Minor Triads Research Articles

Geometrical Methods in Recent Music Theory

[1] The following remarks are prompted by Joti Rockwell's interesting article, Birdcage Flights: A Perspective on Inter-Cardinality Voice Leading (2009). My goal is not to take issue with Rockwell's specific claims but rather to underscore a few details that might escape the casual reader's attention. In particular, I want to stress three basic points.1. Rockwell's discrete birdcage graphs, which represent efficient voice leadings between chords of different sizes, can be embedded within the infinite-dimensional OPC discussed by Callender, Quinn, and myself.(1) In general, discrete voice-leading graphs can always be embedded within the continuous spaces we describe.2. Because of this, Rockwell's graphs inherit some of the complications of C space, and may distort voice-leading relationships among nonadjacent chords.3. This is actually symptomatic of a more general issue affecting a variety of music-theoretical graphs. Roughly speaking, we have no guarantee that graphs whose edges refer to musical motions of a particular type will give rise to an intuitive or familiar notion of distance between nonadjacent chords.[2] This last item is significant because theorists sometimes seem to endorse the following methodology. First, one selects some interesting domain of musical objects and some interesting set of motions among them. (For example, single-voice voice-leading between major and minor triads.) Second, one constructs a graph representing all of these motions between all the objects in question. Third, one interprets the resulting graph as providing a measure of distance. Thus, for example, one might use the graph to analyze music that moves between non-adjacent chords, or claim that larger leaps on the graph are musically disfavored in some way.[3] This last step, however, involves a subtle leap. Consider, for example, the familiar Tonnetz (Figure 1).(2) Two chords are adjacent on this graph if they can be linked by what Cohn calls voice voice leading in which a single voice moves, and it moves by just one or two semitones (Cohn 1996). However, larger distances in the space do not faithfully mirror voice-leading facts. On the Tonnetz, C major is two units away from F major but three units from F minor-even though it takes just two semitones of total motion to move from C major to F minor, and three to move from C major to F major (Figure 2). (This is precisely why F minor so often appears as a passing chord between F major and C major.) It follows that we cannot use the Tonnetz to explain the ubiquitous nineteenth-century IV-iv-I progression, in which the two-semitone motion [arrow right] is broken into the semitonal steps [arrow right][arrow right]. More generally, it shows that Tonnetz-distances do not correspond to voice-leading distances in any straightforward way (Tymoczko 2009).Figure 1. The TonnetzFigure 2(click to enlarge and see the rest)[4] Note that the problem persists even if we try to reinterpret the Tonnetz as representing common tones rather than voice leading: both F minor and E minor are three Tonnetz-steps away from C major, even though C major and F minor have one common tone, while C major and E minor have none. (As before, shorter distances are easier to interpret: two chords are adjacent on the Tonnetz if they have two common tones, and any pair of chords that are two steps away will share exactly one common tone.) Thus, neither voice leading nor common tones allow us to characterize Tonnetz distances precisely. We seem forced to say that Tonnetz-distances represent simply the number of parsimonious moves needed to get from one chord to another-and not some more familiar music-theoretical quality.[5] From this point of view, there is a fundamental difference between the Tonnetz and Douthett and Steinbach's Cube Dance (Douthett and Steinbach 1998). (Figure 3) Like the Tonnetz, Cube Dance depicts a collection of local moves, in this case the single-semitone voice leadings between major, minor, and augmented triads. …

Musical Actions of Dihedral Groups

The sequence of pitches which form a musical melody can be transposed or inverted. Since the 1970s, music theorists have modeled musical transposition and inversion in terms of an action of the dihedral group of order 24. More recently music theorists have found an intriguing second way that the dihedral group of order 24 acts on the set of major and minor triads. We illustrate both geometrically and algebraically how these two actions are dual. Both actions and their duality have been used to analyze works of music as diverse as Hindemith and the Beatles.

Moving Through Triadic Space

Using Julian Hook’s theory of Uniform Triadic Transformations as a point of departure, I construct a network in which the “distances” between major and minor triads may be measured and progressions may be compared. From there, the network is used to analyze a portion of the 1992 composition “A Man in a Room, Gambling,” by contemporary British composer Gavin Bryars, revealing harmonic relationships that may otherwise go unnoticed.

