Neo-Riemannian Analysis Research Papers

A New Theory of Harmonic Motion and Its Application to Pre-tonal and Tonal Repertoire

In the introduction of the paper, the components of tonality, its emergence in the Renaissance period and also important concepts from the tonal theories of last three centuries will be explained, in order to provide a background for the... more

that way, Rameau explains any sonority either as a triad or a seventh chord.

Figure 1.2 : The perfect and imperfect cadences in major, from Rameau’s Traité de in the fundamental bass and implied dissonances.

transformations, P (Variation) R (Paralel) and L (Leittonwechsel), as in Figure 1.3.

Figure 1.4: Hyer’s Table of Tonal Relations (Cohn, 1998: 172) transformations can be seen as vectors between triadic triangles (ibid., 172).

pairs, would give us the one of the three octatonic collections of the system. While minor third related dominant chords are not commonly used in successions or

As it is possible to observe from the figure, all pitch classes are exclusive to one

Table 2.1 : Motion types according to root interval between chords.

Figure 2.3 : Beethoven, Symphony No.7, ii, mm.1-18 (Piano arr. by Franz Liszt)

Figure 2.4 : Mozart, Piano Sonata No.7 in C Major, K309/284b, iii, mm. 1-5 (MSLP) retrogressive motions.

the igniter of the T-S-D-T cadential progression. Figure 2.5 : Mozart, Symphony No. 40 in G Minor, K.550, mm. 1-22 (Piano arr. by August

regular dominant seventh chord with a passing dissonance.

Figure 2.7 : Beethoven, Piano Sonata No.14, Op.27 No.2 (“Moonlight”), mm. 1-5 (MSLP) chord is shown in the excerpt from the theme of Beethoven’s “Moonlight” Sonata. Other chromatic chords such as the diminished triad in the next example from

Figure 2.8 : Beethoven, Piano Sonata No.21, Op.53 (“Waldstein”), mm. 257-260 (IMSLP) » as the raised eleventh (or flattened fifth) of an altered dominant chord.

Figure 2.9 : The four hexatonic systems” (Cohn 1996:17)

basic form. For the sake of simplicity, however, the combined system is presented in its most

Figure 2.11 : J.S. Bach, Prelude and Fugue in C major, BWV 846, mm. 7-9 UMSLP) to its triadic constituents, C major and E minor.

Figure 2.12 : Beethoven, Piano Sonata No.21, Op.53 (“Waldstein”), mm. 274-275 (IMSLP) between the preceeding and succeeding chords.

Figure 4.1 : Tymoczko's map of tonal grammar in major (2010: 9) model, when aforementioned exceptions are taken into account.

Table 1.1 : Riemann's Schritte & Wechsel transformations (Klumpenhouwer 2008: 471) 1.3.3 Neo-Riemannian Theory

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Functional Harmony: Review and Update of the System (English revised and enlarged version)

by Enrique Igoa

2023

The harmonic-functional system is based on a conception of harmony and the chords of the tonality derived from the notion of differential hierarchy, fifth relationships and functional affinity, whose consequence is the recognition of... more

From the basic harmonic intervals (fifths, major and minor thirds), Rameau generated their two basic types of chords, the triad and the seventh chord. All other chords can be derived from these (although in some cases with certain difficulties). On the other hand, he considered that the perfect movements of the bass were those of fifth, followed by those of third (harmonic intervals), being those of second considered like dissonant, for which reason he replaced in the fundamental bass certain sounds of the real progression with others that always allow the exclusive use of the first two types of interval, although in the thoroughbass the seconds were included (see Rameau 1722; Lester 1992 and 2002).

Fig. 2 Basic functions according to Riemann. Coupling of tonic function with dominant and subdominant (Riemann 1893: 8) Riemann begins by isolating as Hauptklang (main sonority or main chord) the tonic chord (T), which is the first basic function o f tonality. From it he derives directly the chord built above the tonic with the higher sound of the tonic as fundamental, that is, the dominant one (D), and the chord built below with the fundamental of the tonic as fifth of the new chord, that is, the subdominant c the tonic the cast of the three basic functions, t expresses it in a graph (Fig. 2) that resembles t hord (S). Both chords complete with he essence of harmony. He himself he one shown above to explain the symmetric relationship in Rameau (Fig. 1), and summarizes it in this sentence: “There are only three tonal functions in Harmony (significant functions within tonality), namely, the Tonic, Dominant and Subdominant. In the change!” of these functions lies the essence of modulation” (Riemann 1893: 9).!3 Dahlhaus puts the basic functions in relation to the cadence to explain the tonality: “In contrast to the theory of degrees [Stufen], the functional theory of Riemann begins with the cadence T-S-D-T in order to establish the tonality, and deduces the scale by analyzing the three main chords (C-E-G, F-A-C, G-B-D = C-D-E-F-G-A-B-C)” (Dahlhaus et al., 2001: 862). The derivation of the basic functions in minor mode is equivalent to that of the major mode, but the three resulting chords (t, d, and s) are minor, since they are taken from the natural scale. It is appropriate to remember, precisely, the relationship that establishes Riemann between the major and minor chords, or, put another way, the explanation of the origin of both chords: “From the combination of a sound with its major third and its upper fifth originates the higher resonance (major chord), and from its combination with its lower major third and its lower fifth originates the lower resonance (minor chord)” (Riemann 1893: 10). This gave validity to the theory of Rameau (strings of F and A flat that vibrate by sympathy below the C) and to the concept of lower resonance, an idea inherited from Arthur von Oetttingen (1836-1920), who formulated in 1866 the theory of «harmonic dualism», by which the minor triad is generated below a fundamental tone, as opposed to the major triad, which is generated above that note.

Fig. 3a Chart depicting the harmonic regions in a major mode according to Schénberg. Those included in the cross are considered as the closest to the tonic (Sch6nberg 1954: 20)

Fig. 3b reproduces the same chart as Fig. 3a using the terminology of the revised harmonic-functional system presented below. It can be seen that the three basic functions of Riemann, together with one of the derivative chords (Tp, the sixth degree are included in the group of the five harmonic regions considered as closest, to which Schénberg adds the minor tonic (t). However, it leaves the chords both on the third and the second degree outside the central cross (Dp, Sp). In any case, the origin of all near and far chords is very visible from the three basic functions, by changing the mode of the basic function itself and/or that of its derivative. In paragraph 3.8.4, dedicated to the levels of intermodal relationship, a similar formulation is proposed, including all the possible combinations of chords within the tonality.

Fig. 4 shows an apparent incoherence in the system which needs to be resolved. As it is evident, the three relative chords of the basic functions (Tp, Sp, Dp in major mode; tP, sP, dP in minor mode) are always chords belonging to the tonality (diatonic degrees of to the key), but two of the counter-chords have accidentals outside the key signature. None of them are derivatives of the tonic function T/t, coming instead from a major dominant (D) and from a minor subdominant (s), and being the relation of counter-chord the cause of this anomaly, it seems to indicate a lesser closeness of this secondary function with respect to the basic function than that of relative function. Nevertheless, the principle governing the construction of relatives and counter-chords requires that the resultant chords must be stable, that is, they must have a perfect fifth, and this is the one of the reasons for the accidentals.”°

The harmonic-functional system, on the contrary, keeps the numbering of the chord sounds in all the positions of the chord, thus applying the concept of function also to the role of each member of the chord, a function that does not change with inversion. That is, the fundamental (first or octave) is always the fundamental, even though it is not in the bass (and even when it is omitted), and the same happens with the third and the fifth.*! Thus, in the first inversion the third is in the bass, and above are the fifth and the octave of the fundamental (and perhaps the duplicated third). In the second inversion the fifth is in the bass, and above are the octave and the third (and perhaps the duplicated fifth). All this is reflected in the harmonic analysis with a 3 or a 5 at the bottom of the letter that represents the chord in question, thus avoiding the combinations of numbers that are used in the traditional ciphering of inversions (Fig. 5). ?8 To be exact, the Neapolitan sixth is not exclusive of Naples (it was used throughout Italy in the expressive context of opera and oratorio), nor is it an inversion, as it might seem today, of a supposed chord (just the sG) which — curiously enough — was almost never used in its root position (at least until 19" century). The origin of this chord will be clear below, in the light of the principle governing the construction of subdominant function.

