the fifth edition. At this point, Rehding’s study did not detail the transition of each harmonic
concept, the characteristics of Riemann’s historical description, his understanding of the related
disciplines, and his later methodological discussions, although it examined his wide spectrum ofmusical thoughts from many perspectives. Additionally, Holtmeier’s reception study (2011) did
not touch on Musik-Lexikon. In this way, despite the current popularity of Riemann studies,interest in Musik-Lexikon itself has not been high.
Therefore, as the first step in grasping this writing systematically, this study focused on
articles on music theory and acoustics because Riemann left behind many related writings.
Presumably, both fields are related closely to each other, and cross-disciplinary arguments areindispensable, especially in the theory of harmony.

1

In the citation, the original German and translator’s (the present author’s) notes are provided within [], while
the English translation is provided within “”. The spelling in German is modernized.
2
Thus, Musik-Lexikon included complements and corrections by multiple persons. This study presupposes thatRiemann, as the editor in the lexicon’s revisions, controlled and approved intentions and actions of others.

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Table 1.
Musik-Lexikon and its type of revisions (boldface) in Riemann’s lifetime with his published writings referred to in this study
EditionYear

Revision type or title of publications

1875“Die objective Existenz der Untertöne in der
Schallwelle [The Objective Existence of theUndertones in the Sound Wave]”

1877

Musikalisches Syntaxis [Musical Syntax]

Edition

5th

Year

Revision type or title of publications

1898

Geschichte der Musiktheorie im IX.- XIX.
Jahrhundert [History of Music Theory in the
IX–XIX Century]

1900“vollständig umgearbeitete Auflage”

1903

Anleitung zum Generalbass-Spielen [Instructionof the Thoroughbass Play] (2nd edition)

1905“vollständig umgearbeitete Auflage”

1st

1882

2nd

1884

“vermehrte Ausgabe [Enlarged Edition]”

3rd

1887

“sorgfältig revidierte und mit den neuesten
Ergebnissen der musikalischen Forschungund Kunstlehre in Einklang gebrachte Auflage[Carefully Revised and with the Newest
Results of the Music Study and Theory of Art
in Accord Brought Edition]”

1905“Das Problem des harmonischen Dualismus [TheProblem on the Harmonic Dualism]”

1887

Handbuch der Harmonielehre [Handbook of 7th
the Harmonic Theory]

1909“vollständig umgearbeitete Auflage”

1891

Katechismus der Akustik [Catechism of the
Acoustics]

1912“ Tonhöhenbewusstsein und Intervalurteil

4th[Simplified 8th

1893

Vereinfachte
Harmony]

1894“vollständig umgearbeitete Auflage[Completely Reworked Edition]”

Harmonielehre

6th[Consciousness of Pitch and Judgment ofInterval]”
1916“vollständig umgearbeitete Auflage”

1916“Ideen zu einer ‘Lehre von den Tonvorstellungen’ [Ideas for a Study “On the
Imagination of Tone”]”

1.3. Method
Next, the selection process for the articles is described.
(1) The articles on music theory and acoustics were extracted from eight editions. Amongthem, short articles, and those only with references to other articles were excluded. Articles withpersonal names were refined to include only famous music theorists. Consequently, 167 articles
were extracted from the first edition. This included many terms specific to Riemann, such as
Klangschlüssel [key of Klang]3 and Klangvertretung [representative of Klang].
(2) The differences among editions were listed. Additionally, added, altered, or deleted
articles were identified.
(3) The list was narrowed to those that underwent major revisions (58 articles). Then, bycomparing altered passages between the latest article and its previous editions, the reasons behind
their alteration were examined.
3

Riemann used the term Klang with respect to sound, resonance, consonant triad, and others. The
Klangschlüssel indicated figures and symbols for chord designation, along with abbreviation, such as T, D, S (tonic,
dominant, and subdominant). Regarding Klangschlüssel, see Nishida and Yasukawa 2019, 100–102. In this paper,
Riemann’s terms and article names in German are provided with English translations within [] at the first appearance.

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NISHIDA Hiroko

In the next section, the overall course of revision is explored by analyzing the tendency to
add articles and delete articles, and new articles are scrutinized. In the third section, articles thathave undergone a major change and the reasons behind their alteration are discussed.