Automatic Transcription of Melody, Bass Line, and Chords in Polyphonic Music

September 01 2008 Automatic Transcription of Melody, Bass Line, and Chords in Polyphonic Music Matti P. Ryynänen, Matti P. Ryynänen Search for other works by this author on: This Site Google Scholar Anssi P. Klapuri Anssi P. Klapuri Search for other works by this author on: This Site Google Scholar Author and Article Information Matti P. Ryynänen Anssi P. Klapuri Online ISSN: 1531-5169 Print ISSN: 0148-9267 © 2008 Massachusetts Institute of Technology2008 Computer Music Journal (2008) 32 (3): 72–86. https://doi.org/10.1162/comj.2008.32.3.72 Cite Icon Cite Permissions Share Icon Share Facebook Twitter LinkedIn MailTo Views Icon Views Article contents Figures & tables Video Audio Supplementary Data Peer Review Search Site Citation Matti P. Ryynänen, Anssi P. Klapuri; Automatic Transcription of Melody, Bass Line, and Chords in Polyphonic Music. Computer Music Journal 2008; 32 (3): 72–86. doi: https://doi.org/10.1162/comj.2008.32.3.72 Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentAll JournalsComputer Music Journal Search Advanced Search This content is only available as a PDF. © 2008 Massachusetts Institute of Technology2008 Article PDF first page preview Close Modal You do not currently have access to this content.

Voice-Leading Spaces

This paper studies networks of pitch-classes that model voiceleading-the progression of pitches to form horizontal voices or lines. While tonal voice-leading depends on distinctions between step and leap, consonance and dissonance, chord-tone and embellishment, voice-leading in post-tonal music simply and nonnormatively describes the motions of pitches, one to another. First, I stipulate some limitations on totally free voice-leading and construct a complete taxonomy of voice-leading motions. Then I examine Richard Cohn's explicit coupling of transformation and voice-leading of major and minor triads as represented on Hugo Riemann's tonnetz. The tonnetz is a traditional example of what I have previously called a compositional space; such spaces are outof-time networks of pcs that can underlie compositional or improvisational action. After studying various transformations of the tonnetz, voice-leading is implemented by the use of another type of compositional space, two-partition graphs. These graphs are generalized so that pairs of pcsets connected in the graph need not be disjoint nor form members of only one set class. Any specified voice motions can be stipulated. The result is a collection of voice-leading spaces, a new category of compositional space. Finally, I describe an algorithm that allows one to construct voice-leading spaces of any degree of complexity or closure. This content downloaded from 157.55.39.35 on Mon, 29 Aug 2016 05:57:48 UTC All use subject to http://about.jstor.org/terms

Sharp low- and high-frequency limits on musical chord recognition

This research is concerned with the recognition of major and minor triads in low- and high-frequency regions. Its specific aim is to determine the limitations on the frequency range within which musicians can recognize minor and major triads. Twelve subjects were tested in a task in which the stimuli were successions of minor and major triads centered on frequencies ranging from 39 to 8372 Hz. Subjects were asked to say, for each triad, if it had or did not have the characteristics of a minor or a major chord. Minor and major chords were presented in separate blocks. Results show that subjects failed to recognize minor and major triads when placed below approximately 120 Hz and above approximately 3000 Hz. Statistical analysis of results also demonstrates that the failure to recognize triads differed at the low- and high-frequency ends of this range — occurring much more abruptly at the low-frequency border. The failures at the borders, and some differences there in the identifiability of major and minor triads, may possibly be related to inadequate spatial resolution of components at the lower-frequency border and to frequency limitations on phase-locking at the higher.

Perception of musical tension in short chord sequences: The influence of harmonic function, sensory dissonance, horizontal motion, and musical training

This study investigates the effect of four variables (tonal hierarchies, sensory chordal consonance, horizontal motion, and musical training) on perceived musical tension. Participants were asked to evaluate the tension created by a chord X in sequences of three chords [C major-->X-->C major] in a C major context key. The X chords could be major or minor triads major-minor seventh, or minor seventh chords built on the 12 notes of the chromatic scale. The data were compared with Krumhansl's (1990) harmonic hierarchy and with predictions of Lerdahl's (1988) cognitive theory, Hutchinson and Knopoff's (1978) and Parncutt's (1989) sensory-psychoacoustical theories, and the model of horizontal motion defined in the paper. As a main outcome, it appears that judgments of tension arose from a convergence of several cognitive and psychoacoustics influences, whose relative importance varies, depending on musical training.

Major/Minor Triad Identification and Discrimination by Musically Trained and Untrained Listeners

The extent to which a computer-synthesized continuum of major to minor triads was categorically perceived was examined using labeling and discrimination tests. The 32 listeners varied widely in " musicality," assessed by an objective test of basic musical skills. There was a strong positive relationship between musicality and ability to label the major/minor continuum consistently ( measured by the slope of the labeling function). Overall discrimination performance varied only weakly with musicality, although the pattern of discrimination performance across the continuum differed strongly among three listener subgroups, distinguished on the basis of musicality. The most "musical" listeners showed a close relationship between the position of the discrimination peak and the category boundary calculated from the labeling function, a strong indicator of categorical perception. On similar criteria, the evidence for categorical perception was nonexistent in the least musical listeners and moderate in an intermediate group. From the evidence that the extent of categorical perception appears to vary in a graded fashion with the degree of musicality, we conclude that categorical perception can arise primarily through a process of learning.

