torstenanders Posted October 6, 2016 Share Posted October 6, 2016 Dear Janusz, We were recently discussing potential microtonal support for Opusmodus. I understand that you suggested a representation based on 12-tone equal division of the octave (EDO) with free cent deviations as a flexible and generic solution. I agree that this could be a sufficient solution in the background, because all microtonal pitches, intervals, chords and scales can be specified that way. However, in my view it would be an insufficient solution as the only microtonal representation at the user-level, because it would be highly cumbersome. Imagine composing a melody in C-major by writing and reading only frequency values in Hz. Of course every tone in C-major can be expressed as a frequency in principle, but as a musician you rarely want to think in such numeric detail. Besides, a slight tuning variation would change all the figures. Composing microtonal music in cent values is equally awkward. I understand that for a CAC environment like Opusmodus it is important to have a generic representation, but there are so many microtonal tunings and notations. Luckily some efforts have been done my the community already towards more generic representations. Basically, there are three main microtonal directions in Western music: Just intonation (JI): Multidimensional tuning approach, where intervals can be represented as whole number frequency ratios, with theoretically an unlimited number of tones per octave. Examples for JI intervals are 3:2 (perfect fifth), 5:4 (major third), 7:4 (harmonic seventh), 7:6 (subminor third) etc. JI thinking is important, e.g., for extending harmony beyond traditional boundaries by intervals that musicians can intonate by ear, and it was useful for that already centuries ago. The Common Lisp support of fractions makes JI support particularly interesting for Opusmodus. Equal temperaments: Equal divisions of the octave (or other intervals). Some equal temperaments are particularly widely used and researched, because they approximate certain just intonation intervals particularly well, including 19-EDO, 22-EDO, 24-EDO (quartertones), 31-EDO (almost quarter comma extended meantone), 41-EDO (almost Pythagorean tuning), 53-EDO and 72-EDO. Other temperaments: Designed to reduce the total number of tones and that the cognitive workload using them, which approximate JI intervals. Examples are the various meantone temperaments, or well temperaments. My explanation already suggested that these different schools of thought are related. Various equal temperaments are widely used, because they approximate certain JI intervals rather well, while at the same time allowing for arbitrary transpositions within a limited number of tones overall. For example, our standard 12-EDO approximates 3:2 and its relatives (4:3 -- 3-limit intervals) almost perfectly, and 5:4 and its relatives (5:6 etc. -- 5-limit) reasonably well, while 7:4 (7-limit) or 11:8 ( 11-limit) are not part of 12-EDO. Some notations aim to present a unified format for multiple approaches. Such notations could be a useful foundation for a microtonal representation of Opusmodus. A relatively simple example of such an approach is the HEWM notation, which stands for Helmholtz / Ellis / Wolf / Monzo notation (Monzo, 2005a and 2005b). This staff notation is an extension of the common music notation designed to support both 72-EDO (a superset of 12-EDO, 24-EDO, and also 36-EDO), and 11-limit just intonation. The notation exists both as staff notation, and ASCII for written communication in emails (and potentially programming code). In this notation, all nominals (pitches without accidentals, like a, b, c, d...) are considered to be tuned in fifth (Pythagorean tuning, 3-limit). In other words, the interval C, E is considered a Pythagorean major third, not a just major third. For each higher limit, the notation introduces a pair of accidentals to raise or lower the pitch accordingly. For example, the JI major third is notated C, E-, where the minus sign (-) represents a transposition by a syntonic comma downwards. That way, the notation can distinguish between a Pythagorean major third and a just major third. In a performance situation, this interval can either be tuned in 72-EDO or justly -- but the notation is the same. Particularly comprehensive notations based on the same principles are Sagittal notation (Secor and Keenan, 2004) and the Extended Helmholtz-Ellis JI Pitch Notation (Sabat and Schweinitz, 2005). Sagittal is explicitly designed to support both just intonation (including highly complex intervals) and many equal temperaments, but Extended Helmholtz-Ellis JI Pitch could do that in principle as well to a certain degree. It is more simple than Sagittal (while more complex and HEWM), and could therefore be preferable. However, only Sagittal offers also ASCII representations for all its accidentals, so for a programming environment it could be a more natural choice. If Opusmodus would support any of these notations both in OMN and the resulting notation, then it would offer a highly flexible environment for composers interested in microtonal music, and it would allow for a variety of tunings with a single notation. The existing ASCII representations could likely not be directly translated into OMN, because many special characters are reserved in Common Lisp, but having them would be a starting point. I would be happy to help designing a notation suitable for Opusmodus, i.e., taking the restrictions of Lisp syntax into account. What do you think? Best, Torsten PS: If you also want to support other temperaments beyond equal temperaments and JI then we should discuss how to represent regular temperaments (Milne et al. 2007), but then the notation gets even more tricky. References Milne, A., W. Sethares & J. Plamondon (2007) Isomorphic Controllers and Dynamic Tuning: Invariant Fingering over a Tuning Continuum. Computer Music Journal. 31(4), 15–32. Monzo, J. (2005a) 72-tone equal-temperament / 72-edo. In: Encyclopedia of Microtonal Music Theory. Available from: http://tonalsoft.com/enc/number/72edo.aspx Monzo, J. (2005b) HEWM notation. In: Encyclopedia of Microtonal Music Theory. Available from: http://tonalsoft.com/enc/h/hewm.aspx Sabat, M. & Schweinitz, W. von (2005) The Extended Helmholtz-Ellis JI Pitch Notation. [online]. Available from: http://www.marcsabat.com/pdfs/notation.pdf Secor, G. D. & Keenan, D. C. (2004) Sagittal. A Microtonal Notation System. Xenharmonikôn, An Informal Journal of Experimental Music. 18. Available from: http://sagittal.org/sagittal.pdf Torsten Anders http://www.torsten-anders.de lviklund 1 Quote Link to comment Share on other sites More sharing options...
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