2. Transition in Articles
2.1. New and Deleted Articles
First, the characteristics were explored using a new article set (Table 2) and deleted articles(Table 3) through editions.
There were 43 new articles, indicating that new articles were added continuously, while 8
articles had been deleted since the fifth edition.
Table 2. New articles (listed in alphabetical order)
EditionYear

New articles

3rd

1887

“Agogik [agogics],” “Agogischer Accent [agogic accent],” “Schluss [close],” “Stumpf, Carl”

4th

1894

“Funktionen [functions],” “Parallelklänge [parallel Klänge],” “Plagalschluss [plagal close],” “Tanaka,
Shohé,” “Tongeschlecht [kind of keys],” “Wechseldominante [dominant of the dominant]”

5th

1900

“Accord [accord],” “arithmetische Teilung [arithmethic division],” “Basse fondamentale [fundamental
bass],” “Common chords,” “Farbenvorstellungen [representation of colors],” “Leittonwechselklang[leading tone exchange],” “Scheinkonsonant [apparent consonance],” “Subdominante [subdominant],”“Tonverschmelzung [tonal fusion],” “Variante [parallel keys]”

6th

1905

“Absolutes Ohr [absolute ear],” “Fondamental [fundamental],” “Gegenklang [counter Klang],”“Generalauftakt [general upbeat],” “Hören [hearing],” “Musikwissenschaft [musicology],” “Nonenakkord[ninth chord],” “Terznonenakkord [third ninth chord],” “Terzseptakkord [third seventh chord],”“Tonpsychologie [psychology of tone],” “Verminderter Dreiklang [diminished triad]”

7th

1909

“Akzent [accent],” “Anschlussmotive [appendage motive],” “Chromatisches Tonsystem [chromatic
system of tone],” “Klangstufen [degrees of Klang],” “Melodik [theory of melody],” “Verwandt [related],”“Zwischendominanten [secondary dominants]”

8th

1916

“Regula dell’ottava [rule of the octave],” “Resonanz [resonance],” “Sprechmaschinen [speaking
machines],” “Tongedächtnis [memory of tone],” “Überstülpung [covering]”

Table 3. Deleted articles
EditionYear

Articles

5th

1900

“Stammakkord [principal chord]”

6th

1905

“Oberdominante [upper dominant]”

7th

1909

“Gegenklang [counter Klang],” “Klangfolge [succession of Klang],” “Tonverschmelzung,” “Wesentliche
Dissonanzen [essential dissonances]”

8th

1916

“Mittöne [co-occurrence tones],” “Telephon [telephone]”

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2.2. General Characteristics
From these articles, the following three characteristics were highlighted.
(1) Especially from the fourth to the seventh editions, new concepts, terms, and denotation
systems on the theory of harmony appeared in each edition.
(2) As the number of articles in foreign languages has increased since the fifth edition, thelexicon has become multi-lingual4.
(3) Since the third edition, new articles on the psychology of tone and physics of tone haveincreased.
Among these characteristics, the first and the third ones were scrutinized.
2.3. Shift of the Theory and Representation of Harmony and Moment of Historization
The crucial changes in the first characteristics were that the Riemannian term “Funktionen”
was registered in the fourth edition (1894), and along with T, D, and S, the denotations, such as“Parallelklänge5” (added as an article in the same edition) and “Leittonwechselklang6” (added as
an article in the next edition), were introduced. The concept of functions that appeared in
Vereinfachte Harmonielehre in 1893 brought about this shift. Furthermore, the denotation of
Leittonwechselklang was updated from the fourth to the fifth edition7.
Additionally, in the article “Funktionen” in the fifth edition, the doctrine that “modulation
always appears as reinterpretation of a function into another [Modulation erscheint stets als
Umdeutung einer Funktion zu einer anderen]” (Riemann 1900, 351l8) was added, complemented
by the passage, “as already Rameau [Jean-Philippe Rameau, 1683–1764] recognized clearly [wiebereits Rameau klar erkannte]” (Riemann 1905, 407l) in the sixth edition, where his intention tohistoricize concepts on the functions could be observed9. However, in the seventh edition, thedescription of the modulation and reference to Rameau were deleted from the same article. This
correction emphasized that the new theory and denotation of harmony were Riemann’s own
invention. In the eighth edition, although there was a difference in wording, he attributed thetheory’s novelty to himself.
In some articles, the concepts were renewed. For instance, “Parallelklänge” was explained
as “Klänge that stand in the relation of relative key with the tonic [Klänge, die im Verhältnis derToniken von Paralleltonarten stehen]” (Riemann 1894, 787r) in the fourth edition, while it was
described as “Klänge that have the third interval in common [Klänge, welche das
4

