again, thank you. thirds and minor sevenths are the devil.
I agree; after reading through, it seems more like âweâve done all this painstaking research, so we think weâve got it closer than beforeâ. But no one really knows, any more than they would know exactly how latin was pronounced back in the day. We can make educated guesses, even very educated guesses, but aural traditions are lost to time.
And this being academia, I have no doubt that other musicologists are at this moment muttering âwhat frauds, Iâll prove them wrong!â
Even in more recent, better documented musical periods like classical and baroque, very well researched groups of period performance practice can come to very different conclusions.
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Do you mean minor thirds or major thirds?
Aieeee! (/CatapultedIntoInfinity)
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In various kinds of mean tone, the comma was distributed to certain fifths, in amounts anywhere from a quarter comma to a 6th comma, in order to make musica ficta usable. That means that certain fifths would be relatively pure, with increasing impurity as you moved away from the home key, until you reached wolf fifths in the notes that were never used at the same time as an interval (e.g., G⯠- Eâ).
Tuning is a vexèd matter; it has always been a matter of compromise. Pythagorean just intonation slights thirds and seconds, given that these are formed by stacked fifths which donât add up to the Ptolemaic proportions, i.e., they arenât pure. They are pretty far from pure in equal temperament as well. Thatâs why I say that, if you are hearing the differences between tunings, you are likely noting the differences between thirds and seconds.
Edit: an example from my own work, using Dechalesâs 1/5-comma meantone. Itâs in B Locrian (contrasted by the relative Mixolydian, and later, the tonic Mixolydian), and ends on a tierce de Picardie. Listen to how the D⯠sticks out in the final cadence.
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are we talking mixolydian scales?
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I thought the augmented fourth was the Diabolis in musica
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Definitely, after your actual âexact octaveâ, your âvery nearly perfect fifthâ is going to be the next thing to aim for. Then the fifth of the fifth (yer major 2nd) is going to be slightly more wobbly, etc. By the time youâve got your full diatonic/heptatonic scale - including the fifth of the fifth of the fifth of the fifth of the fifth of the fifth, (yer tritone), youâre going to be less and less successful at maintaining accurate, transposable, rationalisations. No wonder those guys in the east stopped at pentatonics.
Goodness. Weâve already invoked Pythagoras and Ptolemy. Who invited Plato?
By which I mean to say, who defines âpureâ or âidealâ? Now, as for the nineteen note scales âŚ
(Iâll get my coat).
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Ptolemaic just intonation works with the simple ratios, there being more than one form of just intonation, eh? It has its own problems. 
In some ways, music is only lightly tied to the physics of sound. There are always compromises brought on by competing requirements (especially since the advent of polyphony).
Love it! Its gorgeous âastringencyâ is due to its modulation into a different key, so the 'misâtuning is bound to become more obvious if you do that. The guys in the topical piece stay in the same key, so maybe your contention is that it would not have âroomâ to sound all that odd (to us), even if it had been played in the tuning of the time? I might consider conceding that much, I suppose.
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Thatâs exactly what is happening. I can hear some movement from what weâre used to in the seconds and thirds, but bear in mind one other thing: singers (and string players) also tend to âshadeâ their intervals in the direction of movement, and they probably always have.
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Nggghhhh. Difficult to âproveâ. Wouldnât that also imply a âdestinationâ, which has as much problems as some ideal, pure, ultimate tuning? Like a direction for evolution?
But the system is warning me that Iâm hogging the topic so Iâm stopping.
Not at all difficult: itâs a melodic tendency that is particularly noticeable in leading tones: lower leading tones tend to be raised in practice, and upper leading tones tend to be flattened a bit. The destination, of course, is the following note.
The reason isnât âideal tuningâ, because these tendencies diverge from any ideal; it has to do with adjustment according to melodic function.
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Saw Benjamin Bagby perform Beowulf (or a third of it) at Cal Performances in Berkeley a few years ago, which was a treat.
I havenât a fucking clue whatâs going on above in this thread. Probably terrorism.
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System be buggered!
Sometimes a conversation among a few experts is exactly what the topic requires.
Iâm a very interested and entertained spectator in this thread; I donât have a horse in this race. But it pleases me mightily to observe experts disagreeing in so civil a fashion 
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Yup. Weâre going to break out the shawms, bombards, racketts and serpents, and the world will find out the true meaning of terrorâŚ
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(my bracketed precis)
Oh no, I wasnât saying that. I just mean that - like the idea of some âperfect platonicâ scale, independent of instrumentality and physics - the shading of intervals in the direction of movement (which is how you originally put it) would have had to imply an actual direction of movement known by the performer. The tonic/resolution might be âmiles awayâ at that point. Which seemed to me as strange as the idea that an ancient eohippus was minded to evolve into a modern horse. (That the modern horse is the perfect platonic horse is a quite distinct âdifficultyâ).
Obviously with voice-leading where the target is the very next note, thatâs not an issue.
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As a musician, I say you are spot on.
Iâm convinced that the key to rupturing space-time involves the discrepancies between mathematical ratios and the physical as manifested in sound waves and intonation. At least thatâs what I tell myself to stay sane. âyeah, Iâm not playing out of tune, itâs the laws of the universe fucking with meâ.
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Yeah, but the range can be longer: most melodies imply long-range goals. If I add leaps and changes of register, there is usually an implication that note left hanging before the leap has a deferred destination. A string player might well shade the note according to the implied function. Most of those functions are set by convention anyway, so theyâll be in the playersâ fingers.
Itâs actually hard to combat when youâre singing: if you know youâre going down the scale to a particular note, you have to consciously fight going down too quickly. Hard to explain (for meâŚno formal training). Basically, you have to think of lifting up each note a little higher than feels natural so they donât start going flat in a run to that bottom note.
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Especially in a monotonically increasing/decreasing scale runs up/down to the tonic? Riiight. Thatâs probably why a composer, knowing that performers have been trained thuswise, must force the issue by writing explicit instructions not to do such anticipations. Probably harder (but an interesting challenge) for the performer, but the composerâs the boss!
(If thatâs what the composer wants - they may be happy enough to go with the norm.)