Fig. 6 The authentic cadence. Diagram of possibilities The authentic cadence is melodic, harmonic and rhythmic process used to confirm periodically the main key in a work, to reach a secondary tonic or to deviate to any of the possible tonicizations which take place along it (Fig. 6). The final goal of an authentic cadence is a chord with tonic function in major or minor mode (T/t). The function which announces and prepares the arrival to this tonic is a dominant chord, always in major mode (D). The usual chords which, in turn, precede the D are: a chord with subdominant function (S/s) and/or a chord with dominant of dominant function (PD). In many cases both chords are used, almost ine. in the order S/s > PD — D, because in this way the implicit chromaticism 4-4 # - 5 will be realized according to the usual melodic rule. Of course, there are also examples in which the order changes to PD — S/s > D, and this provokes an irregular melodic motion (4 f - 4 4 5), the usual melodic rule. Of course, there are also examples in which the order changes to

The observation of this figure allows us to answer some of the pending questions. One of them is the denomination of the diminished chord on the VII degree. This chord is considered here as a D’ without fundamental [ D’ ] and, therefore, incomplete, weaker, although its destiny remains the tonic and its function that of dominant.**

Fig. 8 Origin and some variants of chords with dominant of dominant function cannot be considered a basic function, but the construction of its chords forming an extended family equivalent to that of the D (although in this case built on the II degree of the scale) and its destiny in the dominant one, make the PD a function strongly rooted in the cadential process, within the group of predominant functions, which allows to draw in many cases a cycle of fifths that goes from the II degree to the V and the tonic (°D - D - TA). Therefore, the conditions for this function not to lose its identity and the possibilities of combination of sounds are exactly the same as in the dominant function (see Fig. 8).

Fig. 9 Usual chords of the subdominant function without or with minor seventh (II° and II°), which poses two problems. The first is that the IV - V - I melodic sequence in the bass is omitted in the cadence, being as it is by far the most used in this context. The second trouble originates in the apparent contradiction that arises when ciphering the two chords as inversions of supposed chords in fundamental position (II and II’) that are barely used, something that removes credibility to the that derivation. Always remembering that this is the other possible predominant function, being therefore its mission precede the dominant function (with or without an intermediate PD), it is much easier to clarify the origin of the chords. The essential premise in the harmonic-functional system consists of dissociating in this case the harmonic theory (the building of chords by thirds) of the real practice in the composition, at the time that the mentioned melodic movement IV - V - I remains as the most relevant cadential bass. For this purpose it is considered that the bass of the whole chord family with subdominant function is the IV degree (in both modes) and that the third (major or minor) is obligatory. From this basic interval several possibilities are contemplated (Fig. 9):

Of course, not always the secondary dominant and predominant chords appear efore the secondary tonic, but sometimes they do afterwards, and then a stronger confirmation of the secondary tonic closes the process. That is why I think it is sonvenient to add an arrow to make clear to which region the secondary chords in yarentheses are related. In the first case of the examples below the secondary tonic (III legree of F major) appears after the corresponding secondary chords, but in the second ase the secondary tonic (VII degree of A minor) is already heard before, and is sonfirmed after the secondary chords; for this reason, the mentioned secondary tonic should be remembered retroactively as the first announcement of that secondary tonic, something that is clear with the insertion of a pair of arrows in both directions:

In the case of a secondary tonic heard before the secondary D (and S or PD) but then not repeated, Riemann already proposed using a backward arrow to link these determinant chords with the tonic in a retroactive sense, as in the case of next example, where the secondary tonic (VI degree of C major) is heard before its determining chords (taken from Riemann 1893: 143):

On the other hand, secondary chords do not always reach the announced goal since composers, aware of the directional power of a secondary dominant, use thi frustrated expectative as a form of unrealized implication that generates indubitabl tension.*® It is not easy to indicate in the analysis at the same time which was th announced tonic and what is the really achieved tonic. Riemann included the frustrate: tonic in brackets, but at the same level as the other chords. My solution is to open | second level of analysis to indicate through a broken arrow which was the expecte tonic, while the main line shows the chord that actually arrives in its place. Thi differentiation between the real harmonic level and the “virtual” level makes it possibl to clearly express the different levels of harmonic discourse, thereby solving th problem of the implicated but not realized chords. In the following example, th secondary dominant clearly announces the Sp of D major, but instead the subdominan appears. Both things are clearly expressed by the following ciphering in two lines:

The display of two lines of analysis is also very useful to show the critical moment in which the main key of a sonata form movement is abandoned (momentarily) in favor of the subordinate key at the end of the transition, to give way to Secondary Group S, exposed in this last key, an example of modulation in the strict sense (since the new tonic is much more durable as it is associated at least with a complete thematic

group).’? In these cases it is convenient a double line of analysis that shows the functional filiation of the chords responsible for the modulation viewed from the home key (HK) and at the same time from the subordinate key (SK). Once the latter has been clearly established, the main key can be momentarily closed until its reappearance later. In the Fig. 10 — belonging to Sonata No. 12 KV 332 of Mozart — the Transition begins with the tonicization of D minor, the Tp of the main key (F major), which is the Sp of the subordinate key (C major), to then go down through a progression that tonicizes in this case the d of the main key, which is the subordinate key itself in minor mode, closing that space with a half cadence in the dominant of this new key. Typically, the SK appears first in minor mode, changing to major mode with the arrival of the first theme of the subordinate thematic group (Si), a frequent procedure at the end of transitions. The beginning of the Group S in this new key confirms the modulation process and allows closure for the moment of the domain of the main key. In this passage Mozart uses the so called German augmented sixth, which here is analyzed —as it was explained above — as a variant of the PD function without the fundamental and including the seventh and the minor ninth (mm. 35-36). “ The terminology used here for the sonata form comes from La Rue (/1970), Caplin (1998) and Hepokoski & Darcy (2006), revised in my doctoral thesis (Igoa 2014), where P is the first thematic group, T the transition, S the secondary or subordinate thematic group and K the conclusive zone.

5° The explanation of the altered chords, their origin and their level of intermodal relationship can be found in paragraph 3.8.4. In this case, the traditional harmonic theory has never been able to explain satisfactorily the essence of this tonic in relation to the main one, since it is limited to ciphering it as a III degree of C major with change of mode from minor to major. However, the harmonic-functional system allows to see clearly that Beethoven — intuitively, of course — sought to go beyond the typical dominant (G major) as subordinate key, and thought about its relative (E minor), which is what announces at first the transition, but looking for a major mode for its S; theme he changed the mode when it reached it. The theoretical process that leads from C major to E minor and then to E major is found — in the opposite direction and as if it were his musical explanation — at the end of the exposition (mm. 74-86). The following exam ple, taken from Sonata No. 21 op. 53 “Waldstein” by Beethoven (I movement, mm. 18-38), is a summary of several of the processes studied, which converge in the pas second transition (T2) and t of analysis shows the end o sage that includes the end of the first transition (Ti), the he beginning of theme S;. On the one hand, the double line f the main key and the arrival of the subordinate key, which here, surprisingly, is not the dominant (D), but the relative of the dominant, first apparently preserving its own minor mode (Dp), although later enhancing the third (DP), that is, E major,”° which is reached through the tonicization of its subdominant (s) in m. 18-21, a PD and a long extended dominant.

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The Transformational Energetics of the Tonal Universe: Cohn, Rings and Tymoczko

by Kenneth M. Smith

Music Analysis

A further formal curiosity is Schubert’s ‘second subject’s’ F# minor beginning, which Weitzmann would regard as D and Cohn as a downshift to 4.'° But again, Cohn’s analysis disregards the animations beneath. The F major of bar 44 resolves not to the expected B> major, but to a diminished seventh which substitutes Bb for Bi. After recalibrating this symmetrical chord, Schubert orien- tates it towards C#’ rather than B} major and from there lunges at F# minor (Ex. 3). Direct discharge between Bb major and F# minor (the enharmonic parallel (henceforth ‘P’) of Gb major) is mediated by the raw diminished seventh, which, rather than transform one chord to the other, redirects energy from Bb’ to C}#’. These kinships are not registered in Cohn’s voice-leading zones, but in Cohn’s version of Douthett’s ‘4-cube trio’ (or ‘power tower’) (Cohn 2012, p. 158) these minor-third- (rather than major-third-) related chords share the same region: 1, moving to 2 with the diminished chord, and back to 1. What is missing from this model, however closely it illustrates these tetrachordal relations, is the discharge factor: the F’ (bar 44) which precedes these nebulous T3; chords comes from a galaxy far away (region 5); this also requires us to switch between tetrachordal and triadic models.

tig. 2 Schubert, Sonata in Bb, D. 960/i, bars 1-75: model of tonal functionality Fig. 2), in Bb major, G) major is the chord for which the E) major S of bar 5 is the ‘Sp’ (P-related also in the neo-Riemannian sense; this is the relative of E)’s minor mode). Schubert’s F# minor in bar 48 soon functions clearly as an A major substitute (both chords minor-third related, the former being the latter’s Sp), which, after discharging very briefly to D minor in bar 70 and subsequently E major (the parallel Tp of G major which would follow the D minor if the discharging cycle of fifths was continued), returns to A minor and then C major in bar 74 (the chord for which A minor is the Sp). This C major becomes a structural C’ downbeat to the full F major in bar 75. With localised discharges between, we have now heard the four T3 octatonic S nodes: Eb, Gt/F#, A and C, creating a static environment of Sp relations in between T and D pillars (see again Fig. 2). Essentially, this recalls the promise of Lendvai that minor-third rather than major-third relations share functional categories. The progression macro-progression rendered this way is quite clear: T+ S—+D-T, etc., privileged by Riemann and later function theorists.!! We might risk modelling this with new tools. The following hypothetical map is designed to register the functional T — S ~ D —- T rotation which lies at the heart of these chord progressions (Fig. 3). The central triangle represents the