After the fourth edition, English and French translations were published. About the relationship between thetranslated editions and the German original, see Nishida and Yasukawa (2021). This study sights on the German
edition and the relationship among related disciplines in it.
5Tp, Sp, and Dp are the relative chords of tonic, subdominant, and dominant chords, respectively. Regarding
Parallelklänge, see Figure 1.
6
Leittonwechselklang is the chord that changes major or minor nature of a chord, and the prime in minor triad
is half a tone lower than that in major triad. In Riemann’s theory, the prime in minor triad means the uppermost tone
(e.g., in a minor triad A-C-E, the prime is E). In this paper, the pitch names are written in German.
7
The fourth edition showed “0S2>,” for instance, while the next edition used letters of functions andsuperimposed < or >, such as .
8
The lexicon took two-column format; thus, the left and the right columns were distinguished as l and r.
9
Riemann published Geschichte der Musiktheorie im IX.–XIX. Jahrhundert in 1898 that seemed to promote
his attempt to historicize.

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NISHIDA Hiroko

Terzintervall gemeinsam haben]” (Riemann 1900, 835l) with an illustration (Figure 1) in thefifth edition. Correspondingly, it was grounded not at the key level but at the chord level.
Moreover, “Variante” and “Verwandt,” which were set up as an article each in the fifth andthe seventh editions, were necessary concepts for “Verwandtschaftstabelle [table of tonal
relationships]” (Riemann 1916, Ideen, 2010; Figure 2) shown in his treatise “Ideen zu einer ‘Lehrevon den Tonvorstellungen.’” In other words, Riemann attempted to link later editions to his own
evolving thoughts.

Figure 1. Illustration of Parallelklänge(Terz means the third; Riemann 1900, 835l)

Figure 2. Verwandtschaftstablle(Nishida and Yasukawa 2019, 97)

2.4. Toward a Cooperation with Psychology and Physics of Tone
Riemann continued to absorb the latest studies in the psychology and physics of tone. An
increase in his interest can be observed in the third edition (1887). Stumpf (1848–1936) was set
up as an article where his Tonpsychologie, Vol.1 (1883), which discussed human perception andjudgment of tones, was introduced as a development of Hermann von Helmholtz’s (1821–94) Die
Lehre von den Tonempfindungen [On the Sensations of Tone] (1863). In the fifth edition, an
article on “Tonverschmelzung11” was added but deleted in the seventh edition. This deletion was
related to the fact that an article on “Tonpsychologie” as a new field was added in the sixth edition.
In other words, a comprehensive view of the academic field was preferred to the article on an
individual. In relevant articles, a need for the future interaction between this field and music
theory has been commented upon: “the benefit that now comes out from their research for thefoundation of music theory is a very smaller one [der Nutzen, der bis jetzt aus ihren
Untersuchungen für die Grundlagen der Musiktheorie herausspingen, ein sehr kleiner ist]”(Riemann 1905, 1348l).
In addition, the emerging terms in the psychology of tone in the sixth edition werediscussions from cognitive science, such as “Hören” and “Absolutes Ohr.” In his “Ideen zu einer‘Lehre von den Tonvorstellungen’” in 1916, Riemann devoted pages to the arguments on theabsolute ear. In the eighth edition published in the same year, Riemann’s own treatise“Tonhöhenbewusstsein und Intervallurteil” (1912) was listed in the article on “absolute ear,” anda new article on “Tongedächtnis” was set up. To put these facts together, from the 1900s,
Riemann became more concerned with the topics of sensation and psychology in listening tomusic.

10

To distinguish from the eighth edition of Musik-Lexikon published in the same year, references to the “Ideen
zu einer ‘Lehre von den Tonvorstellungen’” was provided as “Riemann, 1916, Ideen.”11
It is the concept to distinguish consonance and dissonance not as physical phenomena but psychologically.

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45

Moreover, along with the article “Farbenvorstellungen,” the article “Scheinkonsonant,”
which was a concept based on the difference between physical and psychological consonances12,
was created. Subsequently, one can see that the results from related disciplines were assimilated
into music theory.
However, “Tanaka, Shohé,” a new article on the physics of tone, was in the fourth edition,
where the instrument harmonium based on just intonation was introduced.
Thus far, we have examined the characteristics of both the new and deleted articles. Whatcharacteristics can be found in the existing articles?

