Responde
This chapter, in my opinion, work as methodology and as historical context to justify the use of the circle of fifths. Nonetheless, there are few new ideas and knowledge to explore for the future. First of all, the notion of going further chords and include the minor-relatives on the harmonic journey. Also, using different transposition intervals, as for example, the third (Figure 2.2b) as vertical transposition journey out the fifths (Figure 2.2b) as horizontal transposition journey - already used.
Figure 2.2b, inspired me to explore new possibilities of transposition of chords that are taken of the harmonic series.
Figure 2.3a, will help me to categorised the stable chords and unstable chords, understanding their relationship between both categories of chords.
Notes
In both music theory and music psychology there are traditions that express the tonal hierarchy by geometric models. These traditions correlate spatial distance with intuitive musical distance. For example, the region (or key) of G major is understood to be closer to C major than Ab major is to C major. Within a C major context, the chord V is closer to I than is ii; the pitch D is closer to C than is Bb. Observe that “closer” does not necessarily mean more proximate in log frequency; in that sense, Db is closer to C than is D. Rather, D is closer in the sense that it lies within the C major scale while Db does not. The distance in question is not acoustic but cognitive. A spatial model is a way to represent this kind of internalised knowledge.
In music theory this approach originates in the Baroque period, when tempered scales came into use, as a device for teaching smooth adulation Johann David Heinichen (1728) proposed the regional circle in Figure 2.1a, in which the major circle of fifths alternates with its relative minor counterpart.
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Figure 2.1a
Such a representation depends on the group-theoretic feature that progression by perfect fifth passes through all the members of the chromatic collection before returning to the starting point. Adjacent moves on the circle stand for close modulations and nonadjacent moves for more distant modulations. However, Heinichen’s circle did not reflect tonal practice - D minor and A minor are not closer to C major that are F major and G major - so alternative proposals were made by Johann Mattheson (1735) and others. All of them took advantage of the circle of fifths but similarly foundered in the handling of major-minor relationships, for which a single circle provides too impoverished a space. David Kellner (1737) made progress by suggesting the structure in Figure 2.1b, which links relative minor-major regions in a double circle of fifths.
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Figure 2.1b
The psychological tradition has focused more on pitches and pitch classes than on regional or harmonic relationships. Pitches close in log frequency are perceived as near to one another, and so are pitches in a 2:1 frequency ratio. To model these two kinds of proximity M. W Drobisch (1855) suggested that pitch height be represented on a helix, which octave recurrences placed proximally on the vertical axis of the turning helix. Roger Shepard (1982) extends this approach to include other closely related intervals such as the perfect fifth. Combining the semitone and fifth cycles, for instance, yields the double helix in Figure 2.2a, which Shepard calls the “melodic map”
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Figure 2.2a
Another pitch space, and one with a long history in music theory, is the lattice structure in Figure 2.2b, in which the horizontal axis is laid out in perfect fifths and the vertical axis in major thirds. It was first proposed by the mathematician L. Euler (1739) as a way of representing jus intonation. If pitches are compressed into one octave, intervals can be figured by multiplying ratios along the lattice.
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Figure 2.2b
The experimental work of Carol Krumhansl and her associates (summarised in Krumhansl 1983, 1990) investigate listeners’ responses for the cognitive proximity of pitch classes, chords, and regions in relation to an induced tonic. For each level, multidimensional scaling yields a geometric solution that perspicuously represents the patterns implicit in the data. The pitch class level patches the form of the cone in Figure 2.3a, in which the most stable pitch classes appear at the bottom and the least stable at the top.
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Figure 2.3a