Well, let’s say pianos are very difficult thing to make but Pianoteq makes good things very easy. I also always like it when there is some curiosity on the real vs physical model synthesis aspect. Personally, I’m not fully aware of all the nuances involved in any case, as I’m far from being a piano technician, nor am I really that math-prone guy, to say the least. But, based on my musical experience, I thought I could share my understanding of the matter and confidently say something useful as well, because I’ve just read so much interesting stuff in this thread, that I’m definitely motivated to give another way of thinking about what do we do with sound, where do we wish to go with that particular scale or not.
I will also try to say something useful independently for anyone who might be reading, but, I must tell you in advance that it will be a bit technical and a very long (maybe boring) post, and it has to be so in a sense due to the fact that one important argument which Joseph brought in here (if not the most relevant) is scale design! Lol. And I’m curious to know more about his idea as, from what I’ve understood with my limited reasoning, his idea of design sounds a bit cloudy to me (not that I always know what I'm exactly doing whenever I try new scales...just saying)
Hence, I will write my post with just background information covering mainly three areas: 1) some useful linked study material of pertinence, 2) not-so-sophisticated observations on "systems" in general and in relation to the "how many pianos do we need" question, so to say, and 3) "deep" analysis of the Maj 7 chord example from Joseph [3,7,10,2], plus a couple of suggestions on alternative tools for scale design best working with Pianoteq (as far as I know)
I think that, as a general rule, if the instrument hosts a string-system by which, along with each own string-design (and the whole of the other means of transmission as well), spectra with prevalent “pure harmonics” are rendered (like, say, a violin, that is, also some instrument where there are less metallic components as possible, etc), then, the tuning system which would better fit the “timbre”, provided that the musician aims at getting a classical “clean” timbre (and not exciting the string at nodes towards the bridge like in contemporary music, for example), the tuning system — I was saying — will be one of those that humanity has always considered the most valuable in terms of “good consonance vs bad dissonance” interval distribution.
These tuning systems, which must result in a balance between “harmony” and “equability”, and also present favourable cardinality per scale, have been revealed indipendently (so to say), as Milena Bozhikova has noted in his book “Music between Ontology and Ideology”, both by mathematical-logical thinking and musical practice, in the course of the centuries. Mike Battaglia, who I think is also the curator of the Xenharmonic Wiki (and an excellent piano player as well as a long-time Pianoteq user whose knowledge on this topic is incredibly wide), employed the Riemann zeta function in order to find those “n-equal division of the octave” (EDOs) that must be the best (if not the only) candidates.
Thing is that in order to conceive such an octave-system which is based on logarithms, you are forced by nature to put some priority onto only one of the most consonant interval that you have in mind, and not the other, whereby, if you take the classical major 4:5:6 triad as the most consonant example provided by nature, like Indians do — and of which, as we all know, the characteristic intervals are the “perfect fifth” (3/2) and the “pure major third” (5/4) — and you wish to dispose of that perfect triad no matter what (like, it has to be available in a more diffuse chromatic setup, and not just a diatonic scale), then, you must decide which must be the only consonant interval at needing more representation.
And so, the octave-logarithmic tuning systems which end up giving the utmost priviledge to the “perfect” fifth, apart from giving it to the perfect octave (to which they only really do in fact), they basically fall into the Zeta EDO list (more strict EDOs in this list go like 2, 5, 7, 12, 19, 31, 53, 270, etc). But, then, further distinctions are to be made, since that 31-tone system, for example, gives priviledge to “pure” thirds, rather than “perfect” fifths, as meantone systems tend to do, representing partial n. 5 more than partial n. 3 (and so goes that story and many other ones leading to all sorts of speculations about the golden section since the root of 5 is at play) and some of those EDOs might also hint at “exotic” things like 7-limit harmony, 11-limit, 13-limit or more consonances beyond such prime-limits (and yes, all are consonances by definition, otherwise surrogates of the same intervals wouldn’t systematically appear in the Zeta list)
And then to cut things short, because the 12-tone system is also a meantone system, namely, the first one that comes with the least de-tuned fifths in that category, and which then sorts intervals in the chromatic scale in such a way that “consonant” major triads somewhat resembling to 4:5:6 are always available everywhere, it has to be the most played or the most playable. And so it is obvious that almost the entire global world has ended up with adopting that type of compromised solution for matching the tuning with the timbre of classical instruments, that even the Indians did not resist, but imported the western harmonium and contaminated their singers, who now are trained with that system for the most part.