Fig. 4 Schubert, Sonata in B), D. 960/i: exposition Table 1 Schubert, Sonata in Bb, D. 960/i: comparison of exposition/recapitulation

Cohn’s other analyses work likewise. His excerpt from Brahms’s Second Symphony progresses LRLR to produce a cycle of fifths, rerouted through R substitutions. This is transparent in Cohn’s cube dance, but the discharging cycle of fifths cannot connect up directly (Fig. 5). The wheel turns here only because of the major- and minor-third alternations, whose L transformations (with leading notes) resolve indirectly around the cycle of fifths. Cliinkewls weenie: eemewedla je KEe eee erie tel weeeienime Seen Alalawess feens

ix. 4 Schubert, Symphony in C major, D. 944/111, bars 101-142: voice-leading graph the primary major triads are the P and PR relations (lp and “1, etc.) and the central D}/C# minor. A diatonic T+ S—+D-T cycle rotates twice and the central chromatic drift is carefully directed back home (Fig. 6). This Schubert example helps us reconceptualise Ex. 5, the old neo- Riemannian war horse which is explored by Kevin Mooney (1996), Harrison (1994), David Kopp (2002) and many more, and from whom Audacious Euphony derives its name (see Ex. 5). The passage comes from Riemann’s definition of ‘tonality’ from his Musik-Lexicon of 1909 (Cohn 2006, p. ix). As Avo Somer recently observed, ‘It provides an important late-nineteenth-century admission that a tonic preceded or surrounded only by chromatic mediant or submediant chords may still create a coherent tonal structure’ (1995, p. 215). Although Somer feels it ‘bizarre for omitting the dominant’ (p. 218), we might hear the missing G major as substituted by E major (its T; relative). The potent driving force is the leading-note rise to the B pitch; the consonant bass leap of a fifth is replaced by another function-confirming jump — the major third. Matthew Bribitzer-Stull, examining “The A}-C—E Complex’, recalls the famous anecdote about C.P.E Bach: ‘[w]hen asked “Who but yourself would dare go directly from C major to E major?” C.P.E. Bach replied, “Anyone can and will assuredly do it who knows that E is the dominant of A, and that A minor is very closely related to C major” ’ (Bribitzer-Stull 2006, p. 167). Ahead of time Bach pinpoints one of the tensions between neo-Riemannian transformation theory (where C major moves to E major because it transforms smoothly) and Funktionstheorie (where E major is a replacement for the dominant G major). Of course, Ab major would fulfil an S function, just as an augmented sixth would do, the lower Ab (+6) imbuing a Phrygian quality. This progression reads: T 4 S(> T) ~> DT. Lendvai, we recall, stipulates that C major, Ab major and E major best represent the T+ S—D-T, rather than the more intuitive C major, F major and G major (1993, pp. 8-11). In fact, Alexander Rehding demonstrates that Riemann stops strikingly short of this exact interpretation (Rehding renders the more nuanced paleo-Riemannian functional interpretation as T— °Sp oT — D(Tp) — T) to avoid concatenations of Scheinkonsonanzen (Rehding 2011, pp. 167-90), which are disallowed in Riemann’s theory, although seemingly permitted in his analyses.

Chord progressions like these in Riemann’s little epitaph abound in the nineteenth-century literature and push right into the twentieth century. A short passage from Strauss’s Elektra demonstrates how these progressions overlap with borderline atonal repertoire and in fact highlights the ‘discharges’

the performance of Janus-like différances between sounds and concepts is in fact a liberating compositional tool and a suggestive (if inconclusive) way of concep- tualising nineteenth-century animated harmonic progression. For this reader, Rings’s book is to be applauded for establishing (a) the plurality of models and the dialectics between them, (b) the integration of ‘energy’ into the GIS, (c) the retroactivity of his hearing and (d) problematising subject-object relations. The answer to the book’s fundamental question — ‘Can we reconcile energeticist metaphors with the transformational attitude?’ (p. 104) — is ‘yes, but the book is only a step closer’. While Ringsian transformation theory shuttles between subject and object positions, between retroactivity and forward motion, between energy and transformation, between one hearing and another, it doesn’t recon- cile fully.

Chords seem to obey specifically harmonic laws ... . This is one of the features of Western music that is most suggestively language-like. For just as in English, the subject normally precedes the verb, which in turn precedes the object, so too does Western music seem to have a harmonic ‘grammar’ according to which subdomi- nant chords precede dominants that in turn precede tonics. (p. 226)

After considering the rise and fall of the ‘chord as object’, Tymoczko intro- duces the ‘major-third system’ and the ‘minor-third system’ (2011, p. 280). But his Mars and Venus come perilously close to collision in his reading of a progression from Schubert’s D major Sonata, D. 850, in which, following a D major tonic, a hexatonic progression (D — F major > C# major > A’) ‘leaves traditional tonality in favour of the major-third system’. The final chord’s seventh is considered ‘inessential’ and ‘decorates the more basic triadic relation- ship’ (p. 280). A (as a triad) is considered close to F (purely based on proximate voice-leading parsimony), and for Tymoczko the C} is the alien triad. I would contend that the seventh 7s essential, even if it causes Mars and Venus to collide. The basic progression for me is T ~ S ~ D 4 T, with F (as the RP of D major) discharging to C# via enharmonic leading-note resolution; C# likewise discharges to A (via the leading note G#), whose seventh secures a full discharge to the true T of D major. And Tymoczko almost slips into Cohn’s black hole: ‘since major-third-related triads can be connected by efficient voice-leading, they can substitute for one another without much disrupting the music’s contrapuntal or harmonic fabric’ (p. 283). But, as proposed in my extended reading of Cohn’s Schubert analysis, hexatonically related triads contain each other’s leading notes and can therefore discharge into each other and alter function; octatonically related chords have no leading notes and can only store up functional energy.'® Tymoczko finds his two polarised ‘systems’ exchanging spaces in Chopin’s E major Prelude, Op. 28 No. 9, bars 5-8 (2011, p. 218). His B major (chord 2 of Ex. 8) is a major-third relative of G major but is also a parallel minor third Dominant! of FE. so that the B maior provides a deceptive embedded cadential

‘Mars’/‘Venus’ segregation (pp. 97 and 220). For Tymoczko, the subsequent bars outline the minor-third system with the dominants (or dominant substi- tutes). This is an octatonic succession of functionally synonymous chords. The B} minor (which Tymoczko’s analysis glosses over) now leads to a mock E major discharge as a T,; retrograded parody of the earlier E major > B major progres- sion. Tymoczko’s assertion that nineteenth-century music ‘abandons functional norms in favour of direct chromatic voice-leading between triads and seventh chords’ (p. 217) is thus only partially correct. The functional tectonics here move T>S—54D-T again with a full mixture of tetrachords and triads. Despite my protestation, Tymoczko is nonetheless correct: octatonically related triads have sevenths because sevenths are available in the collection; hexatonically related triads are denied sevenths. Geometrically, this makes perfect sense. Musically, it doesn’t always work, because these planets are in a diatonic solar system and are still subject to the same gravitational forces. Adrian Childs, the first neo-Riemannian to take account of seventh chords sufficiently, claims that ‘the composers whose works seem best suited for neo-Riemannian analysis rarely limited their harmonic vocabulary to simple triads’ (1998, p. 181). The irony is that octatonic seventh chords are full of ‘drive’ but the absence of leading notes prevents them from discharging, and thus they remain static; hexatonic chords can’t have sevenths but can nonetheless discharge into each other as leading notes. Both are gravitationally bound, needing each other to hecqme animated

(F—G#B-E) is nestled between instances of an F” tetrachord (Ex. 9), which refracts and later ‘resolves’ to Ab major. The progression is relatively straight- forward, but a geometric interpretation is complex. Because both sonorities contain tense tritones, Tymoczko adopts the well-chosen vocabulary of ‘resolu- tion’ and ‘discharge’. However, the language is misleading. Since when did a ‘root position’ Tristan chord on F want to ‘resolve’ to another F major—rooted seventh? When, for that matter, did it want to resolve to Ab major? We could look instead to Rings. His technology might model the opening progression as both a ResV (in which the Tristan chord resolves to V’) and a ResI (with a resolution to the Tristan chord as ‘IT’ [Fig. 12]). Flaws immediately materialise. The D in Brahms’s score, which Tymoczko ignores, transforms the chord into a full Bi” chord, replacing F’’s desired B) with a prohibitive Bi. Rings’s technology copes with standard abstract resolutions but collapses in situations when (a) we wish to express that the resolution given was not the desired one; (b) the tonic centre is veiled; (c) we want to outlay each pitch’s multivalent resolution tendencies (which they always have); (d) chord x ‘resolves’ to y, which then ‘resolves’ back Fig. 12 Brahms, Intermezzo, Op. 76 No. 4, opening chords: a Ringsian digraph