3. Transition in Descriptions
3.1. Articles that Underwent a Major Change
Along with new and deleted articles, the procedure indicated in Section 1.3, 58 existingarticles were excluded. Approximately 45 articles were scrutinized based on the characteristics
described in Section 2.2.
3.2. Gradually Prevailing Theory and Representation of Harmony and Revised Historization
The theory and representation of harmony reached a tipping point in the fourth edition(1894), following the publication of Vereinfachte Harmonielehre (1893). However, as presumed
in the previous section, these changes did not occur simultaneously. The denotation based on
harmonic dualism, such as c+ and 0c13, the representation using functional symbols, such as T, D,
and S, and new terms were gradually identified to have a close connection (Table 4).
Table 4. Articles that showed a change in theory and representation of harmony
ArticleAlteration

Auflösung [resolution]

Functional symbols were added in the 6th edition.

Chroma [chroma]

Functional symbols were added in the 5th edition, and the name of the article was altered to“Chromatik” in the 7th edition.

Dissonanz [dissonance]

Chord symbols, such as g7 in the 4th edition, were replaced with functional symbols in the 5thedition.

Dominante [dominant]“Oberquintklang [Klang of upper fifth]” and “Unterquintklang [Klang of lower fifth]” were,respectively, replaced with “Dominante” and “Subdominante” in the 5th edition. Riemann insisted
his novelty differed from his predecessors, such as Rameau, in that “he takes the initial letter of thethree functions (T, D, and S) as sole basis [er die Anfangsbuchstaben der drei Funktionen (T, D,
and S) als alleinige Grundlage durchführt]” (Riemann 1916, 254r) in the 8th edition.

Dreiklang [triad]

Chord symbols by Arthur von Oettingen and Riemann., such as c+ and 0g, had introduced in the
4th edition.

Durtonart [major keys]

T, D, S, Tp, Sp, and Dp were elucidated in the 4th edition. In describing Parallelklänge, thepassage, “either as par a l l e l Kl änge or as l e a di ng t one e xc ha nge [entweder als

12

For example, the transposed C–E–A of a triad A–C–E consists of physically consonant intervals, but it wasconsidered as a dissonant chord.
13
c+ means the major triad C–E–G, while 0e means the minor triad A–C–E.

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NISHIDA Hiroko

Pa r a l l e l kl ä nge oder als Lei t t onwechs el kl änge ]” (Riemann 1900, 281l), was added in the
5th edition. The symbol of “Mollsubdominate (0S) [subdominant in minor keys]” (Riemann 1905,327l) in major keys was added in the 6th edition.
Fétis, François-Joseph[1784–1871]

The passage, “Der moderne Begriff der Tona l i t ä t [the modern concept of t ona l i t y]” (Riemann
1900, 318l), included until the 5th edition was deleted, and Fétis was evaluated as “in diesem
Gebiete […] nicht eigentlich produktiv [not really productive in the theory of harmony]” (Riemann
1905, 369r) in the 6th edition.

Fundamentalbass[fundamental bass]

The explanation about Rameau was deleted in the 4th edition. Rameau’s fault was pointed out inthe 5th edition. It was corrected and replaced with the description that functional symbols were “afurther formation useful for the practice of Rameaus’s fundamental bass [eine für die Praxis
brauchbare Weiterbildung von Rameaus F.]” (Riemann 1916, 337r) in the 8th edition.

Fünfstufige Tonleitern[five-degree scale]

Functional symbols appeared in the 5th edition and “(pentatonisch) [pentatonic]” was added to thearticle name in the 6th edition.

Generalbass[thorough bass]

Riemann’s Anleitung zum Generalbass-Spielen (second edition, 1903) was explained as a bookequipped with both thorough bass figures and functional symbols in the 7th edition.

Harmonielehre[theory of harmony]

The criticism against Rameau’s fundamental bass was deleted and replaced with the passage inwhich he proposed the theory leading to Klang and tonality in the 5th edition. The description “thatthere are only t hr e e f unc t i ons i n ha r mony , […] and that modul a t i on is nothing more thanan exchange of these functions [dass es nur dr e i Funkt i onen der Har moni e gibt, […] unddass Modul a t i on nichts anderes ist als ein Wechsel dieser Funktionen]” (Riemann 1909, 568r)
was added in the 7th edition.

Hauptmann, Moritz[1792–1868]

The negative evaluation was reduced in the 3rd edition. The description about the connection
between Hauptmann and Oettingen as well as Riemann was deleted, and Hauptmann’s fault was
bespoken in the 5th edition. The positive evaluation was added in the 7th edition.

Kadenz [cadence]Six-four chord was denoted by functional signs in the 4th edition. Progressions of T-D-T, T-S-DT, and T-S-T were elucidated in the 7th edition.

Klangschlüssel

An explanation of functions was added in the 4th edition.

Modulation [modulation]

The reinterpretation of functions was added in the 5th edition.

