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Microtonal notations

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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? 




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.




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


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Dear Janusz,


Thanks for your response. What you are suggesting here are special accidentals for 1/4 tones and 1/6 tones (24-EDO and 36-EDO) -- simple subdivisions of the standard 12-EDO proposed early on in the 20th century -- and you are suggesting a numeric pitch notation based deviations from 12-EDO in cent. 


Microtonality and microtonal notations offer to rethinking harmony from an acoustic foundation, like the figured bass notation offered to think about harmony in a different way than in Renaissance counterpoint, and the later thinking in triads and their inversions offered a next level of abstraction. But such rethinking needs suitable foundations. Notating all microtonal pitches as deviations of 12-EDO is about as cumbersome as trying to compose Wagnerian harmonic progressions in Baroque figured bass notation. Sure, that is possible in principle, but not suitable. 


If you take the 1/4 tones and 1/6 tones of your proposal, complement these by 1/12 tones, add some memorable ASCII representation (not cent deviations), and arrange these in a systematic way then you already have 72-EDO, a possible rendering of HEWM, the simplest notation I was proposing above, which is suitable for both notating 11-limit JI and at least a few equal temperaments. 


While HEWM is restricted (e.g., many widely used equal temperaments could be expressed in that notation; and JI intervals beyond 11-limit are not included), at least it is based on acoustic foundations and is therefore more future-facing than 24-EDO or 36-EDO. 


Please don't get me wrong. I don't want to appear here as a zealot for some obscure temperaments. I just suggest to have the microtonal pitch representation based on *musical* (e.g., harmonic) considerations, not just some cent figures. As I said in my first post, OMN supports pitch *names*, because they are more memorable etc. than mere frequency values. Microtonal pitches should be treated the same (while having numeric cent values as an option is certainly welcome as well). 


What would be truly great is if you would allow users to add such functionality (extending the OMN parser by adding new pitch symbols or accidental names, and allowing for defining how these accidentals are notated in the score). 


Anyway, if you feel that supporting a microtonal notation along the lines I sketched above is beyond what you have time for or are interested in then I understand that of course. After all, one should not aim for a perfect system, but one that is good enough for the purposes of the majority of its users.    ;-)




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On 2nd thought, users might be able to define some custom representation on top of OMN so that user-defined pitch and accidental symbols could be translated into the cent-based representation you suggest (e.g., C4+50). However, if at least users could define their own accidentals for specific tones, that would be charm. For octave-repeating tuning systems (which are the most common) it would be sufficient to have users define mapping of specific tones without octave component to accidentals (or accidental combinations).


Basically, if users could freely define accidental mappings like those of 1/4 and 1/6 tones you used above, e.g. by mapping to special positions in certain accidental fonts. I know that this cannot be directly exported to MusicXML (although with some tricks using things like templates and Sibelius scripts certain things are indeed possible...), but the Opusmodus notation can also be directly exported to PDF already :)


Such a proposal is likely more easy to realise, and it is also fully user-customisable. 



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I think it is not feasible to introduce too many new symbols. They are not practical because hard to remember and therefore not commonly used. Furthermore they will not be compatible with other notation software.


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> I think it is not feasible to introduce too many new symbols. They are not practical because hard to remember and therefore not commonly used.


It would be ideal to let users define what accidental symbols they want to introduce. They can then decide themselves how many symbols they want to handle 😉


Several notation software packages do already support a wide range of microtonal accidentals. There are too many notation systems to list them all here, but the highly comprehensive sagittal notation alone is already supported by various notation software, e.g., built-in in the upcoming Dorico via its  SMuFL (Standard Music Font Layout) alongside many other accidental sets (http://www.smufl.org/version/latest/, e.g., http://www.smufl.org/version/latest/range/spartanSagittalSingleShaftAccidentals/); built-in in MuseScore (https://musescore.org/en/node/41001); available in Sibelius via a script (http://dkeenan.com/sagittal/) etc.   


Yes, automatically exporting microtonal scores to notation software is indeed tricky, but even with the limited support we have with current notation software there are ways (admittedly hacks, but nevertheless -- see attached), and with the new Dorico, which internally represents pitch more flexibly, there may in the future be more strait-forward ways too.  


I don't know how Opusmodus internally creates its notation, but if at some stage in the future it can perhaps support arbitrary fonts, e.g., for user-defined articulations (e.g., various string techniques http://www.smufl.org/version/latest/range/stringTechniques/, or percussion beaters http://www.smufl.org/version/latest/range/beatersPictograms/ etc.) then why not also allow for arbitrary accidentals?


