Talk:IFDO
Critique of naming system
Recently, Flora introduced the idea of conceptualizing equal-step tunings in terms of various well-known mathematical means such as the geometric mean, arithmetic mean, and harmonic mean. To be clear, these were not novel tunings she proposed; rather, she introduced yet another way of identifying and reasoning about this existing equal-step family of tunings which have already been studied and named ad nauseum throughout xenharmonic history.
Flora wisely chose not to suggest identifying tunings by the "harmonic mean", due to the established and contradictory meaning of "harmonic" in our musical field. I have pushed back against her use of "arithmetic mean" for similar reasons; Shaahin Mohajeri introduced it to the xen lexicon in 2006 via his Arithmetic Division of the Octave (ADO) tuning (see here: https://yahootuninggroupsultimatebackup.github.io/makemicromusic/topicId_13427.html#13427), where Shaahin used the word as a reference to the mathematical concept of an arithmetic sequence/progression, which is a sequence formed by repeatedly adding a constant. So what Shaahin did was assume the constant was in terms of frequency. Another thing he did was use a word for an open sequence to describe a tuning which is a periodic division of an interval. In 2021, along with Billy Stiltner and Paul Erlich, I developed a naming system for equal-step tunings which handles both periodic divisions and open sequences. In this system, the former are "Equal Divisions", as in the overwhelming popular Equal Divisions of the Octave (EDO); the latter are "Arithmetic Sequences". So we borrowed Shaahin's use of "arithmetic", but repurposed it in a more appropriate way: for sequences, not divisions. And another thing we improved over Shaahin's concept was to treat pitch, not frequency, as the default musical resource; this agrees with the overwhelming popularity of EDOs, which are in fact equal pitch divisions of the octave. In short, we extended the established basic ideas of equal divisions and arithmetic sequences of pitch to the two other fundamental musical resources of frequency and length. We felt comfortable with the fact Shaahin's ADO made different assumptions than our system, because Shaahin was clearly not attempting to develop a full, internally-consistent system like we were (one need only look to his other tuning, EDL, which is neither consistent with ADO nor EDO). However, Flora does indeed seem to be attempting to develop a rivalling full internally-consistent system to the one we developed already. And my concern with Flora's proposal is that it uses the word "arithmetic" to describe tuning systems in this family but in a different sense than "arithmetic" has been being used historically. In the case of Shaahin's ADO, the two meanings happen to coincide. But when she extends the idea to "inverse-arithmetic mean", as a substitute for the even more unusable "harmonic mean", her meaning of "arithmetic" comes into conflict with our system's meaning. And we can see another negative consequence of Flora's proposal in that a page cropped up for "Arithmetic MOS", where CompactStar has gotten comfortable with treating "arithmetic" as synonymous with "of frequency" (see: https://en.xen.wiki/w/Talk:Arithmetic_MOS_scale).
Recently on the Discord server, CompactStar took a step toward generalizing Flora's concept of using mathematical means to describe equal-step tunings when he proposed the concept of an RD, where an interval is divided into equal parts according to the root mean square (RMS). Flora took the next step, showing that any power mean (p-mean) could be used in this way to define an equal-step tuning. The geometric mean is p = 0, the arithmetic mean is p = 1, the harmonic mean is p = -1, and the quadratic mean (same as RMS) is p = 2. So p = 0 corresponds with ED's/AS's of pitch, p = 1 corresponds with ED's/AS's of frequency, and p = -1 corresponds with ED's/AS's of length (of strings, resonating chambers, etc.). But this p = 2 corresponds with a new type of musical resource, on the other side of frequency yet from pitch. This is a novel tuning system and worth studying.
In my opinion, originally, Flora's idea of using mathematical means to describe these popular equal-step tunings was unnecessarily obscure, in comparison with the system I helped design a couple years ago which is much more directly accessible to practicing musicians, through its use of the three actual physical/psychoacoustic resources of frequency, pitch, and length. However, this new idea which leverages mathematical power means to explore new tunings, and describe this family of tunings along a smooth continuum — e.g. there's potential for tunings defined somewhere between equal divisions of frequency and pitch like 1/2MD, or between pitch and length e.g. (-1/2)MD — finally justifies Flora's concept. And so, if mathematical means are to be used in this way, then I would strongly prefer use of the power mean's power in the name, rather than "arithmetic" and the new coinage "inverse-arithmetic" (which is a bit strained anyway) as in AD and ID. I hope that in consideration of all of this, Flora and CompactStar will consider switching to refer to AD's as 1MD's and ID's as (-1)MD's. --Cmloegcmluin (talk) 19:06, 27 March 2023 (UTC)
- A few points to add:
- I disagree I used the term arithmetic differently from the sense of Shaahin Mohajeri (2006), altho I admit it differs from Billy Stiltner et al (2021). As one way to see it, frequency is implied as the default measurement – but more appropriately the measurement is determined by the specific word in the place. Using arithmetic for anything other than frequency would bug me, just as harmonic mean bugs us since its defined in terms of length.
- I suppose the letter M stands for mean, such that the full name is m-mean division? Not that I disapprove of a change of the page title, at this point I hesitate to say it's the right word choice. Specifically, whenever there's a mean there's also a sequence, including the fractional-order ones, since the mean is really an element of a special case of a sequence. So this concept is based on not exactly means but sequences. I'd consider the full name is order-m division or order-m sequence division.
- I believe m = 1 and m = -1 deserve verbal names (even tho they're disputed) in addition to the formal one, just as m = 0 does.