Mollakkord [minor chord]It was insisted that the symbol of minor chords that Oettingen introduced be first used consistently
by Riemann in the 3rd edition. The symbols, such as 0e, were supplemented in the 5th edition.

Molltonart [minor key]

Functional symbols were used in the 4th edition.

Neapolitanische Sexte[neapolitan sixth]

It was explained as “the leading tone exchange of the subdominant [Der Leittonwechselklang der
Subdominante]” (Riemann 1900, 778r) in the 5th edition.

Nebendreiklänge[secondary triads]

Functional symbols were added in the 4th edition.

Nebenseptimenakkorde
Functional symbols were added in the 4th edition.
[secondary seventh chords]
Nebentonarten[secondary keys]

Along with relative and dominant keys, relative keys of dominant and parallel keys of the main key
were included in the 7th edition.

Phyrigische Tonart[phyrigian key]

Functional symbols were added in the 6th edition.

Schlüssel [clef]

Functional symbols were added in the 5th edition.

Septimenakkord[seventh chord]

Riemann’s own definition as “only the seventh chords that attach the upper seventh (of thefundamental tone) to the major chord or attach the lower seventh (of the highest tone) to the minor

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chord [nur die Akkorde S., welche dem Durakkorde die Oberseptime (des Grundtones) oder dem
Mollakkord die Unterseptime (des höchsten Tons) beifügen]” (Riemann 1900, 1046r) was addedin the 5th edition. Seventh chords on every degree in C major were represented in the 6th edition,
which contradicts the definition in the 5th edition. Another definition was provided in the 7thedition.
Stimmführung[voice leading]

Klangschlüssel, such as < and >, were explained in the 5th edition.

Stufe [degree]

Riemann described the following, “since Gottfried Weber […], the numbers of degrees also form
the basis of an analytic denotation of the chord, from which H. Riemann has developed his
denotation of function [seit Gottfried Weber […] bilden auch die Stufenzahlen die Grundlagen
einer analytischen Akkordbezeichnung, aus welcher H. Riemann seine Funktionsbezeichnung
entwickelt hat]” (Riemann 1909, 1374l), in the 7th edition.

System [system]

Functions were explained in the 4th edition. The explanation that this concept dated back to
Rameau was added in the 5th edition. “Oberdominante [upper dominant]” and “Unterdominante[lower dominant]” were, respectively, replaced with “Dominante” and “Subdominante” in the 6thedition.

Tonalität [tonality]

It was defined as “the modern concept of the key extended by Fétis [der moderne durch Fétis
erweiterte Begriff der Tonart]” (Riemann 1894, 1080l) in the 4th edition. This was deleted andcorrected as a concept proposed by Rameau and named by Fétis in the 5th edition. Additionally,
the French “Tonalité” was attached to the article name and the functional concept was explained
as a recent theory.

As shown in Table 4, all functional symbols were introduced in the fourth edition. Somesymbols first appeared in the fifth and later editions. In the fifth and sixth editions, in place of
preceding dualistic terms (“Ober-” and “Unter-”), functional terms, such as “Dominante” and“Subdominante,” came to be used.
How about other terms? In the article “Durtonart” in the fourth edition, the functions
including Parallelklänge were elucidated, while Leittonwechselklang was first explained in thefifth edition. “Neapolitanische Sexte” was also reinterpreted by Leittonwechselklang in the sixth
edition. Given that Leittonwechselklang appeared as an article, it has been demonstrably
recognized as an authoritative term since the turn of the century. In the prevailing terminology
process, cases existed in which a gap in definition was found among editions, such as in“Septimenakkord.”
In later editions, in the article “Nebentonarten,” its range was extended to the “Variante.”This concept became one of the bases of the abovementioned “Verwandtschaftstabelle.”
As further characteristics in later years, there was an attempt to historicize the functional
concepts, as in “Dominante” in the eighth edition, “Stufe” in the seventh edition, and “System”
and “Tonalität” in the fifth edition14. Especially in the article “Tonalität” in the fifth edition, byplacing functional concepts as recently invented, it was suggested that they had never been
influenced by Fétis who died in 1871. This revision, when the novelty of Riemann’s concepts
was emphasized in the article “Funktionen” in the seventh and eighth editions (see Section 2.3)
and “Dominante” in the eighth edition, reinforced such novelty. Moreover, since the fifth edition,

14

Riemann’s evaluation on Fétis and Rameau was partly discussed in Section 3.3 in Nishida and Yasukawa(2021); thus, what was not mentioned there is examined here.

