Here is a video I made showing the actual measured beat rates.
(Note: Test notes below F3 not tuned to temperament)

https://youtu.be/2vXXd-XUquA

Well done! You can't get much better than that!

OK, so the next step Hakki is to spread the temperament out of the temperament octave, while still keeping its basic character.

For me, this has worked well until the last treble octave or so, where, if you choose, as I do, to use 2:1 octaves rather than pure 12ths (which is what I use, in general, for the rest of the piano) or something else, the 2:1 octaves destroy any temperament, but sound good.

Really? I would think expanding a temperament with P12s (which I also do) would move the temperament around the circle of fifths each 12th, while tuning with octaves would retain the temperament.

In the case of my piano, the iH changes over the compass seem to favour holding the temperaments. I haven't looked at the math to show me why, though I know there are some iH scale designs that allow numerous partials to align. I don't think my BB is one of them.

The 2:1 octaves in the C6/C7 to C7/C8 range really destroy any temperament, and yet my ears still hear all the normal interval types. The high treble intervals themselves, and not the beating (inaudible) give me the sense of temperament, whereas, to my ears, the beating gives the sense of the temperament in the middle of the piano.

Anyway, given iH, I like pure 12ths tuning.

Thanks but the credit goes to my ETD.

It has good ears!

Hmmm, they say the first taste of an ice cream cone determines what you think the rest of the cone tastes like. The idea is that your tongue becomes numb due to the cold and doesn't taste very well after that, so your mind relies on memory. I dunno, maybe, it might explain why I don't care for neapolitan - conflicting sensations.

Anyhoo, I have to consider that your memory of the temperament in the middle of the piano might affect how you hear the temperament at the ends. There really isn't much math to consider when expanding the temperament with twelfths, just a concept. Like if E3 is, say, 4 cents flat of ET and A3 is 2 cents flat, E5 will be only 2 cents flat, not 4 cents if you expand with pure 12ths.

Would it be an interesting idea to construct temperaments separately to suit specific compositions and calculate the beats and frequencies just for a one off recording of that piece ? It would keep musicologists and tuners busy . Or is that a bit far out ? The technology here would be ideal.

I retuned some classical guitars to uneven temperaments last year . They have wiggly frets which are glued on with Bostik . My favourite is Kirnberger III which seems much more resonant .
If the octaves on your piano are stretched the physical resonance would start to decline. Do you think so ?
A separate idea would be a violin with wiggly frets to test key colours which would not be possible with standard violins . Vibrato would be possible but not quite so much portamento . That might work though as Julian Bream can do it .

I think this is absolutely the case. We don't like and composers don't write M3s in the A0-D1 range because they sound grungy. Even a C1/G1 interval is not all that pleasant. I would argue that the temperament, no matter of what type, doesn't matter all that much at the extremes. Octaves do.

Just a quick picture of the M3s on my piano in ET. you can see that from about C2-C6, using P12ths produces a nice progression. Stretching the bass for better resonance with the midrange and using 2:1 octaves for volume and clarity in the high treble throws the M3s way out.

I should add that the order of notes on the graph within each octave is in a cricle of fifths - C1, G1, D1, A1, E1...

I saw an article where the player tuned his own piano and absent mindedly ended up with part Young and part Werckmeister . He said he quite liked it .

Greetings,
Since we don't hear much in the way of beats in the 6th octave and up, differences due to temperament would certainly be hard to discern. However, if the harmony an octave or two below were to be more (or less) consonant, our perception of what that third up there sounded like might change. Much like the background of a photo can really change how you sense the overall effect. It isn't a black and white division, either, but, rather, dependent on how intervals are used. The clarity of the minor third E-G, in the fifth octave is certainly different in a conventional, medium contrast, unequal temperament. Same is true of the C-E. In, in a WT, we play one, then the other, the lack of tempering begins to change how this sounds....

What might be more interesting is that the highly tempered thirds WILL sound more consonant lower in the scale than the more pure ones. This has to do with critical band, I think, but on a Young temperament, as one goes lower and lower, the F# third (21 cents), will still sound like a usable third when the F-A next to it has become a cloudy mess.
Regards,

I basically agree with your thoughts. I certainly don't hear beats unless the partial freqs are close together, as they are in a M3 in the deep bass, or a partial match from a D1 and F#4, or, of course, most of the intervals in the middle of the piano.

Outside that though, the fundamental width of an interval - a m2 or M2 or m3 - (not its upper partials), is imprinted on my memory, so that an ET M3 - C7/E7 or D7/F#7 is still clearly a Major 3rd, not a minor 3rd, because it's fundamental ratio is still close to an ET M3 width (about 4 cents narrow of ET) even though the 5th and 4th partials, which we can't hear, are wildly wrong.

This idea of extending a temperament via 12ths vs octaves was interesting enough that I tried doing a little math to see what's going on.

First, here's a picture of the "Railsback" curve of a typical stretched tuning with the Thomas Young unequal temperament applied to it:


The temperament is being expanded by octaves, and the octaves are stretched for inharmonicity. But the temperament stays intact in each octave, which you can see by superimposing the tuning curves of the 2 neighboring octaves onto the F3-F4 temperament octave.


Just for fun, I also tried shifting the tuning curve by perfect fifths instead of an octave to see if it still kept some of its personality. It did. So one cycle around the circle of fifths and the temperament hasn't actually changed that much.


The last test was to shift the tuning curve by the perfect 12th (octave + fifth)


As with the 5ths, there are differences, but the spikes and valleys are in roughly the same places, so the intervals will generally sound similar outside the temperament octave.

So I think Jeff/UprightTooner is correct that you'll be altering the temperament each iteration, gradually cycling it around the circle of 5ths, but perhaps these differences aren't as obvious as you'd expect because each iteration is fairly similar to the previous. Of course I think my little analysis is an oversimplification, since I think tuning outward by pure 12ths will be doing weird things to the beat rates of the 5ths contained inside those 12ths... it kind of makes my head hurt thinking about that.

Thanks Anthony for that analysis. It is really interesting to see the persistance of the temperament. I produced a graph some years ago of my piano in Young using the same format as the ET graph I posted above and it showed the distinct characteristics of the temperament being carried throughout the compass in a manner very similar to your results.

As I mentioned in my response to Ed, once I get out of the range where the beats tell me the temperament, it becomes the actual interval widths that I rely on for a sense of temperament.

In the end I find that I hear only about four kinds of temperaments - Meantone style, Strong Well, Mild Well, and ETs, and within each of those temperament groups, I can't tell the difference, nor do I care about the differences between them.

Cheers

Perhaps that is because virtually all the "wells" follow the same basic philosophy (centering around the key of C) but simply to different degrees. AND according to Thomas Young, by following his rules of tuning to personal taste you can produce an extreme well to a Victorian well (almost EBVT) and everything in between by altering the speed of the C-E M3rd, still maintaining the essential key coloration pattern throughout. The guy was pretty smart.

Pwg

That seems to be the case for musicians that I have had conversations with about the matter. It makes the whole practice of being specific about named temperaments such "Vallotti" or "Werkmeister" etc all seem academic and historical rather than practical or musical. DoelKees on this forum once proposed a range of theoretical and ideal unequal temperaments that varied by degree rather than by shape.

Indeed it does have very accurate ears.