Ex. 9 Brahms, Intermezzo, Op. 76 No. 4, bars 1-3

to x, thus requiring two asymmetrical profiles of tension and release; and (e) the tonic resolves away from itself. These five issues render my Ringsian modelling in Fig. 12 faulty. Chromatic though the voice leading is, no leading notes signal discharge between chords, nor do they develop in the second ‘resolution’ to A major (the Ab major also contains a delicate F#, omitted from Tymoczko’s reduction, that assembles a German sixth). F’ and Ab’ major chords are minor- third relatives and share two pitches each with the F’’. The only Tymoczkan technology to model the sequence is his three-dimensional four-note chromatic space, which can demonstrate the two ways of parsimoniously connecting F” with the piece’s ‘resolutions’.!? These are unfilled particles in my Fig. 13. Tymoczko’s space proves that (a) the chords are two parsimonious steps apart,

Fig. 14 Douthett’s 4-cube trio, modified by Cohn (2012, p. 158). By permission of Oxford University Press, USA

progression — the paradigmatic ‘discharge’ motion — connects the two planets in the most obvious way. But on reflection, it seems that Tymoczko’s two ‘planets’ are actually two different functions or forces: one force releases tension while the other stores it. Ironically, while we might imagine the discharger to be the seventh chord and the storer to be the triad; the opposite is true because of the association between triads and hexatonics and between seventh chords and octatonics, the former allowing discharge, the latter arousing the desire for but prohibiting it. This theory is latent in Tymoczko and Cohn, but only Rings really attempts to model the crucial X factor: the discharge motion. In order to stage a true dialogue, we must gaze once more into the night sky, bringing our three theorists into a synthetic analysis. For me, the high-resolution lenses for gazing at nineteenth-century night sky are (a) Cohn’s ‘voice-leading zones’, (b) the Ringsian digraphs and (c) the Tymoczkan geometries of three/four-note space,

Strauss’s song is broadly triadic (in contradistinction to Chopin), the extrac reproduced as Ex. 11 maps well onto three-note chromatic space, and we migh’ ‘ignore’ sevenths when they arise as auxiliary in function. Fig. 16 demonstrate: the symmetry of this purple patch, the E> major and C# major appearing a: discharge outlets for the central hexatonic triads. Voice-leading stages betweer chords can be counted readily on this graph, and V and I lie three step: apart. But is this useful? We could have learned this from Cohn’s 2-D book and the Tymoczkan 3-D models are occasionally a clumsy way of rendering progressions.”! If the aim of the geometric models really is to ‘clearly illustrate musical principles that can otherwise be quite difficult to grasp’ (p. 64), then we sometimes fall short with these cluttered 3-D diagrams rendered on Oxforc University Press’s woefully 2-D paper. 2 ee OE mee Gf, eee ee ea ee a a Se: Pe Meme. (Ey. i deem te eM ee AA

LO Tlls slapilicds LOECMmIaue, Crucial to the piece, therefore, seems to be the perpetual T+ S—> D-T rotation of Ex. 10, whose functions are inscribed below it. Occasionally some- thing like the Funktionstheore equivalent to a neighbour-note motion will occur. where a chord will leave a functional category and return to it — usually to the same chord. This happens only twice in the piece, which otherwise rotates functions no fewer than eleven times. Each and every hexatonic transformation discharges leading notes (F — G}/C# > D), pushing around the circle, and in fact this usually occurs because a hexatonic cycle is being rotated ‘downwards (i.e. by downwards Ts transposition rather than T4). "Thinoe change clightly in the archectral nanctlide when the enddean olaha’l

Fig. 17 Strauss, “Twilight’, bars 33-39: a Ringsian GIS model Tymoczko!). Tymoczko would doubtless approve of the juxtaposition of the minor-third system and the major-third system as two separate planets, but the bigger picture emphasises their functional difference: these octatonics are non- discharging (lifeless? objective?) and confirm for us the fully discharging (living? subjective?) hexatonics which coincided with rotations of tonal function. After approaching E) major obliquely (a D minor chord: PSP), the C) major b VI “Tod’ erupts as the only proper reversal of the T — S ~ D > T paradigm.

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Lewinian Transformations, Transformations of Transformations, Musical Hermeneutics

by Andrew J Chung

2012

Figure 1.1 Lewin's schematic of the transformational situation situation modeled by transformational theory, is shown below.'’

Figure 1.2 Lewin's figure 7.1 from GMIT, analytical representation of Schoenberg op. 19 no. 6 piece highlight some of the natural perceptions a player can form while performing it.

Figure 1.4 Opening bars of Beethoven Op. 58 piano concerto with transformation labels

Figure 2.2 Some examples of P, L and R transformations

Figure 2.3 Binary transformations compounded from P, L and R transformations

Figure 2.4 A typical two-dimensional tonnetz of modern NRT

minor triads and have been collectively called the hyper-hexatonic system shown 1

octatonic system.”* Each octatonic cycle (Op, O, and O,) contains the pes of the familiar

Figure 3.1 Lewin's network of networks for “Nos qui vivimus” reproduced here as figure 3.1.

Figure 3.4 a) Growth of the falling semitone analyzed with TRANS+ 2 transformations b) Transpositional development of the falling semitone gesture c) Intervallic structure of the opening right hand chord

Figure 3.7 Some Scarbo motifs and their component gestures represented as pitch-transpositions

Figure 3.8 The development of Scarbo motif forms analyzed with transformations of pitch: transpositions

equivalent to adding P transformations to either the beginning and/or end of a given same interval class. Arrows specify transformations of triadic transformations that are

Figure 4.2 (continued) b) Transformations between ic 4 root motion PLR transformations c) Transformations between ic 5 root motion PLR transformations

Figure 4.4 Space of pitch-classes organized by alternating major and minor thirds

Figure 4.5 Score reduction of line 1 with segment of RP/PL cycle realized in line 1 of the anguish and personal distress of the speaker.

Figure 4.7 Score reduction of line 2 with segment of R/L cycle implied by line 2

Figure 4.8 Segments of cyclic transformation graphs corresponding to RP/PL cycle of line 1 and R/L cycle of line 2

Figure 4.9 Graphs of figure 4.8 filled in with appropriate triadic mode information (+ : majot, — : minor)

Figure 4.10 Segment of a more inclusive transformational space with RP/PL path and R/L path

Figure 4.14 Score reduction of line 4 with transformations between “dolor” and “meus” nodes

Figure 4.15 Network of transformations of PLR group transformations applicable to line 4

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Modal four-note pitch collections in the music of Dvořák’s American period

by Benedict Taylor

The use of four-note extended triadic harmonies in late-Romantic music is still under-researched, despite their great significance within this repertory. This article explores the use of four-note collections in the music of Dvořák,... more

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Skryabin's Revolving Harmonies, Lacanian Desire, and Riemannian Funktionstheorie

by Kenneth M. Smith

That Skryabin's harmonic language is rooted in dominant functionality is commonly acknowledged. However, the flow of his tensile dominant-based sonorities has not been adequately explored. This article seeks to correlate his harmonic... more

Example 1 Skryabin, Etude, Op. 42, no. 2, bars 12-16.

Example 2 Skryabin, Flammes sombres, Op. 73, no. 2, bars 1-4. (and later D and A}) can be interpreted as refractions of a single chord complex — a “French sixth’ configuration of interlocking tritones, exemplified here by the pitches F, B, A, and D+ (E}). Dernova calls this complex the ‘tritone link’, because the seventh chord with a flattened fifth reproduces its pitch-class content under the t6 operation: thus F—A—C}—E) becomes B—D#-F—A in Example 3. Many instances of this are found in Skryabin’s output, and the pliability of tonal implication that it enables in respect of tritone-related keys makes the chord a highly flexible resource, particularly when so much of this music is organized by t3, t6, and t9 sequences.

But each sonority expressly comprises ‘tritone link’ chords, meaning that a chromati chord such as B’” (of the B>B) progression) also embodies its tritone relative — an F chor in this instance, which thereby produces F>B} motion, commensurate with the circle o fifths. This enharmonic ‘tritone link’ is a more discreet version of the ‘tritone substitution common in jazz, and can be mapped successfully onto the collapsed spatial model. Figure - ‘a and b) lays out Berg’s paths along the tritone-linked circle of fifths, notwithstanding th apparent chromatic chord progression of Example 4b. Squares here represent the chord neard, while arrows show their progression. Unlike Berg in this instance, Skryabin does not always adhere to ‘French sixth’ chord

For all that this particular ‘French sixth’ chord is utilized seemingly inexhaustibly by skryabin, nowhere in his own output is it used as exclusively as in the first of Alban Berg’s arly settings of Alfred Mombert’s Der Gliihende (Op. 2, no. 2). The beginning and end of he song are reproduced in Examples 4a and 4b. In Example 4a the bass circle of fifths stretches from Bj) to E (with C}/B providing an enharmonic bridge). In the final lines, by -ontrast, the central circle-of-fifths passage (moving again from B} to E in Example 4b) is surrounded by chromatic progressions.