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NISHIDA Hiroko

the criticism against Rameau was deleted in the article “Harmonielehre”; instead, the linkage tohim had been stated. Thus, contradictions arose among editions when Riemann related himself
to Rameau in the history of the theory of harmony and simultaneously emphasized the novelty of
his own functional concepts.
Various factors underlie the oscillations in his descriptions: He might have been influenced
by research trends on Fétis and Rameau within France in the 1900s (Nishida and Yasukawa 2021).
By contrast, Riemann had already criticized Fétis and amplified Rameau’s theory of harmony inGeschichte der Musiktheorie im IX.–XIX. Jahrhundert in 1898. He argued that “it is probably
finally only the word [of tonality] that we owe to FÉTIS and maybe an extension of the meaningof the concept (including the modulations); however, even this is doubtful [Im Grunde ist es wohl
schliesslich nur das Wort, was wir FÉTIS verdanken und vielleicht eine Erweiterung der Bedeutung
des Begriffes (einschliesslich der Modulationen); selbst dies ist aber fraglich]” (Riemann 18998,
451, 456f.). This means that the historicizing descriptions observed in Musik-Lexikon since the1900s became summaries that disseminated the output of studies on the history of theory thatRiemann conducted before the turn of the century beyond a country (see also note 9).
3.3. Conjunction with the Fields in Acoustics
Next, the conjunction with the field of acoustics is to be examined. The articles where thedescriptions related to psychology and physiology of tone were found were “Hauptmann” and“Klang.” In the former, it had been insisted until the second edition to take not Helmholtz’s
physicological method, but the psychological method based on “eine logische Aktivität desGeistes [a logical activity of mind]” (Riemann 1884, 384l). However, this description was deleted
in the third edition (1887). In the fifth edition, Stumpf was introduced as a person who showedideas contrasting Helmholtz’s theory.
Coincidentally, in the article “Klang” in the fifth edition, Stumpf’s Tonverschmelzung was
explained, although the name Stumpf disappeared in the seventh edition. The reason was
discussed in Section 2.4. Additionally, Riemann was skeptical of Stumpf’s method, which
presumably influenced this deletion. In fact, in the seventh edition, the term physiology waspreferred to psychology, evident in the rewritten title of the article “Tonpsychologie” to“Tonphysiologie (Physiologie der Tonempfindungen) [physiology of tone (physiology ofsensations of tone)]” and he stated that “all efforts to give a physiological or psychologicalfoundation to music theory have fallen into water [alle Anstrengungen, eine physiologische oderpsychologische Fundamentierung der Musiktheorie zu geben, sind ins Wasser gefallen]”(Riemann 1909, 1433l). Furthermore, Stumpf was criticized in Riemann’s “Ideen zu einer ‘Lehrevon den Tonvorstellungen’” (Riemann 1916, Ideen, 1–2).
In his “Ideen zu einer ‘Lehre von den Tonvorstellungen’” in 1916, Riemann achieved aturnaround from logical thinking, as seen in his Handbuch der Harmonielehre (1887) andVereinfachte Harmonielehre (1893), where he attempted to thoroughly apply musical logic, toharmonic theory, which emphasized psychological thinking (Nishida 2019). However, thispsychological thinking in his later days should not be equated entirely with his interest in
psychological methods expressed until the second edition. Considering a series of his

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49

abovementioned criticisms against Stumpf in the 1900s15, one can interpret that Riemann came
to express “Ideen zu einer ‘Lehre von den Tonvorstellungen’” as a new method that overcame
Stumpf’s psychology of tone.
The articles related to the physics of tone are listed in Table 5.
Table 5. Articles that showed a change in physics of tone
ArticleAlteration

Akustik [acoustics]

Recent works of literature were added in the 5th edition. Reference to the Tonspsychologie wasadded in the 6th edition. Works of literature on room acoustics were introduced in the 7th edition.
The field was provided as a subsection within the article in the 8th edition.

Chroma [chroma]In the 3rd edition, it was explained that the difference of comma (interval difference) in justintonation was unnecessary in practice, and the enharmonic interpretation depended on the context.
Jankó clavier by Paul von Jankó was introduced in the 4th edition.

Enharmonik[theory of enharmonic]

The description that reinterpretation was impossible in just intonation in the 6th edition. Works byquarter-tones were introduced in the 7th edition.

Harmonium [harmonium]

Works of literature by Shohei Tanaka and Riemann were added, and music on just intonation was
evaluated as aesthetically unacceptable in the 4th edition.