Support for a simple mapping of the notation Janusz proposed to accidentals would allow users to define which pitch + accidental etc. to notate for which OMN pitch. E.g., 

;; Notation of 31-EDO with common accidentals plus quarter tones (alternatively, this temperament can also be notated with double-sharps and double-flats)

;; I left out octaves in notation for simplicity and I rounded pitch deviations to full cent 

;; Explanation of mapping notation 

;; (<OMN notation> <glyph in SMuFL>+ [<pitch nominal if deviating from OMN notation>])


 (c (nil U+E261))    ; Natural

 (c+39 U+E282)         ; quarter-tone sharp

 (c+77 U+E262)         ; sharp

 (cs+16 U+E260 d)      ; flat

; ...



The notation for quarter and sixth tones listed above could be encoded in the same way, but if users could encode their own notation this way, then this system would not be hard-coded to any notation. Using rounded cent values might not be enough for very high precision pitch notation, but is likely enough for practical cases. 







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  • 4 years later...
  • 3 weeks later...

I purchased Opusmodus only thirty minutes ago and need to catch up re. its current microtonal notation ability and potential. (I'm a composer finishing a doctorate in artistic-research musicology with a focus on non-octave scales, e.g. the Bohlen–Pierce system. I've also been teaching a seminar on microtonality for a few years at the Hochschule für Musik und Theater Hamburg.) We use a library for Max/MSP called MaxScore, by Georg Hajdu and Nick Didkovsky. A feature is that the user can select from various notation styles and instantly change the look of the score, depending on what the reader is accustomed to, and the system is open so that users can create their own notation styles. This is perhaps a practical solution for Opusmodus because, in my experience, having composed many microtonal scores, nearly every performer requests a different notation style either created by them or conventionally established in their field or for their particular instrument, e.g. a 19-tone trumpet uses standard accidentals but distinguishes G# from Ab, a 41-tone guitar uses slashes and back-slashes in front of accidentals, and the Hamburg Bohlen–Pierce notation uses no accidentals but a six-line staff. So instead of Opusmodus being responsible for choosing and implementing various microtonal notations, I suggest they somehow keep a customizable library available so the user can map their own set of symbols, as required.

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> artistic-research with a focus on non-octave scales

Could you share some more information on what kind of notation schemes exist for these kind of tunings, and what you would like to do in principle -- independent of Opusmodus, e.g., what you are doing with MaxScore so far? Does there perhaps exist any notation scheme that provides principles suitable for a certain or even wide range of such tunings, or do you use custom/independent notations for each such tuning?


Officially, microtonality support in the Opusmodus notation OMN is limited to quarter tones (e.g., c4+) and eighth notes (e.g., c4.) and explained at the front page (https://opusmodus.com ). (Seemingly, there is not even a separate eighth flat accidental symbol, those pitch deviations depend on combinations with standard accidentals).  


However, according to Janusz Opusmodus additionally supports (or will support) arbitrary microtonal deviations of 12 EDO (or 24/48 EDO?) expressed by cent values to be added as technique, but this is not yet documented. So far, I have not yet delved into the existing microtonality support, but instead I am waiting for this feature (arbitrary cent deviations) to be documented. Then I would add for my own purposes by and by a richer microtonal notation on top of the builtin OMN with custom functions.


For example, I plan to define for myself tuning table support, which would allow to arbitrarily detune (in an octave repeating way and/or freely for each octave) the supported pitches incl. all quarter and eighth tones. For example, 31-EDO (septimal meantone) and various other temperaments can be notated with only the accidentals already available in Opusmodus, but we would need to tune all pitches differently. That could be done simply by a function that maps OMN pitches (incl. their cent value deviations relative the the tuning table) to 12 EDO pitches with cent value deviations. The tricky bit with this approach is that enharmonically distinct pitches (like C# vs Db) could be tuned differently in such tuning table (e.g., these are different pitches in 31 EDO), while many pitch-processing functions in Opusmodus seemingly disregard enharmonic distinctions. So, such Opusmodus functions must then be avoided. 


At a next stage, I might define my own pitch representation from scratch that would support regular temperaments (i.e. incl. just intonation for arbitrary limits, arbitrary equal temperaments, but also things listed at https://en.xen.wiki/w/Tour_of_Regular_Temperaments ) using perhaps accidentals like sagittal notation (which was designed to support a wide range of tunings). For example, I might represent additional accidentals as custom OMN techniques, which could be translated into cent deviations for MIDI playback, but would be included as some ASCII technique text of these accidentals for MusicXML output (which I could then manually translate into proper accidentals with notation software like Dorico).


Now, all of this would not directly cover those non-octave scales you are interested in, but perhaps you can use a similar approach, where you translate some custom OMN notation developed for your purposes into the relevant OMN with cent deviations for playback etc. Perhaps we might even work together on something like this at some stage. 




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