. It would have been strange if this thread did not contain at least one example of a feudal comma. You have given several.
I'd like to clarify a few things that have come up in email and facebook conversations:
It is so far only a conjecture
, I don't have a proof, that all noble numbers are the quotient of two feudal primes or units
. But it is certainly the case that not all such quotients are noble.
The conjecture goes further to say that the primes involved are only, and all, non-wooden primes, which are the factors of the ordinary primes that split or ramify in the feudal domain, i.e. those primes named with "f" and "F" above. And that except in the case of the noble ϕ, which requires the unit 1, the units involved are only ϕ⁻¹ and ϕ, which can be called f1 and F1. And that except in the case of the nobles ϕ and ϕ², the numerator is never a unit and the the quotient always consist of a small f divided by a small f, or a big F divided by a big F, where f5 can also be written as F5 since they are both equal to √5, and the units f1 and F1 are included as potential denominators. And if this all be true, we know there is still a further restriction on which of these quotients are noble, given by the "unimodular" requirement, whereby, when the wooden and golden parts of the numerator and denominator are treated as a 2×2 matrix, its determinant must be ±1.
(or anti-JI) dyads
(dyads approximating simple noble number frequency ratios) should not be expected to be tuned by ear
by the slowing of beats or "locking in" of partials in the same way that JI dyads or CT (combination tone) triads can. Their whole shtick is in maximally avoiding
coinciding harmonics. So there should be no advantage in tuning them precisely, provided they don't stray so far that they cease to be approximate simple nobles and become approximate simple rationals instead. So there should be no need to give decimal places of cents for nobles, but they probably need to be tuned within say ±5 ¢.
However, Mike Battaglia responds:
You claim in this that noble numbers are not sensitive to mistuning in the sense that there are no partials to try to get to cohere, but I don't really agree with that. In particular the thing you are now calling n5/4 seems to be pretty fragile. There is a real sweet spot somewhere in the 330-340 cent range where things are maximally "stanky," let's call it, and then things kind of fall off pretty steeply, particularly as you tune sharp.
I guess we should expect that, as with JI dyads, the more complex an MI dyad is, the more accurately it will need to be tuned, if it is to retain its identity at all. But perhaps there is something happening there that is not predicted by our simple model of coinciding partials.
I contend that humanity has always used MI with JI
. It's only RTT (regular temperament theory) that hasn't had a way to deal with MI until now. And frankly it's not that important, because near-enough approximations of MI usually turn up anyway.
People used to argue over whether the minor seventh interval in the dominant seventh chord in 12-edo was an approximation of 4:7 or 5:9 or 9:16. The Noble Mediant paper says it's none of those. Its purpose is instability. Notice that 9:16 is the mediant of 4:7 and 5:9. I claim that the ideal tuning of that minor seventh is the limit as you keep taking mediants 4:7 5:9 9:16 14:25 23:41 .... Mind you, because of the lack of a requirement for accurate tuning of mercifuls, and the lack of audible justness once rationals become too complex, either of those last two rationals would be perfectly good tunings of that noble (which is f31/f11 = n7/4 ≈ 1002 ¢). The "n7/4" notation is described in the following post
If you ever see ratios involving prime numbers like 17, 19 and above, as explanations for the tuning of some interval, it's worth considering whether a better explanation might be that it is approximating a simple noble number, for the purpose of maximally non-coinciding harmonics. Another possibility is that it may have nothing to do with harmonics, coinciding or otherwise, neither JI nor MI. It may instead have to do with combination tone effects. The give-away for CT harmony is that the effect is easier to hear with sine waves than with harmonic timbres. And even easier when distortion, such as asymmetrical soft clipping, is applied to the sine waves after they have been combined.
4. MI is not CT
. The golden ratio has a reputation for producing interesting combination tone effects. So you could be forgiven for assuming that all noble numbers are capable of producing similar combination tone effects, but perhaps less strongly as they get more complex. But this is not the case, except in in the sense that any interval is capable of producing combination tone effects
. It only needs to be combined with the right, precisely-defined, third note, or rather one of a very few such notes.
The golden ratio (833.1 ¢) is only special in relation to CT harmony because the right third note for it
happens to be obtained by stacking another
golden ratio interval above or below it.
Other "celebrity irrationals" that have this CT-self-stacking property are the silver ratio and to a lesser extent the bronze ratio. These are in general called metallic ratios
. These are very different from the noble numbers of MI. The golden ratio ϕ just happens to be a member of both clubs.