Figure 3 Berg’s path through tritone-linked circles of fifths in (a) bars 1-5 (b) bars 14-18.

bombastic opening Dernova claims, ‘It is difficult to see how any tonal orientation other than a motion from D to G; could be deduced.’”° In response to this calamitous under- statement, the analytical method now adopted organizes the fascinating linear chain of seemingly disjointed harmonies into a clear circle of fifths, though this fifth progression is rerouted by minor-third and tritone relationships. Plotting the path Skryabin takes in the opening four bars of his Poéme puts this in a nutshell (see Example 5).

Figure 4 Path traced by Poeme, Op. 32, no. 2, bars 1-5. In spite of its key signature, Skryabin’s opening sonority is highly ambiguous. An inter- pretation of the chord as D is problematized by the seventh (C) in the bass, the diminished fifth (A}), and the delayed third (F#). Nevertheless, this reading is retrospectively validated when Skryabin moves to a G major chord (albeit in first inversion with the suspended F# above it) at the end of bar 2. This retroactively consolidates the function of the opening chord as a dominant seventh on D that drives towards G, regardless of the fact that the diminished fifth (A}) creates the potential for the chord to act additionally as ‘tritone link’ on A) (hence implying Dj).*’ The next two bars are slightly more straightforward. An E’ chord is struck in bar 3 which, despite the suspended Ft, the chromatic flirtation with E#, and the delayed fifth (B) in the melody line, is projected forcefully by the firm bass E and the D—G+ tritone. Following this, the subsequent protracted harmony is consolidated as a chromatically elabo- rated A’ chord by its resolution onto a full D major triad. This is now posited as a firm tonic for the piece, or at least for the opening phrase. Using my modified version of the Weber/ Lerdahl ‘table of tonal relations’ this procedure could be modelled as in Figure 4. In this four-column grid squares still represent chords (rather generally to include both triads and sevenths) and the double conjoining line (A) = D) demonstrates the synonymity between the ‘tritone link’ chords. Although this A} is only potentially operative at this stage, it plays its part in both the chord’s aural ambiguity and the piece’s tonal narrative. Notwithstanding the chromatic uncertainty of this opening passage, the progression seems relatively clear once the initial chord is retrospectively understood: D>G+E’>A’>D, or IsIV>II>V>I. once the initial chord is retrospectively understood: D>G-+E’>A‘’>D, or Io IVsHsV-1

Figure 5b An extension to illustrate Riemann’s Funktionstheorie.

example, the idea that a tritone-related chord can substitute for the true T chord cannot always be applied directly to diatonic harmony. Lendvai rather overconfidently asserts that “counterpoles’ (by which he means tritone-related chords) are ‘more directly related to each other than the relative keys of classical harmony’, and thus a ‘pole can be replaced by its counterpole without any change in its function’.*° An extreme reading of these words would imply that a simple diatonic tune in C major could end satisfactorily on an Fé triad and still achieve tonal resolution. And yet in specific ways ‘tritone substitution’ can func- tion perfectly. As any jazz pianist knows, a dominant seventh can be readily replaced with its tritone counterpart (for example a G’ chord replaced by a Dj’ chord in C major) at a cadential juncture. Both chords, when imbued with this dominant-seventh driving quality, contain the same potent tritone that pulls towards the established tonic. But this potential

Figure 6a Skryabin, Poeme, Op. 32, no. 2, bars 5-7.

G} triad, I draw an arrow on the graph to show an actual progression to C’””’. From this position Skryabin moves to a clearer E} triad (bar 9), which in turn progresses to an A} that shares a tritone link with D’”’. The Aj then switches to an F chord in bar 10, followed by an F/B tritone link. The entire T space is thus filled out using each of its potential repre- sentatives (D-F—A}—-B). Thus Figure 6b exhibits a dense network of chord relationships, despite its clear D> T motion. Of course Skryabin now has the problem of establishing the true T chord of D. After all, substitutes are only substitutes and do not carry the same functional weight as the primary triads that they stand in for. In order to achieve this, he first consolidates F as a full seventh chord and resolves it to a B} in the following bar. The tritone link is once again exploited as he shifts quite forcefully from the initial B} to an E’ chord and activates an A’ in bars 12-14. This chord is drawn out over two bars to confirm itself as the true D function, thus indicat- ing the true T (see Figure 6c). But, after flexing and stretching this A as D in order to fully confirm the D as T, Skryabin denies us the satisfaction of an actual D triad until the end of the piece, offering instead an ambiguous return to the opening D’’’. The full course of Poeme, Op. 32, no. 2, yields two progressions similar to bars 1-14 before the final resolution of the full D major T chord, but the procedures hitherto discussed are charted in Figure 6d.

Figure 6d Skryabin, Poéme, bars 1-14, according to both four- and two-column models.

seventh chord, we witness the drive’s failure and its return to its circular path. Lacan’s model corresponds neatly to Skryabin’s practice, with a crucial difference. While the tonic chord (in this instance D, or one of its substitutes, F, B, or Aj) is clearly posited as an object of our desire, Skryabin does achieve this object at the close of the piece when he displays a full D major triad, and this appears to be successful. But for Lacan — and seemingly for Skryabin, who says “When I have no desire I am nothing’ — this is a hollow gesture, a mis- recognition of the drive’s true goal.*! This is where an examination of Skryabin’s late style becomes imperative. Here the tonic chord is at once both implicated and avoided, forcing tonal function to operate quite differently.

Example 7 Skryabin, Sonata no. 6, bars 13-15. Example 8 Skryabin, Sonata no. 6, bar 39 ff: second subject or transition?

the S is foreshadowed when the seventh chords on B} (bar 43) resolve to chords on E} (bar 47 — mediated by an S substitute on G} in bars 45-6 with the B} pedal retained in the bass). This new chord on E} undergoes tritone substitution with an A from the same S region. But in this transitional passage the listener is pulled back and forth between two functional areas, for Skryabin retreats to the initial T region in bars 54-60 with seventh chords on B), F}, and D} (see Example 9). A reconstruction of bars 43-7 sees a return to G}, E}, and A in bar 64 to stabilize the S, and by bar 69 the S region is utterly secure.

Example 6 Skryabin, Sonata no. 6, bars 10-12.

The ‘modulatory’ aspect of this passage is analogous to the transitional theme of a classi- | sonata model, and thus I designate it as a transition to a more distinct second theme at ur 82, with its prompt change of character. This reading finds encouragement in Skryabin’s orid performance indications. My transition — Willmann’s second subject — is accom- anied by ‘le réve prend forme (clarté, douceur, pureté)’ (the dream takes shape: brightness, ntleness, purity). This crystallization of the poetic ‘dream’ confirms the function of tran- tion; the consequent ‘avec entrainement’ implies the newly shaped, tonally secured idea — e full second subject. And one must not undervalue melodic stability here; despite its any connections with previous motivic content, the five-note ascending cell is now pre- nted in fully developed form, supporting the new S matrix. However, one would expect tonally functional model to utilize the D region rather than the S in the second subject, id in this regard Skryabin changes the tonal position of the archetypal lyrical second ibject. As is so often the case, the tonal and thematic processes in the movement’s recapit- ation can shed light on the issue. Tha pacanitilatian Ahagine uith the D matriv and ice nracentead a whale tane lair than the

Example 10 Skryabin, Sonata no. 6, bars 244-5. piVViIttts LIddoli©s Ul CIAGVUUTALE UICTIIAUL ValliallOll Lil illo VYLALIIIL LUCaPIUIAUU TI, A more significant departure from the exposition occurs in the reorganized transition section of the recapitulation. Whereas the transition’s function in the recapitulation is generally to preserve the tonic key between the first and second theme groups, this example operates as a homecoming that returns us to the tonic. The new transition also carries struc- tural weight, as the climax of the work is articulated here in the space between the two themes as a kind of secondary development. This apex is, however, rather subdued, charged by the cumulative effect of ostinatos.

Example 11 Skryabin, Sonata no. 6, bars 254-5.

Example 12 Skryabin, Sonata no. 6, bars 126-30. = “ — ‘neo-functionality’, a few preliminary considerations are in order. The sonata largely eradicates the possibility of local diatonic progression, directing our full attention to the global progression. The absence of possibilities of discharge within strict minor-third-related harmonies results in clearly demarcated blocks that exploit single matrices and, given the excessive length of these vast stretches involving the claustrophobic alternation of only four chords, an attentive listener could reasonably process the global discharges into a new matrix as a functional shift. Naturally, the literal discharges (i.e., generally V’ +I gestures) at local level give the listener the clearest chance of registering the modulations between matrices. The intersection between the exposition’s concluding S function and the development’s D (shown in Example 12) begins with just such a direct discharge motion from a tritone-linked F#/C to B/F(E¢), which places the developmental restatement of the opening theme firmly in the “dominant key’. Note also how, after the period of transition in the exposition, the final consolidation of the new second subject that occurs at bars 63-4 (a modified repeat of bars 29 ff, shown in Example 8) moves from Bj’ to E}, and although a G;’ chord is nestled between the discharging chord and its resolu- tion, the B) pedal underlying it strengthens this T>S motion. Through a dialectic between local periods of tonal inertia and concentrated structural discharging motions, Skryabin forces us to switch from an octatonic foreground to a pseudo-diatonic background level, structured by fifth motion that, albeit in drastically reinvented form, mirrors the relation- ships between micro harmonic progressions and macro key operations in the prototypical sonata dvnamic.