Klirrtöne [clank tones]

In the 2nd edition, the following was added: “still, the editor of this lexicon succeeded in producing evenlower undertones [doch gelang es dem Hearusgeber dieses Lexikons, die Untertöne noch weiter hinab
hervorzubringen]” (Riemann 1884, 467r). The subject in this sentence has been deleted and replacedwith “it is certainly successful [es gelingt auch wohl]” (Riemann 1905, 679r) since the 6th edition.

Komma [comma]

The symbol of syntonic comma by Oettingen and others was added in the 7th edition.

Konsonanz [consonance]

The definition by physicists in the 5th edition was introduced. The word “Physiker” was replacedwith “physicists and psychologists of tone [Physiker und Tonpsychologen],” and references toTonpsychologie were given in the 6th edition (Riemann 1905, 697l). It was insisted in the 8thedition that Helmholtz’s “Lehre von den Tonempfindungen” should be developed into the “Lehrevon den Tonvorstellungen.”

Oettingen

It was added that Oettingen advocated introducing just intonation into practice in the 7th edition.

Reine Stimmung[just intonation]

The attempts to increase the number of keys by temperaments other than equal temperament werereferred to, and the same kind of attempts in the 17th century were supplemented in the 7th edition.

Schwebungen [beats]

In the 5th edition, Rameau and Helmholtz explained that dissonance was based on beats, which,however, was proven uncertain.

Temperatur [temperament]

The skepticism against the practicality of the harmonium was shown in the 4th edition.

Tonbestimmung[tone designation]

A table and algebraic denotation by Q (fifth) and T (third) were added in the 4th edition.
Additionally, the frequency of vibration was elucidated in the 6th edition.

Untertöne [undertones]

The existence and the audibility of undertones were preached until the 3rd edition. In the 4thedition, it was explained that as the output in Katechismus der Akustik, undertones were inaudible
because of the interference of overtones.

Thus, his arguments on the physics of tone are divided into articles on just intonations andnew instruments based on them and those on physical phenomena such as beats and temperament.

15

About Riemann’s other criticism against, see Rehding 2003, 52–54, 108–109.

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NISHIDA Hiroko

In the first type of articles, such as “Harmonium” and “Temperatur,” his personal view was thathe could not accept music based on just intonation aesthetically, and he was skeptical of its
practical utility.
Subsequently, it could be assumed that Riemann strongly recognized the distance between
physical studies of just intonation and music-theoretical studies on the “interpretation” of tones
and harmony. This is clearly seen in the article “Konsonanz” in the eighth edition. Here, thenecessity of “Ideen zu einer ‘Lehre von den Tonvorstellungen’” was insisted, though with theprovision that “if the acoustics should keep full contact with the theory of music [wenn die
Akustik vollen Kontakt mit der Musiklehre erhalten soll]” (Riemann 1916, 576r). To paraphrase,
Riemann confirmed that acoustics could scarcely contribute to music theory if it remained inphysiology and physics.
The second argument relates to harmonic dualism, which is one of the importantconjunctures between music theory and the physics of tone, will be discussed in the next section.
3.4. Shift of the Relationship between Harmonic Dualism and Undertones
Riemann published “Die objektive Existenz der Untertöne in der Schallwelle” in 1875 andcontinued to insist that the undertones be audible in Musikalische Syntaxis in 1877. However, in
Katechismus der Akustik in 1891, he acknowledged that the undertones were not audible andrejected his attempts to derive a minor triad from the undertones in “Das Problem desharmonischen Dualismus” in 1905. How the undertones were described in Musik-Lexikon would
become an even greater concern in Riemann’s studies because his claim on their audibility
aroused criticism16.
He conceptually discussed harmonic dualism in the articles “arithmetische Teilung,”“Dualismus, harmonischer [dualism, harmonic],” “Moll,” and “Oettingen.”In the article “arithmetische Teilung,” in the seventh edition, the description was added that
Gioseffo Zarlino (1517–90) constructed a “dual foundation of the theory of harmony [duale
Fundamentierung der Harmonielehre]” (Riemann 1909, 50l) from the contrast between arithmetic
and harmonic divisions. This is an attempt to trace the concept of harmonic dualism back to
Zarlino.
About “Dualismus, harmonischer,” although only references were provided until the third
edition, a specific explanation was added in the fourth edition. However, this description is a
general description without individual names. Only since the sixth edition, the advocators havebeen listed, such as Zarlino, Rameau, Hauptmann, Oettingen, Riemann, and so on, and the terms“Obertöne [overtones]” and “Untertöne” appeared (Riemann 1905, 318r). However, in the eighth
edition, the terms of overtones and undertones disappeared, and they were replaced with the
problem of interpretation: “Dursinn [sense of major]” and “Mollsinn [sense of minor]” (Riemann
1916, 264r). Riemann abandoned the derivation of the minor triad from undertones in 1905.
Therefore, he could have deleted the relevant terms earlier (sixth edition). In contrast, in the article“Untertöne” in Table 5, the description was altered already in the fourth edition just after the
publication of Catechism of the Acoustics (Riemann 1894, 1111l). Here, the evolution of
16

For example, see criticism in Ellis (1877) and Schafhäutl (1878). Cf. Hui 2012, Chap.2.

