>Relationship between the three main harmonic functions in a tonality

>Each ellipse gives information about

>It isnoteworthy that some chords have different function depending on its previous chord in the progression

>Chord progressions are widely used in music. The automatic generation of chord progressions can be challenging because it depends on many factors, such as the musical context, personal preference, and aesthetic choices. In this work, we propose a penalty function that encodes musical rules to automatically generate chord progressions. Then we use an artificial immune system (AIS) to minimize the penalty function when proposing candidates for the next chord in a sequence. The AIS is capable of finding multiple optima in parallel, resulting in several different chords as appropriate candidates. We performed a listening test to evaluate the chords subjectively and validate the penalty function. We found that chords with a low penalty value were considered better candidates than chords with higher penalty values.

>In this paper we present a 12-dimensional tonal space in the context of the Tonnetz, Chew’s Spiral Array, and Harte’s 6-dimensional Tonal Centroid Space. The proposed Tonal Interval Space is calculated as the weighted Discrete Fourier Transform of normalized 12-element chroma vectors, which we represent as six circles covering the set of all possible pitch intervals in the chroma space. By weighting the contribution of each circle (and hence pitch interval) independently, we can create a space in which angular and Euclidean distances among pitches, chords, and regions concur with music theory principles. Furthermore, the Euclidean distance of pitch configurations from the centre of the space acts as an indicator of consonance.

>これも納得感ある

> 7つのコードに本質的な境界などなく円環状の構造であって、人間が適当に境界を引いただけということ