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A New Geometric Model and Methodology for Understanding Parsimonious Seventh-Sonority Pitch-Class Space

by Enoch Jacobus

2012

Parsimonious voice leading is a term, first used by Richard Cohn, to describe non-diatonic motion among triads that will preserve as many common tones as possible, while limiting the distance traveled by the voice that does move to a tone... more

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From Uncanny to Marvelous: Poulenc's Hexatonic Pole

by David Heetderks

Tu vois le feu du soir qui sort de sa coquille Et tu vois la forêt enfouie dans la fraîcheur Tu vois la plaine nue aux flancs du ciel traînard La neige haute comme la mer Et la mer haute dans l'azur

Francis Poulenc, “Tu vois le feu du soir” [You see the fire of the evening] from Miroirs brilantes (1938), coda (m. 4’ Text by Paul Eluard

« “FixP” = a triad that is in a hexatonic pole relation with a minor tonic.

Figure 12. Reading of “Tu vois” through metaphor of tonic as self.

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Perspectives on Tonality and Transformation in Schubert's Impromptu in E-flat, D. 899, no. 2

by Steven Rings

2007, Journal of Schenkerian Studies

In addition to a pdf of the actual publication, I have attached my own submission files for the article (RingsJSSText_rev2.pdf and RingsJSSExx_rev) as I've never been pleased with the layout of the actual publication.

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AETHER/OR? THE CHANGING CONCEPT OF ABSOLUTE SPACE, 1854-1905

by James (Jim) E Beichler

It is generally believed in the scientific community that no changes in the physical concept space occurred during the period of time just before the onset of the Second Scientific Revolution except for Mach’s successful logical argument... more

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HYPERSPACE CONTINUUM: The Five-Dimensional Continuum Approach to a Unified Field Theory

by James (Jim) E Beichler

2012, Hyperspace was a Masters Thesis at San Francisco State University, 1980, Continuum added in 2012

Modern physics is based on two paradigms that emerged from the Second Scientific Revolution - relativity and the quantum. Since their inception these two theoretical giants have advanced science and society far beyond anything that their... more

Modern physics is based on two paradigms that emerged from the Second Scientific Revolution - relativity and the quantum. Since their inception these two theoretical giants have advanced science and society far beyond anything that their originators could have even dreamed up, but in a sense they seem to be progressing in different directions and are regarded mutually incompatible. Yet they come to a point in history where the next step in their own advancement would be to unify into a single overwhelming physical paradigm. The concept of unification is well known in relativity theory with many attempts dating from just a few short years after Einstein first developed relativity. But the hyperspace unification that seems to have originated with Kaluza in 1921 and was modified by Klein a few years later has become the best known attempt. This book has two parts. The first, Hyperspace, deals with the original Kaluza theory in all of its aspects and incarnations up until the 1960s. After the 1960s the idea of unification was adopted by quantum theoreticians based on their belief that the quantum is more fundamental than relative space-time. However, since the 1980s Klein’s adaptation of Kaluza’s theory has been used to unify physics under the quantum banner as has the purely quantum theory Standard Model. Yet both of these quantum attempts have fundamental problems that can only be solved by returning to Kaluza’s original theory as modified by Einstein and Bergmann in the 1930s. So the second part of the book, Continuum, tells the rest of the story and offers a new theory, a five-dimensional single field theory, the merges the best of both paradigms together. By merging the two paradigms instead of replacing one by the other, single field theory preserves all of the many successes of both theories. This is the book that every physicist, physical scientist and non-professional science advocate should read and every theoretical physicist and physics students must read.

Assume each intersection is a dimensionless point in 3-D space and there is no space between points. At each dimensionless point in 4-D space-time a perpendicular line called an A-line extends into the 5th- D and closes back into the point in 4-D space-time. Each A-line or loop is equal and constant length. Since all of the loops are are of equal size in the 5th-D and parallel to each other, they form a cylinder (A-cylindricity) of sorts in the higher dimension. The 5th-D of space-time is not itself closed with respect to 4-D space-time, but each individual point is closed in the higher dimension with respect to 4-D space-time.

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Proof of the Riemann Hypothesis part I

by Nikos Mantzakouras

Mantzakouras Nikos

The Riemann zeta function is one of the most Euler's important and fascinating functions in mathematics. By analyzing the material of Riemann's conjecture, we divide our analysis in the zeta function and in the proof of the conjecture,... more

Let us calculate #5.Extension of ¢(s) from {Re s > 1} to {Res > 0}

Holomorphic extension domain from the right half-plane to the left half-plane Before we give more information about the function C(s), we mention that each of the two previous equations, give an extension of C(s) on the entire plane C except s = 1, as itis illustrated in Fig. 4. The gamma function was alreadv known to Euler. It generalizes the factonal n!. namelv The gamma function was already known to Euler. It generalizes the factorial n!, namely

generally as shown in Figure 6: ReAs’. = Res). 1/2 = Oypyy¢-—1/ 2 = 0.1228 and ReAs,=1/2-—Res, =1/2-—o,,, = 0.1228 iv. If we accept the non-trivial zeroes of the Riemann Zeta Function C(s) as $,= 0) +it, anc

b) Solution of the second equation: f=Log[1-z]/Log[z]-2*(2*m)4(-z)*Cos[m*z/2]*Gammal[z];ContourPlot[{Re[f/.z->x+l y]==0,Im[f/.z->x+l y]==0},{x,-15,15},{y,-10,10},ContourStyle->{Red,Blue}] Clear [2]; Schematically we illustrate figure 7 with the program of mathematica: Using the correlation theory and after using Lemma 2 ie.. Log[lim ¢(1—s)/¢(s)]=0 anc Hence the root s and we see a good 20 digit approximation, of upper Re s = 0.5000049 of the order 10°" relations with 1 from 3 groups fields and therefore for our case we will get, if we replace

With this analysis, we see that the lower and upper limit converge in the value Res = 1/2, which are the expected results, proving theorem2.

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Roman numeral and neo-Riemannian analyses of Blackwood's Suite for Guitar in 15-note Equal Tuning

by Lillian M Hearne

Easley Blackwood was a pioneer in the use of small equal divisions of the octave other than 12 for musical composition. Such works present an obstacle for musical analysis. This essay presents analyses of the second and third movements of... more

Where in the 19" century the tonnetz was of just intonation, Hyer re-interprets it in equal

Blackwood gives his notation of 15EDO above 15 Notes: In the score to his Twelve Microtonal Etudes for Electronic Music Media, Op. 28 (1980-19:

Movement by fourth and fifth where the triad species remains constant can be described by neo-

section is given below, where the progression the B section follows is the inverse of the progressic

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Motive, Collection, and Voice Leading in Coltrane's "Giant Steps"

by Rich Pellegrin

2020, Jazz Perspectives

Coltrane's “Giant Steps” is a tightly-woven, theoretically-dense composition containing zero-sum characteristics, hexatonic and nonatonic properties, and various types of patterns on both the surface and deeper levels of the piece.... more

Figure 2. The Western nonatonic collection as a combination of two overlapping hexachords — the hexatonic and whole-tone. The overlapping tones form an augmented triad.

Figure 4. The four nonatonic collections and their pitch-class content, shown as discrete, unor- dered sets.

Figure 5. The four nonatonic collections and their pitch-class content, shown as intersecting, unordered sets.

Figure 6. Augmented triads extracted from Figure 5 and their relationship to the four hexatonic and two whole-tone collections.

Figure 7. Figure 6 superimposed back onto Figure 5 — both with generic designations — to show the relationships between augmented triads, whole-tone collections, hexatonic collections, and nonatonic collections.

observations — the limitation to a certain pitch-class collection, in this case a symmetrical scale that contains all the melody notes and chords roots.*° An additional feature of the Southern system is that it accounts for the (1,3) whole-tone collection used by bassist Paul Chambers during the first half of the head of GS on the original recording (see Figure 16). This whole-tone aspect is yet another piece of the GS puzzle.

Figure 16. The (1,3) whole-tone bassline performed by Paul Chambers during first half of the head of “Giant Steps.”

Figure 18. Sample hexatonic improvisation using hexatonic shell voicings. (The bass note A lies outside of the Hex(2,3) collection).

SRASAE RENEE A NE MS SE NEN IR, AINA) NRA ey SENG NN OU REN ey Ch he Table 2 summarizes the GS realizations I have explored above in terms of the four properties we have been concerned with - inclusion of major sevenths, inclusion of ii chords, use of at least three voices, and zero-sum voice leading.°' “Redefined” systems are those where I have kept the existing chord structure of Santa’s or Cohn’s systems, but changed the voicings of the chords. “Rearranged” systems are those where I have departed entirely from Santa’s or Cohn’s systems, creating new systems by rearranging the notes of the associated nonatonic or hexatonic collection (and using new chord voicings as well). The cross-type realization from Figure 21 uses the redefined system from Figure 19 and its modification, given in Figure 20. As can be seen, each

Figure 23. Melody and root relationships in the first half of “Giant Steps.” All pitch-classes belonc to Hex(2,3) except for the A’s from the modified Slonimsky cell. In the second system the roots o' measures 1 and 5 have been reversed to illustrate the quasi-canonic structure.