On the contrary, apart from the timbre of the instrument towards which the tuning will be destined, if you want your system to be made of, like, “bad consonance vs good dissonance”, while also still not touching the octave purity, but put as much as “dissonant intervals” in its span as possible — not to hurt anyone’s ears, but just have fun — then, a good way to start (in my opinion) is to look at EDOs which split the “fifth”, or the “fourth”, into two “fifths” and/or two “fourths”, in a “major or minor” style…example that comes to my mind now being 23-EDO…but there are infinite other alternatives (and if you are looking for more clarifications in this sector as well, explore free pdf book “Alternative Tunings: Theory, Notation and practice” from Kite Giedraitis)
To Joseph: I repeat myself, I’m not that math-prone guy, and so I can’t give you nice mathematical formulas tout-court but, eventually, as far as I can see in this respect, I say that one direct consequence of those EDOs which allow you to achieve what I’ve just said is, to my ears, a more evident scale distribution of so-called “interseptimal intervals”, to the detriment of the “non-interseptimal intervals” (such as 3/2)
And the inherent “inharmonicity” of acoustic pianos (which is “une grosse affaire” with Pianoteq as well) will, like, “poke” that person with (maybe) a more curious ear in such a way, that if that person was met with the possibility of putting hands onto a grand piano and apply some tuning without any exterior constriction (say, no one is obliged to make that piano “suitable” the moment there is no need to play sonata per pianoforte and all that wonderful stuff), then, that person would most probably prefer to leave the tuning lever after finding a nice rough “supermajor third”, after putting more tension and going “beyond” the pythagorean third of 81/64, rather loosen the string in the flat direction.
Just saying, this is not proven, I’m only guessing here. The demand of finding a region of “metastable” intervals and/or intervals convergent with acoustic phi and those siblings, well, maybe it is incremented by the feedback provided to the ear by the inharmonicity of the string-system in general, and more maybe by the complexity of pianos. I don’t know if this can be called entropy.
This whole western degenerated idea now also very well applies onto the question of “how to deal with whatever it is that is emerging in your face (or rather going althrough your body) when it comes to the tuning affecting the timbre and viceversa”, that this idea has been recenty baptized as “anti-just intonation”, “merciful intonation”, “noble intonation” etc. It is so funny. To be honest I think there is something not always noble, like when you patent “your” tuning system. How can you even think for one second that it is a good idea if you really know what you are doing? It’s not, like, selling an instrument coming with some fancy setting. Is it music in itself? I don’t think so, but in my mind, I would never do it. It is just too much, makes no sense, totally illegal. But whatever…“the west is the best”, Jim said. And to think that all this is not considering any “octave-strech” as of yet. But, don’t worry. I won’t discuss that.
Back to Joseph’s request, what certainly will turn more useful than my comment and offer a mathematical understanding of the possible relationships between "tuning" and "timbre" to the reader is, without any doubt, the research made by William Sethares. He published peer-reviewed I think, either free articles or books under Springer-Verlag editor, specifically for this matter, "Tuning, Timbre, Spectrum, Scale" (2005, 2nd edition). For an interesting alternative application of the theory he covers, just watch the 5-minutes video named "Inharmonic strings and the hyperpiano" on YouTube. Also his website is what I exstimate as a website with a sober interface and is presenting, with enough details, the contents of his research (you know, just in case you can't afford the book...)
Now, is this "hyperpiano" timbre the complete opposite of the timbre you want to achieve? In Pianoteq, a similar sound like that of the hyperpiano, or vaguely bell-like, as you've probably guessed, could result by abusing the "String tension" mode, or by setting the "string length" parameter at minimum, whereas "Full rebuild" should maintain the original timbre of the instrument. This means that, if you, instead, actually need the timbre to sound like something "stable" and as close as possible to the tuning you want to match, while still sounding like a traditional piano, you are put in front of a paradox.
The paradox is that you need to reduce "inharmonicity" down to zero. Which, is literally impossible on real systems, mainly because of string stiffness, and above all on physical pianos, as it would require infinitely long strings (and a huge soundboard). There are, very rarely, constructors which try to get remotely close to something like that, like Luigi Borgato & his Grand Prix 333.