Internal tuning and external harmony of carillon bells

Bells produce sound spectra with frequencies that are not necessarily harmonically related. In good carillon bells, however, several of the lower partials are tuned to a typical pattern given by the frequency ratios 0.5 (hum), 1.0 (prime), 1.2 (third), 1.5 (fifth), and 2.0 (octave). The fact that the third is minor might explain why most carillon music is written in a minor key, because the internal tuning of the bells agrees with the minor triads of tonic, dominant, and subdominant in such keys. It has been proposed in the past that for music in major keys it might be preferable to tune the third to a major third (1.25) or, maybe, to a neutral third (1.225) to allow music in both types of keys. Apart from the question whether or not such a tuning is physically possible, a psychophysical experiment was performed with two short harmonized melodies, one in D‐minor and an equivalent transcription in D‐major, played on three different computer‐generated carillons. Each bell of these carillons was a pulse code‐modulated recording of a real carillon bell, from which the third partial had been electronically removed and replaced by a partial of similar temporal structure and appropriate frequency (1.2, 1.225, and 1.25, respectively). Many subjects, uninformed about the sounds they heard, were asked to select the best‐sounding melody in each of the 15 pairs that can be formed from the six melody versions. Rating scales produced by various groups of subjects will be discussed.

Perceived Structure of Triads

This study explored the perception of major, minor, diminished, and augmented triads in two experiments. In the first, subjects judged whether pairs of these triads were of the same or different chord type. In the second, subject rated the similarity of these pairs of triads. Subjects were grouped by degree of musical training. We looked for effects of several variables on the subjects' judgments: the "musical key" suggested by major and minor triads, whether a triad was presented in root position or in first inversion, and whether the notes of the triad were presented successively (i. e., arpeggiated) or simultaneously. Among the three groups of subjects, only the judgments of those who were highly trained appeared to be influenced by the above factors. For these subjects, judgments were influenced by relative position in the circle of fifths. In addition, judgments appeared to incorporate the perceived degree of consonance or dissonance, rather than relative interval sizes.

Perception of musical and nonmusical triads

Fundamental frequencies of just major triads are in ratios of successive integers 4 to 5 to 6 which produces many coincidences of harmonics. This triad has an unambiguous “fundamental bass” (Rameau and Terhardt. We examined judgments of major (4:5:6) and minor (10:11:12) triads as well as nonmusical triads using ratios of nonsuccessive integers (3:5:7 and 5:7:9) which likewise have coincidences of overtones and unambiguous fundamental bass notes. By using nonmusical triads, we can observe intonation sensitivities apart from biases due to musical experience. We varied the middle note of each triad by −30, −15, +15, and +30 cents thus producing beating harmonics and ambiguous fundamental bass notes. Subjects were presented with pairs of triads and selected (1) the triad that is more “in tune,” (2) the triad they prefer, and (3) the triad that is more smooth. Listeners divided into two distinct groups: one showed an inverted V function, in which ratings are highest for the “just” triads; the other showed an inverted W function in which the triads deviating from just intonation by 15 cents are given the highest ratings.

Tonality in Britten's Song Cycles with Piano

As an harmonic technique, expanded tonality may act either as a decisive stage in a composer's approach to atonality, or as a means whereby he can preserve the essential hierarchies of tonality itself in a context of pervasive chromaticism. For Schoenberg, the expansion of tonality was a process which did not stop until tonality had been ‘exploded’, chromaticism ousting diatonicism just as new types of chord ousted major and minor triads. For Bartók, as, later, for Britten, tonality could survive even without the triad, without fundamental diatonicism. The use of tonal centres—a tonality of single notes—is the hallmark of Bartók's third string quartet, and the tonal centres control the chromaticism, harmonic progressions functioning non-systematically but logically in terms of the schemes of tonal conflicts and relationships appropriate to the work.

A Just Scale for Music

In addition to the well-known just major scale, Robert Smith described another that he regarded as equally harmonious. The second scale is like the well-known one except that its first two intervals are interchanged. It is now shown that this scale may be obtained from three consecutive minor triads, in a manner similar to that in which its well known counterpart is often obtained from major triads. A comparison and experimental test of the two scales shows little to choose between them. Sometimes one is preferred, and sometimes the other. A satisfactory scale should include the pitches from both.

The Nature of Harmony. Major and Minor Triads

The Nature of Harmony. Major and Minor Triads