Figure 24. Motivic parallelism in the first half of “Giant Steps.” The descending major-third motive appearing at three different structural levels.

Figure 26. The two-bar hypermetric downbeats of “Giant Steps” and “Dual Duel.” In “Giant Steps,” these downbeats all consist of perfect fifths and therefore involve two different augmen- ted triads. In “Dual Duel,” these downbeats are all major thirds and only use one augmented triad, creating a canon of hypermetric downbeats. Prolongations are also indicated, following Demsey. Hypermetric downbeats in the first half of “Giant Steps” mirror the surface-level root motions in the same section (illustrated in Figure 23).

7°There are a fixed set of collections, but a number of ways to count them. For any given approach to this calcu- lation, it must necessarily be the case that each collection is ultimately counted once and only once. 77\umerals are normally written in descending order, with the 1s omitted, to facilitate solving; | have written them in ascending order here, with the 1s included, to facilitate understanding of the construction itself. The expression can now be re-notated and evaluated. There are nine unique eight-note subsets of the nonatonic collection - there are nine ways of ignoring one note — and each of these eight-note subsets occurs once. This is expressed with 9/1. For each of these eight-note subsets there are eight unique seven-note subsets. Each seven-note subset occurs twice, and thus we divide by two, arriving at 8/2. (9 « 8) / (1 « 2) expresses the number of unique seven-note subsets that need to be added back after being subtracted too many times with the eight-note subsets. The process continues in similar fashion until one-note subsets have been considered. eae: "7 - cou ss ~ 2

Table 1. Cross-type transformations in nonatonic systems. D=Dominant seventh 135 or 137 with various doublings. M = Major seventh 1357 voicing.

Table 2. Summary of “Giant Steps” realizations. “Redefined” systems are those where the chord voicings have been changed from the original systems. “Rearranged” systems are those created by rearranging the notes of the associated nonatonic or hexatonic collection (and using new chord voicings as well). The cross-type realization from Figure 18 uses the redefined system from Figure 16 and its modification, given in Figure 17.

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Hugo Wolf's Harmony as Weitzmannian Critique: The Augmented Triad and its Hexatonic Shadows

by Frederick Reece

2014, Mosaic: Journal of Music Research

Before abandoning his career as a critic in 1887, Hugo Wolf was active as one of the most voluble proponents of New German Music in Vienna’s notoriously partisan concert press. Eduard Mörike’s poem Abschied, set by Wolf in 1888, reflects... more

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A Generalized Intervallic Approach to Triads

by Andrew Aziz

2016, Sonus

Chordal distance is a topic of great theoretical debate, as it suggests space in both the physical and perceptual realms. The current paper introduces new tools for understanding chordal distance, applying the concept of a Lewinian... more

Figure 4: Cube Dance Graph’ belong to augmented triads (sums 0, 3, 6, and 9).

riad can be viewed as either an upshift or a downshift of an augmented triad, generating four original pair of triads— Bb+ and F#-are in the same hexatonic region. ‘°

of one of the four augmented triads, or, on the other hand, as “floating” in one of the fou

another; this is confirmed by the (W;, W2) value of (3, 2, 4). This directly ties into how one hears moved into a new tonal area— potentially suggesting the onset of a secondary theme group.

Example 2: Beethoven, “Hammerklavier,” on set of recapitulation It shows the interpolation of the confounding B + triad (m. 201) interjected after an arrival on D+

Figure 8. Boretz spiders (from Cohn 2012, p. 157) Thus, in the same manner as earlier with regard to triads, any seventh chord can be characterized voice-leading enjoyed by the privileged members of set-class (037), we can bridge the 2-

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Parsimony and Jazz: The Major Seventh Inclusive Tonnetz and Cuff Torus

by Timothy R Mastic

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Anton Webern's Use of Guitar in Op. 18 and its Influence on His Late Works (PhD Dissertation, Chapter 1 "Introduction")

by Adam Shanley

This is the first chapter of my dissertation entitled "Anton Webern's Use of Guitar in Op. 18 and its Influence on His Late Works."

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Tchaikovsky Violin Concerto, 3rd Mvmt. - Neo-Riemannian Analysis

by Leonard Fu

"The Third Movement of Tchaikovsky's Violin Concerto Explained By Neo Riemannian Analysis" is a harmonic analysis paper using Transformation Theory to explain a phenomenon focusing on the relations between chords.

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The Music of Radiohead: New Approaches to Popular Music Analysis (Syllabus)

by Adam Shanley

Office hour: M-W 11am-noon, 2pm-3pm Course Description: An increased interest, resulting in an increased critical inquiry into popular music, substantiates it as a genre worthy of study. In this class we will be listening to and analyzing... more

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Simetria intervalar e rede de coleções: análise estrutural dos Choros nº 4 e Choros nº 7 de Heitor Villa-Lobos (2014)

by Joel Albuquerque

2014, Dissertação de Mestrado em Música, PPGMUS/ECA/USP

Selected for analysis in this dissertation the Choros No.4 (1926) and Choros No.7 (1924), chamber works for the period of the 1920s, at which time the composer has shown great interest in a tilted to a high content of post- tonal... more

Selected for analysis in this dissertation the Choros No.4 (1926) and Choros No.7 (1924), chamber works for the period of the 1920s, at which time the composer has shown great interest in a tilted to a high content of post- tonal modernist aesthetic harmonic complexity concatenated surface exposure of arguments coming from the Brazilian popular culture, in this particular case referencing the urban carioca choro from the first decades of the twentieth century. In summary, the language used by Villa-Lobos in the two Choros is closely to the composition technique in 'layers' applied in works such as Stravinsky's Rite of Spring (1913), Petrushka (1911) and The Firebird (1910). Villa-Lobos uses the interaction between independent harmonic materials (simple structures generated individually with traditional scales not managed by the tonality inherited of classicism). In many instances these scales are stacked forming large sets (supersets) with eight, nine, ten, eleven pitch-classes or the complete chromatic; structures related by invariances. At other times these harmonic complexes appear juxtaposed, related by gradual intervallic transformations, operated around invariants pitch-class sets and discrete movements between remaining pitch-classes not common (primarily by movements of tones and semitones), a procedure called by the theorists of the 'neo-Riemannian' (as Richard Cohn, Jack Douthett & Peter Steinbach and others) as 'parsimony'. We also noticed that Villa-Lobos often relates these stratified layers by symmetrical intervallic arrangements, managing pitch-class sets that make up the parts or whole harmonic around axes of symmetry invariants (reflection). We also noted invariances often repeated on ostinato (translational symmetry), while remnants variants arise and disappear as segments of scales into different layers along the stretch led by ostinato, completing the entirety harmonious by juxtaposition. From this preliminary diagnosis, we choose in this research two distinct and concurrent analytical currents: the intervallic symmetry (invertional and translational) from studies on Pitch-Class Set Theory developed mainly in the writings of Joseph Straus and João Pedro Oliveira; and the intervallic transformation working as conductor in the relationship between scales arranged in network and in building of large harmonics sets (supersets), approaching our research to neo-Riemannian studies about scale networks and parsimony, in particular the texts by Dmitri Tymoczko, Richard Cohn, Jack Douthett & Peter Steinbach.

Resumo em português
Selecionamos para análise nesta dissertação os Choros nº4 (1926) e Choros nº7 (1924), obras de câmara compostas por Heitor Villa-Lobos (1887-1959) no período em que o compositor demonstrou grande interesse pela estética modernista que hoje chamamos "pós-tonal". Villa-Lobos mesclou em seus Choros - bem como em grande parte de sua obra - elementos do modernismo europeu com outros oriundos da cultura popular brasileira, neste caso em particular o choro urbano carioca das primeiras décadas do século XX. Em síntese, a linguagem utilizada por Villa-Lobos nos dois Choros se aproxima muito da técnica de composição "em camadas" recorrente em obras de Stravinsky como A Sagração da Primavera (1913), Petrushka (1911) e O pássaro de fogo (1910). Villa-Lobos utiliza a interação entre materiais harmônicos independentes (estruturas simples geradas individualmente e calcadas em escalas tradicionais não gerenciadas pela tonalidade herdada do classicismo). Em muitos momentos esses materiais escalares estão empilhados formando grandes conjuntos (supercoleções) com oito, nove, dez, onze classes de altura e até o conjunto cromático completo, relacionados por invariâncias entre essas estruturas. Em outras ocasiões esses complexos harmônicos aparecem justapostos, relacionados por transformação intervalar gradativa, operados em torno de classes de altura invariantes e movimentos discretos entre as remanescentes não comuns (prioritariamente por movimentos de tons e semitons), procedimento chamado pelos teóricos da vertente "neo-Riemanniana" (como Richard Cohn, Jack Douthett e Peter Steinbach, entre outros) como "parcimônia". Notamos também que Villa-Lobos com frequência relaciona essas camadas estratificadas por disposições intervalares simétricas, gerenciando os conjuntos de classes de altura que compõem as partes ou a totalidade harmônica em torno de eixos de simetrias invariantes (reflexão). Notamos ainda invariâncias frequentemente reiteradas em ostinato (simetria por translação), enquanto que variantes remanescentes surgem e desaparecem como segmentos de coleções em outras camadas ao longo do trecho conduzido pelo ostinato, completando a totalidade harmônica por justaposição. A partir desse diagnóstico preliminar, elegemos nesta pesquisa duas correntes analíticas distintas e concomitantes: a simetria intervalar (reflexão e translação) a partir de estudos sobre Teoria dos Conjuntos desenvolvidos essencialmente nos textos de Joseph Straus e João Pedro Oliveira; e a transformação intervalar atuando como regente na relação entre coleções dispostas em rede e na construção de grandes complexos harmônicos (supercoleções), aproximando nossa pesquisa de estudos neo-Riemannianos sobre redes de coleções e parcimônia, em especial os trabalhos desenvolvidos por Dmitri Tymoczko, Richard Cohn, Jack Douthett e Peter Steinbach.

