Research that Pianoteq users can help with

Yes indeed, very interesting. In considering the effects of different temperaments we would surely need to factor in other parameters available in Pianoteq: unison width, stretching (I notice that Davey Jones' examples use 1.01 Harmonic), and string tension vs. full rebuild.

I'm nowhere near specialist enough on the subject of tuning (let alone the maze of Bach temperaments over the centuries) to either properly or meaningfully defend or critique Lehman.  I would just offer though, as an educator & performer, the admittedly poor & entirely subjective observation that I find his scheme to be a pleasing temperament on the harpsichord and clavichord for Bach... and Lehman himself seems to be arguing for a rather subjective basis to a temperament scheme derived from the WTC (using the squiggle as mnemonic) which I think does have a fair amount of practical appeal & value for the musician at home with his/her clavichord:

but at least one commentator (Byrnak, 2011) has found that his scheme shares similarities with Werckmeister's temperaments:

so perhaps Lehman isn't entirely out in a little silo all on his own...?

of course, it would appear that there's plenty to critique in his method, as in >here< for instance, but if nothing else (aside from its "if it sounds good, it is good" merits) Lehman's Bach scheme(s) certainly has/have prompted a great deal of interesting debate!  (Concerning which debate, here's a very nice survey of the various historic temperaments to date that concludes, under "dissonance theory", that Kirnberger II is actually probably the closest thing to a "Bach temperament" out there ==> https://www.recercat.cat/bitstream/hand...sequence=1).

In any event, it's all a delightful maze to get lost in and constantly compounds one's sense of total awe at Bach's celestially divine gifts to the world! 

cheers,
dj

These are all lovely illustrations, and I'd be the last person to argue against using some sort of well-temperament for Chopin!  But, as far as Chopin goes, I guess I'm scratching my head a little (again, "if it sounds good, it is good" considerations aside... 'liberty hall here & I suppose one can always do whatever one wants) as to why Kellner particularly?  If one's playing Chopin then why not go with a well-temperament that's more historically appropriate (ie Jousse)?   And of course, Kellner is not above critique (especially as a Bach temperament)... I'll lean on Lehman a little more here (geez, I'm starting to sound like a fanboy... I'm really not, I assure you! ) and some of the observations on Kellner he made in the Bach-Cantatas list post back in 2004:

whereas, concerning Jousse I'll turn to Jorgensen who, in the section on "The Practice of Well Temperament in 1842 through 1848" (specifically de Morgan's scheme), writes:

(from "the bible": Jorgensen, Tuning, p. 456.)

now, de Morgan is not Kellner, but to my thinking at least, being concerned with historically-informed performance, Jousse would certainly seem to offer the advantage of being a historically appropriate well temperament, and has the added benefit of sounding awfully nice to boot...

Happily. 

Here's a piano-roll recording of Paderewski (source midi file >here<) rendered in Pianoteq using the 1835 Pleyel (set at the Pleyel factory's diapason of 1836: a¹=446) in

a) Jousse well temperament (1832)

b) Jousse quasi-equal temperament (1840)

c) Kellner's "Bach" temperament (Pianoteq's "well temperment" setting)

and d) modern ET

the wonder of Pianoteq! being able to bring these great old pianists back and have them play Chopin, if not exactly HIP, at least on historically appropriate instruments & tunings! 

cheers,
dj

Ah yes, DJ, I can hear why you favour Jousse for Chopin - very satisfying. Kellner less convincing, to my ears (sounded a bit sour).
On the subject of piano roll midi files, I have found that quite a few of them seem to sound better with the 'slow keyboard' velocity curve. Has that been your experience?

Jake: I presume that DJ probably just got the Jousse tunings from the scale archive: http://www.huygens-fokker.org/scala/dow...tml#scales . Here's the contents list (it's huge!): http://www.huygens-fokker.org/docs/scalesdir.txt

It's not hijacking at all - these are important and helpful experiments although I hope that people will do the resonance experiments that I started with.