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Riemann’s own studies is immediately reflected. While, as seen in the article “Klirrtöne,” hewithdrew his insistence not until the sixth edition. Thus, his attitude toward the undertones was
not consistent. Riemann presumably maintained harmonic dualism as a framework for his
thinking and interpretation, even if he could not verify the existence of undertones by acoustic
experiments.
In the article “Moll” in the seventh edition, it is explained that the major and minor triads
are the Klänge that consist of tones in the simplest relationships in frequency and sound wave,
respectively (Riemann 1909, 933r). This would be a new foundation for dualism, instead of acontrast between overtones and undertones. In the article “Oettingen” in the same edition, theexplanation of harmonic dualism was added, where the history of harmonic dualism from Zarlino
to Riemann was traced.

4. Conclusion
4.1. Musik-Lexikon in Riemann Studies
From the abovementioned investigation, it became clear that Riemann, feeling the gap
between constantly advancing studies on temperament and hearing and music-theoretical studies,
gradually constructed and reinforced the theory of functional harmony and the history of dualistic
harmony. Thus, one can read the minute alterations in the eight editions of Musik-Lexikon as an
essence of the development of thought in his enormous oeuvre.
On the one hand, discrepancies were observed between the descriptions in his other writings
and those in Musik-Lexikon (as discussed in Sections 3.2 and 3.4). Even if his concepts wereseemingly renewed, one can grasp the process by which his theory advanced and became
established with oscillations. On the other hand, before the concepts were proposed in his books
and treatises, Musik-Lexikon sometimes prepared the necessary concepts and updated theirdefinitions, which could not be read in his main products (as seen in Section 2.3). Furthermore,differences in psychological thinking between the early and later editions were suggested, which
could be traced to the descriptions in Musik-Lexikon (as seen in Section 3.3).
Such fluctuating shifts represent knowledge that can only be found in lexicons with multiple
editions. At this point, Musik-Lexikon can become one of the systematic materials to survey theintellectual connections between music theory and other related disciplines.
4.2. Self-historicization of Riemann
Inconsistencies in his attempts to criticize others and historicize himself emerged as his
descriptions of Fétis and Rameau concerned twisting the history of the harmonic theory. This
inconsistency came to a head not in the description of facts but in their “historicization.” Thesechanges may be considered inevitable results reflecting the research outputs in each phase.
However, alongside it, considering that this lexicon had already been disseminated outside
Germany through multi-language editions during his lifetime (see note 4), one should feature therole of Musik-Lexikon as the “essential tool of Riemann’s ‘propaganda’” (Holtmeier 2011, 4).
This nature of Musik-Lexikon would partly encourage Riemann to correct the history throughlater editions to position his theory of functional harmony, which systematized the logic of52

NISHIDA Hiroko

harmony into a bigger history over countries (see Section 3.2).
Finally, Riemann seems to arrive at the more fundamental, methodological skepticism thata music theory could not be grounded by the contemporary physiological, psychological, andphysical methods through his explanations and criticisms of concepts in related disciplines, suchas “Tonverschmelzung” and “Reine Stimmung.” Notably, from them, the new method of “Ideen
zu einer ‘Lehre von den Tonvorstellungen’” was proposed in the eighth edition (as shown in
Section 3.3) as an action to self-historicize Riemann. To shift one’s perspective, Riemann’sability to develop methodological arguments with related disciplines could be attributed to his
environment, in which he was continuously engaged in editing the multi-disciplinary lexicon.
Considering everything, constructing the history of the harmonic theory and editing articles in
the lexicon formed the background of the new methodology in his later period, as if they wereinterconnected.

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Cochlea,” Hermann Helmholtz, On the Sensations of Tone as a Physiological Basis for the
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Helmholtz, Hermann von. 1863. Die Lehre von den Tonempfindungen als physiologische
Grundlage für die Theorie der Musik. Braunschweig: Friedrich Vieweg und Sohn.
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Acknowledgement
This study was supported by JSPS KAKENHI Grant Number JP20K00128.