Twelve-tone Equal Temperament Research Articles

편종의 음색 분석 연구

편종의 소리합성을 위한 연구의 첫 단계로 FFT 분석을 통하여 그 음색을 분석하였다. 현대 편종 음계의 음고는 서양의 12평균율의 음고와 큰 차이가 없었다. 편종의 음색은 시간의 흐름에 따라 변화한다. 타격시 모든 주파수 대역에 걸쳐 생성된 노이즈들은 단시간에 사라지고, 고유 배음만이 남는다. 그 후 높은 주파수 성분의 배음부터 차례로 소멸되고 여음부에는 두개의 배음만이 남는데, 이 때 남은 배음의 주파수에 의하여 맥놀이 현상인 발견된다. 본 연구에서 사용된 편종은 비교적 최근에 제작된 편종으로 추후 고대 편종의 음색에 관한 연구가 병행된다면 편종 원형의 소리 합성이 가능할 것이다. As the first step in research for sound synthesis of Pyungjong, timbre was analyzed by FFT analysis. There was no significant difference in the pitch of scale between modern Pyungjong and western twelve-tone equal temperament scale. The timbre of Pyunjong chagnes over time. When striking Pyunjong, the noises to be generated across all frequencies will disappear in a short period of time, and a unique partial only is left. Then it disappear from the partial of high frequency components in order, and two partials remain at release time. At this time, the beating is occurring by the frequency of the remaining partial. Pyungjong used in this study is relatively a recently manufactured Pyungjong, and for original sound synthesis of Pyungjong, a study of ancient Pyungjong's timbre should be additionally conducted later.

A fuzzy framework to explain musical tuning in practice

A theoretical tuning system is a set of pitches that can be used to play music. It is a fact that the human ear perceives notes with very close frequencies as if they were the same note. Therefore, in our approach a musical note and its pitch sensation are modeled as L–R fuzzy numbers with a modal interval and a bounded support. We pay particular attention to the 12-tone equal temperament (12-TET) for being the most widely used tuning system and we define the fuzzy 12-TET composed of 12 fuzzy notes. A similarity relation between a fuzzy note and a theoretical note can be defined, and subsequently a similarity class associated to each one of the fuzzy notes in the fuzzy 12-TET arises. Finally, we study the similarity between the theoretical notes in a given score and those notes played by a musician making use of an OWA operator. Some numerical examples illustrate our approach.

On Pitch-Class Set Cartography

Recent music-theoretical research has proposed two ways of mapping the pitch-class set universe. Fourier spaces, constructed by Quinn, relate sets to one another based upon their composition from members of interval cycles, reflecting what might be called “harmonic quality.” Voice-leading spaces, generalized by Callender, Quinn, and Tymoczko, illustrate voice-leading relationships between sets. Though many researchers have noted hints of a relationship between these two types of space, their exact relation remains murky.One way of relating these two spaces involves associating voice leading with motion through Fourier space. Voice-leading displacements of each of the six interval classes can be associated with specific changes of position in each of the six Fourier spaces for twelve-tone equal temperament. Likewise, displacement spaces, showing all of the sets that can be related by voice-leading displacements of a particular interval class, model predictable motion through each of the Fourier spaces.

Alternative tunings, alternative tonalities

Several approaches to the design and use of alternativetuning systems (defined as systems other than 12-tone equal temperament)are described. Each of these tuning systems has been created inorder to preserve or extend qualities associated with tonal musics.This is by no means a comprehensive catalogue of such approaches,but rather a selection among those speculative theories that appearto offer greater potential as resources for a wide community ofcomposers. Atonal or purely timbral uses of alternative tuning systemsare not examined in this article nor are historical tunings or schemesfor retuning existing repertoire.

On Coherence and Sameness, and the Evaluation of Scale Candidacy Claims

We begin with a view of the universe of pitch classes in 12-tone equal temperament, shown in (a) in Example 1. Pitch class n is represented by the fraction n/12, for n = 0, 1, ... 11. Next, we expand the field of discussion to the set of all real numbers greater than or equal to 0 and strictly less than 1, shown in 1(b). This represents an infinite pitch-class space that can be understood as having been constructed by filling in the blank spaces of 1(a). In this paper, I would like to inquire into this question: which pitch-class subsets of 1(b) might function as scales?