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Teoria dos Conjuntos versus Teoria neo-Riemanniana: duas abordagens interdependentes na análise dos Choros nº4 e Choros nº7 de Villa-Lobos (2014)

by Joel Albuquerque and

2014, VI Encontro de Musicologia de Ribeirão Preto

Este trabalho propõe um paralelo simultâneo entre dois parâmetros distintos de análise musical os quais em o nosso estudo se mostraram interdependentes e necessários na construção de um arcabouço teórico suficiente para compreender de... more

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Anton Webern's Use of Guitar in Op. 18 and its Influence on His Late Works (PhD Dissertation, Chapter 2 "Methodology")

by Adam Shanley

This is the second chapter from my dissertation entitled "Anton Webern's Use of Guitar in Op. 18 and its Influence on His Late works." This chapter describes the various methods used in order to make a case for the connection between... more

Figure 2.1: Variance in hand shapes that exists between set-class 3-11 on the piano keyboard (top), while hand-shape is related to set-class consistency in guitar hand- shape. Note that in the tablature staff held strings remain consistent while fret number values, r, increase by one. shown in figure 2.1 below.

Figure 2.2: 3-3 trichords root-6 (left), and root-5 (second from left). 3-5 trichords root-6 (second from right), and root-5 (right). Koozin-shapes are shown in figure 2.2 below.

Figure 2.4: A section of the 3-5 Tonnetz Figure 2.3: A section of the 3-3 Tonnetz.

and 2.4 below. set-class. Sections of the 3-3, and 3-5 Tonnetze are shown in figures 2.

inversion space for set-class 3-5, which can be seen in figure 2.7.

Figure 2.7: Set-class 3-5 based Contextual Inversion Space modeled after Straus’ 3-3 based space.

Figure 2.9: Horizontal alleys of a 3-3 derived Tonnetz contain pitch collections equal to the four members of set-class 6-20. class 6-20 generating alleys on the 3-3 Tonnetz are shown in figure 2.9. the Tonnetz, or PL chains, contain each hexatonic collection. The set-

Figure 2.10: The 3-3 trichordal Hyper-hexatonic system superset. My 3-3 based hyper-hexatonic system is shown below in figure

Figure 2.11: 3-3/3-4/3-5 tri-trichordal hyper-hex(enne)atonic model. 2.11 below shows my tri-trichordal hyper-hex(enne)atonic model.

2.12, with set-classes for each tetrachord labeled. is flanked by two 4-3 tetrachords. The Op. 30 row is shown in figure

Figure 2.13: Set-classes 4-3 and 4-17 appear as a right-pointing parallelogram and a diamond when mapped onto a 3-3 Tonnetz. figure 2.13. “diamond” shape. These shapes are shown below on the 3-3 Tonnetz in

Figure 2.14: Set-class 4-7 appearing as a left-pointing parallelogram on a 3-3 Tonnetz. below in figure 2.14. From any point on an LP generated alley if we begin to move East

Figure 2.15: Octatonic generating LP chains run Northwest/Southeast, while hexatonic generating RL chains run East and West. figure 2.15 below how each octatonic (LP) alley generates the four

Figure 2.18: Hexatonic hanger graph generated from Octz,3. From left to right pitches in each orbit hanger consist only of pitches from the North, East, South, and West hexatonic poles.

subsets from each set-class coming from the 6-20 superset.

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Review of Audacious Euphony: Chromaticism and the Triad's Second Nature (2012) by Richard Cohn

by Steven D Mathews

2013, Notes

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Análise da seção A do Estudo No 12 para violão de Heitor Villa-Lobos

by ciro visconti and

Abstract: This analysis verifies the section A’s chords progression of Vila-Lobos Etude No 12 with triadic post-tonality theories tools, as triadic transformations (described by Neo-Riemannian theory) and motivic progressions. This effort... more

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Hypothesis Riemann part II

by Nikos Mantzakouras

Mantzakouras Nikos

Distribution and finding intervals contained of the zeros of the Riemann Zeta function. Abstract.. One of the most celebrated problems of mathematics is the Riemann hypothesis which states that all the non trivial zeros of the... more

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Review of Timothy A. Johnson, "John Adams’s Nixon in China: Musical Analysis, Historical and Political Perspectives"

by William Guerin

2014

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Octatonic Ambiguities

by Nikola Komatović

2020, Musical Representations of East and West (proceedings)

This paper analyses the original use of octatonic, its reception among the Slavic composers, as well as its different evolution in the musical language of the western composers from the second half of the 19th and the 20th centuries. The... more

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Conversão Diatônica entre Sistemas Riemannianos Não-Redundantes

by Liduino Pitombeira

Apresentamos uma ferramenta para análise dos aspectos verticais e horizontais de uma obra diatônica. Trata-se de um mapeamento entre sistemas sonoros, baseado na função tonal generalizada de David Lewin. Propomos, então, um esquema de... more

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Rede de projeções por inversão: relações entre Tonnetze de diferentes tricordes (2017)

by Joel Albuquerque and

2017, Revista TeMA, MUSICA THEORICA 201617

Esta é uma proposta metodológica para análise harmônica fundamentada essencialmente em torno da inter-relação entre duas correntes prioritariamente desenvolvidas para o estudo de obras pós-tonais: a teoria dos conjuntos (STRAUS, 2005;... more

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Choros nº7 de Villa-Lobos: rede de coleções simultâneas (2013)

by Joel Albuquerque

2013, XXIII Congresso da ANPPOM, Natal/RN

Resumo: Este artigo pretende analisar o uso de coleções em rede no Choros nº7 de Villa-Lobos, mostrando a aplicação dos estudos desenvolvidos por Tymoczko em Scale Networks and Debussy (2007) e A Geometry of Music (2011). Nestes textos o... more

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Harmonic Function and Modality in Classic Heavy Metal (postprint)

by Esa Lilja

2019, Metal Music Studies

This article explores harmonic function, modality and guitar distortion in so-called classic heavy metal. I suggest a dual framework for analysing heavy metal, including such mirror-symmetries as ‘major’/‘minor’, ‘sharp’/‘flat’,... more

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Review, R. Cohn - Audacious Euphony, 2012

by Antonio Grande

Full title: “Robert Cohn, Audacious Euphony. Chromaticism and the Triad’s Second Nature”, Oxford University Press, New York 2012, 256 pp.
Paper review published on RATM, Lim 2015-1, pp. 154:163

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2 Proofs In 1: P<=NP and The Riemann Hypothesis

by Ohad Manor

2021, Manor Ohad

this paper is a collection of previous proofs in their updated form. The PNP proof and the Riemann Hypothesis. Both originally published in 23.3.21 by the author. Those are updated to 14.5.21 and meant for readers to have both in one... more

Define: P™ as the set of {2, 3} as "minimal primes" The first two combinations are by the natural maps and one used them to build the othe combinations. Overall, there are eight such combinations and additional one arrow combination, which yield (333)

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Climactic Preparation in the Adagio from Bruckner's Symphony No. 7 (1885)

by Samuel Gordon Huston

The Adagio from Anton Bruckner’s Symphony No. 7 (1885) has received limited scholarly analysis, although the Symphony continues to be widely performed. Indeed, hermeneutic studies of the Symphony, such as that by Martin Pulbrook, focus on... more

This table summarises movement towards C major in the Adagio:

The opening statement of the subject area can be rationalised as such:

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On the Riemann Z Function Zeros in an R2 Space and its relation with Irrationality

by Alfonso De Miguel

Projecting the square 1 through the diagonal of its hypotenuse we can build a new prime square 1 with an irrational symmetry. Combining the rational and irrational symmetries we can get new prime squares which roots will be irrational.... more

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Neo-Riemannian Analysis

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