I must be honest and repeat that I guess, that the factors which must be taken into consideration in order to really take care of the intonation inasmuch as possible on a real grand piano, they are largely hidden to me. But in Italy, when we talk about the intonation ("intonazione") strictly regarding acoustic pianos, we refer to a completely different and intermediary phase of setting the instrument — perhaps, the most tough phase, mostly based on, how can I say, hammer care. This comes just before, that maybe even asserting that it regards "unison" is a bit of an overstatement, because it is a phase in which the tuning lever is simply not an option. It is about sound.
The third and last phase — when you tune, temper, stretch, etc — that is not "intonazione", but "accordatura", which merely is, as a piano tuning master said, "only the icing on the cake". I heard from this same master that Borgato model, apart from mounting four strings for high-unisons compensation (not to talk about the soundboard), deals with intonation in all kinds of way, by also mounting hammer heads which have a pyramid-like rather than "oval/rounded" shape. Or that the legendary pianist Michelangeli made his trusted Fabbrini (personal piano builder and tuner technician) apply hazardous settings of "intonazione", since the hammers were to rest dangerously close (like less than 1 mm) to the strings after being struck (for reasons which I won't discuss, sorry, miss the lexicon)
Qexl, to whom I'm grateful for the much passionate and prompt responses, gave plenty of indications and made good points, most of which I agree on. I own Pianoteq Pro Studio but unfortunately I'm far from knowing the in-and-outs of it. I also don't think these very particular aspects of the hammer side for example, that I've just hinted at (and which indeed affect the timbre), are implemented for regular users as of yet. But things alike must be somewhere in Moddart equations.
Hard for me to say what you need to change to get to the timbre you desire. My suggestion is (if you own the Studio version) to access the "Note Edit" window, by double-clicking on "Spectrum profile" (under "Hammer hardness", in the middle section "Voicing", main panel). Then, check out the per-note values of those parameters which you think they might be responsible of badly affecting both the tuning and the timbre as you intended to be (whatever you mean, you know). In the Studio version, a big list of parameters will appear as you expand on the left. Just to name a few of interest here: "Detune", "Unison Width", "Unison Balance", "Stretch Points"...but also "Spectrum Profile" (per-note), "String Length", "Sympathetic Resonance", of course, etc.
I have the feeling that Pianoteq's "Stretch points" parameters will drive you crazy. Many many interesting options in there for evaluating the tuning/timbre relationship. I repeat, I do not have much experience on the practical side of acoustic piano tuning, but if you want to go into theory, inharmonicity and all that stuff, I can tell you first that...well...I thought that the "Circular Harmonic System" (C.Ha.S.) by Alfredo Capurso, a proposal of — I'm simplyfing — slightly tempering the octaves too, was, I mean, reasonable enough to be at least taken into consideration. If I remember, he also claimed that the strings-system seems, like, to fit more naturally (like the instrument requires less overall tension or something), but you know how it goes with this stuff, no one really cares so much...just play...sounds good.
And then, I don't know how, but I ended up on this "Psycho-Acoustic Temperament Hierarchy (PATH)" page by Ben Woolley, I think his name is, and if you go and read about his system [url: spatiotemporal.io/clavitone.html], well, now that is quite something also so extreme that must be taken into consideration. As a side note, this guy is one of the very, very few people who acknowledges that the "53-comma" (the difference between 53 iterations of 3/2, "perfect fifths", and 31 iterations of 2/1, "perfect octaves") had already been calculated, with really good precision, by a Chinese man, like, two thousands years ago, AND calls that interval in his honor, the Fang's comma.
Whereas, among westerners and their territory, where everything has to be centered on the logarithmic division of the octave, the same interval still bears the name of prominent logarithmic users of the Illuministic era, and so it is either known as the artificial comma of Mercator (from Nicholaus "Mercator" Kauffmann) or, among the Turkish people, as the Holderian (sometimes, Holdrian) comma, from William Holder. However, this 53-notion goes on-and-off all throughout the history, problem being providing the most exact *mathematical* calculation, in order to obtain as much as consonant intervals within one octave as possible. And so, during more or less the same Illuministic era, also Pier Francesco Valentini knew about that, Jean Gallé knew about that, Isaac Newton knew about that, etc.
But I will talk a little about designing scales (of maybe interest from yours) in the next part. As a general rule in Pianoteq, start with a preset which sounds the most decent already, and only after that, tweak with those per-note parameters (if your version allows you to do so) with both "Full rebuild" mode engaged and your custom tuning, until you are satisfied with the result. Forget about math-precision, you will always need your ear for sound-check on piano physical modeling. Same principle goes for real piano.