The Jousse quasi equal sounded much better to me than the Jousse well temperament, although I've got to do some more research on Jousse. I tried tuning one of his temperaments on a harpschord and it was basically three sets of four perfect fifths or four sets of three - don't remember - so putting Pythagorean thirds everywhere and not sounding nice at all.

The problem with Kellner is that over years with real instruments is that I've experienced very very much better results. The reason is a fundamental problem with running Midi files through computer simulations . .  . . It seems objective but the reality is that it isn't. The performer, if a good performer, reacts to the sound and works with the sound of the instrument and its tuning. So a perfomer using Kellner will perform and emphasise differently to a performer with ET. This may well be why the Quasi ET sounds so good here because the performer was performing on an ET instrument.

So it's complicated.

This is the reason for looking at things like resonance and asking ourselves how we exploit what is presented to us to best effect in performance.

Best wishes

David P

DJ:  "... then perhaps the player and the work being performed are the software, while the instrument and it's diapason/temperament are the hardware?  certainly there's a dynamic between the two..."

I am certain in the case of a digital piano the above requires accurate proximity sensing for outputting data from the keys. The MIDI9 optical sensing system qualifies.

Lanny

It's been my experience most poignantly with a Kawai KG-8C but with Steinways and other pianos too, and audible on the Pianoteq simulations, that a well tuned unequal temperament can improve the tone of the piano, making it sound more whole, coherent, musical, warm, less strident so I've started to look at the mathematics of why that might be. The other effect is to reduce confusion with sustaining pedal effects and resonance.

I've put on the PTG forum a spreadsheet upon which others no doubt may improve but which might be a start of some helpful analysis. No other forum seems to enable upload of such a spreadsheet :-(

Hopefully it will be accessible from here - https://my.ptg.org/HigherLogic/System/D...506fc8721e

It sets out the frequencies of fundamentals and harmonics up to the 11th in Equal, Meantone, Kellner and Kirnberger III temperaments.

The 11th harmonic I've ignored, collapsing the columns as it's effectively 1/4 tone off in all temperaments and irrelevant to resonance.

The 9th harmonic I've tended to colour red as it adds to the metallic sound of the instrument.

Generally, coloured green are the frequencies that are on the harmonic within one beat.

Coloured blue are frequencies between one and five beats, which add to the glistening nature of the piano sound and can be near enough to resonate.

Coloured orange are frequencies between 5 and 10 beats which may add colour but certainly glisten or add to disturbance of sound and may or may not resonate.

We can see the way in which equal temperament gives smooth progression of everything adding glistening and providing off-resonance, the collection of such near and universal vibrational consistencies giving the maximum confusion of sustain sound.

When we go to Meantone we see how the 3rd and 6th harmonics resonate and resonances of other harmonics are fragmented.

Kellner and Kirnberger less so, and particularly we see less universal capability of 9th harmonics to be resonating, and also different responses in different keys, particularly showing different tone colours or tonalities resulting in different keys where in some the 9th harmonic is spot on and capable of resonance, whilst in others supplying only supporting glistening or not at all. The home keys, C F Bb G tend to excite more the concordant harmonics whilst the more strident keys such as C# the more strident harmonics.

It's far from a perfect illustration and takes no stretching into account but perhaps it's a start.

Perhaps the assumption that the lack of taking stretching into account might be valid as the idea of taking inharmoncity into account is to align harmonics better. So neglecting that and looking at the system as if the harmonic relationship to the scale can be considered as undisturbed is not invalid.

Perhaps others might have better ways of illustrating what's capable of happening and what those of us who tune unequally can hear, or have other thoughts about how close harmonics and scale pitches have to be to resonate strongly stably or provide wobble to the resonating sound. Hopefully it's a start and not cause to be murdered by the staunch equaltemperamentists.

Perhaps developers at MODARTT might already have looked at such frequency analysis . . . ?

Best wishes

David P