A Venerable Temperament Rediscovered

N THE HISTORY OF Western music, the keyboard temperament with the longest run was meantone, which was the nearly universal standard from 1350 or so until it was almost totally eclipsed by twelve-tone equal temperament (twelve-equal) in the first half of the nineteenth century. Meantone and the development (and ultimate success) of keyboard instruments are inextricably intertwined: because its sonorities are so close to those of just tuning, meantone was able to emulate the sweetness of postPythagorean, pure triadic intonation characteristic of vocal music from around the time of Dunstable-the earliest keyboard pieces were in fact transcriptions of vocal works. The present brief panegyric on meantone, however, is not meant as a rehearsal of its established merits for earlier music; the recent, very successful reintroduction of meantone as an intonational standard by many of

A Proposed Extension to the MIDI Specification concerning Tuning

The Musical Instrument Digital Interface (MIDI) specification was conceived originally as nothing more elaborate than a way to trigger one synthesizer's sound from another one's keyboard. It is therefore not surprising that the continuing and expanding use of MIDI in a wide variety of musical environments has revealed a number of deficiencies (Moore 1988). Although still called MIDI 1.0, the specification allows for extensions and addenda, and the current official version (International MIDI Association 1988) is much expanded from its initial publication in 1983. One area that has received little attention so far is that of tunings other than 12-tone equal temperament (12-ET). Perhaps owing to the clavier-based nature of MIDI synthesizers-or to their preponderant use in commercial music, where such subtleties are seldom explored-tuning traditionally has been a low priority for manufacturers. However, with five manufacturers (Emu, Ensoniq, Korg, Kurzweil, Yamaha) currently implementing one form or another of user-defined tuning in their instruments, the time for extending the MIDI specification to embrace this area may have arrived.

A Mathematical Model of the Diatonic System

In this article a mathematical model of the is presented, that is, the notion diatonic system is defined in mathematical terms. We believe the proposed model to be of considerable descriptive power, and potentially at least, of considerable explanatory power as well. That is to say, the model not only successfully captures the intuitive sense of diatonic but also opens possibilities towards answering deeper questions regarding the system, such as the source of its remarkable persistency through centuries, if not millennia, of human musical history (Western musical history, at any rate). In Agmon (1986) it is argued at length that the model represents a cognitive theory of diatonicism.' While this point is crucial from a broader theoretical perspective, we shall not pursue it again here except for pointing out that the empirical domain which the model describes is not a perceptual domain of pitches, but a more abstract domain of mental representation. In particular, to a considerable (and possibly surprising) extent, the model is independent of the question of how it may or may not be implemented in terms of pitches, so that the whole issue of intonation (including, for example, the status of 12-tone equal temperament) is for the

Paratactical Tuning: An Agenda for the Use of Computers in Experimental Intonation

Several factors have significantly affected, and perhaps more importantly, limited, the choice and usage of intonational strategies in music. Recognition of the existence of these limitations, as well as a discussion of procedures adopted to circumvent them, may help us to design computer-based tuning systems with greater flexibility and increased experimental possibilities. One of these limiting factors is the classification of instruments into two categories: those of fixed pitch (like keyboards) and those of variable pitch (like unfretted strings). Even though most instruments do not securely fit into either category (like the fretted strings, the keyed winds, or countless other instruments from around the world), it has been a consistent tendency among composers, theorists, and musicians to assume that they do. Or it has been assumed that their idiomatic usages preclude the possibility of performance in the mode of the other category. For example, it is generally conceded that string quartets and vocal ensembles develop sophisticated intonational adjustment procedures because of the nondiscrete pitch scale available to them, while other ensembles of so-called fixed-pitch instruments (those with a presumed limited set of pitches available, like the piano) have a simpler set of procedures for intonational performance. (In modern Western music, this is assumed to be twelve-tone equal temperament.) Although most contemporary Western performers on fixedpitch instruments have developed methods of extending the pitch resources of the instruments' twelve-tone equal design, these methods are not

Musical Instruments in Nineteen-Tone Equal Temperament

Yasser has proposed a theory of evolving tonality to account for the structural relations of the pentatonic scales used by various primitive peoples to the seven-tone scales used in modern European music. He postulates a series of organically connected scales of increasing complexity and predicts a “supra-diatonic” scale as the logical next stage in tonal evolution. This scale has twelve regular degrees and seven intermediate auxiliaries. The intonation of this scale which appears most practical is nineteen-tone equal temperament. This temperament has been known for centuries as an intonation of the seven-tone scale. The fact that it serves for both scales is a consequence of their interrelations. In this paper, the theory of evolving tonality is sketched, instruments built for testing the theory are described, and both new and old scales and chords in nineteen-tone equal temperament are demonstrated aurally and compared with the nearest equivalents in the conventional twelve-tone equal temperament.