Mathematical distinction between "equal" and "flat" temperament?

My understanding is that "equal" is equal temperament with detuning enabled and "flat" is equal temperament without detuning or other tuning customizations.

I think the equal temperament is the one that would be elaborated by a human piano tuner. It takes into account the inharmonicity of the strings and will produce because of that slightly stretched octaves and other shifts to make the piano sound. The flat temperament is the one that would be realized with an electronic tuner and that would set the theoretical frequencies of an equal tuning (for example the all A will be tuned to 55, 110, 220, 440, 880 etc...). A human piano tuner never set a flat temperament on a real piano I think.
Egal temperament is close to the "cordier" temperament with Perfect fifths and with stretched octaves.
Very interesting link here http://www.temperamentcordier.org/ it is in french but I think it exist in english.
I hope "cordier" temperament will be added one day in pianoteq.

Thank you, and what about "cordier" temperament

On peut à partir du tempérament "flat" créer le tempérament "cordier" avec l'outil d'étirement d'octave.
En théorie le rapport de fréquences entre octaves devrait être de 2.00387 pour obtenir des quintes justes avec un rapport de 1.5.
En mesurant les fréquences à partir du steinway D c'est avec un étirement d'octave de 1.25 que j'obtiens ces valeurs.
Cela est théorique et il est vraisemblable qu'un accordeur tiendra compte d'autres facteurs même en utilisant la méthode "cordier", et obtiendra un accord plus précis.
Merci à Philippe si il lit de confirmer ou non cette méthode.

From the flat temperament we can create the "Cordier" temperament with the octave stretching value.
In theory the ratio of frequencies between octaves should be 2.00387 to obtain perfect fifths( with a ratio of 1.5).
By measuring the frequencies from the steinway D it is with an octave stretching of 1.25 that I get these values.
This is theoretical and it is likely that a tuner will take other factors into account even when using the "Cordier" method, and get a more accurate tuning.
Thanks to Philippe if he reads to confirm or not this method.

Translated with www.DeepL.com/Translator (free version)

Thank you... I didn't know 3:2 and 6:4 could be different... I will make search on this mystery...
Here is a very intesting text about inharmonicity and Cordier temperament :
http://www.temperamentcordier.org/inhar...ite%20.pdf

Thank you very much... Wonderful world of piano and tuning where there is so much to learn. Moreover pianoteq is great to test the tuning, the inharmonicity, to hear the beats etc...

Equal temperament is 12 equal steps, in theory. To me the text might say..

"Equal Temperament[:] The standard Tuning _theory_ where the octave is divided in 12 equal steps." ?

Set the stretching to 1 - that disables all stretching (not 0) - hope that's your fix @RA.

With real pianos, it's impossible for a theory to be implemented 100% perfectly, against all that immense pressure of wood/steel etc. in the frame/case. There are different issues with the points of contact (ends of strings), some pianos have multiple wood parts, not just one big straight piece and so on.. different forces across the harp.. each manufacturer has different methods - but there are similarities enough, that a piano technician might be able to tune similarly from one to another. But.. like real world physics, it's very different to a fully theoretical sine wave per note Kind of amazing to me how Pianoteq does all this.

Generally speaking, Flat gives the closest thing to a real piano tuned for close tuning with MIDI in mind/synths etc. without so much stretching and issues with inharmonicity.

But.. that's really most likely to be a desireable thing in music where you're hitting the center of the mainstream audiences.. with auto-tuned vocals and you know you're 'going for' this. Otherwise.. other tunings might breath desirable 'life' into a fairly generically tuned MIDI instrument mix. I love a piano which is full of character, and only rarely have I really wanted to go with Flat to double-down on the General MIDI clean lines created by all other tracks.. I think we'd know when we'd want Flat or not though in any context.

But.. it can come down to being measures too small for us to hear. We may think 12 exact notes should be easy to tune a piano to But.. no. A piano note is not a sine wave - but has markedly complex spectrum living deep beyond the initial tone. Each string effects surrounding ones, diminishing with distance etc. So, altering one string's tuning can sour or sweeten

Piano technicians might have similar ideas, backgrounds but still have some personal preferences about working on 'which octave first'.. or to their ears, it may be ornamentally positive to have a certain amount of beating in certain places.. some may stretch a little more/less than the next technician might. However, in very close numbers, they may each be theoretically tuning "to Equal temperament".

Some lovely old pianos might seem vivid and with great tonality, when tuned with one temperament - but might sound stilted, when tuned to another temperament. Subjective though! 

But.. in Pianoteq-land, it IS possible to impose quite exacting numbers on things.. which could take many hours - instead a click or 2.

But, physics blows me away any time I imagine what's happening in any real piano, and how Pianoteq has modelled this all so well, and improving all the time.

Even though I always have thought "Flat tuning" within Pianoteq is closer to standard MIDI - and can be more easily tuned without so much internal 'real piano' behaviours, it's not 1:1 exactly - same with any VSTI - there are real world reasons a saxophone may be tuned exactly to a note, but still not be as 'exact' to a number, as a sine wave. On an oscilloscope you can see 1 wave form.. and hear clearly one tone.. but real instruments have kind of hundreds of deeper, distant cousins crushed in with the initial or significant tone. An oscilloscopic picture would maybe show a more fuzzy set of lines moving as notes develop, with even the main line displaying contextual movement from its transient to full fade.

My basic uses of Flat along those lines talked about above did seem to give me much 'cleaner' 'modern mix piano' tuning which, when desired, can be achieved. Personally I prefer more realistic pianos - and often times, finding ways to 'work in' even a lovely flavor of a Well Temperament tuning.. that can lead to a more characterful outcome.

But in much music, there are few/no rules, if making our own music - indeed we can benefit creatively by trying out "Flat" in a situation where you need piano to match cleanly other standard MIDI tuned instruments, and some other tunings.. something more interesting can happen for you, not to think "the correct tuning for this, is that".

General MIDI specifies exact frequency per note - each VSTI will implement something of their own, close to it.. pianos may sound not their own best self, if tuned in that very static way, solo. But.. in a modern music mix context, it may be fortuitous. Each musical situation presents different choices to try. I find theoretically, I can begin with one inspiration but end up with a different outcome, just because of decisions along the way like swapping through some different pianos and tuning types.. each giving something different, at least in 'vibe/feel/emotional effect'.

It's possible to kind of 'auto-tune' a whole mix these days - so having an exactly static Gen MIDI Flat piano temperament, though helpful, may not be as incredibly important as some might think.

It's a fascinating subject, and I always feel I learn more, to my benefit each time it comes up. Because, there are so many moving parts, there's not much "black & white" about it.

(If I made any errors about anything above, happy to learn how/why - cheers!)

Je crois avoir compris : Pour accorder la quinte LA1-MI2 par exemple, 3:2 veut dire que l'on aligne le 3eme partiel du LA1 (mi3) avec le 2eme partiel du MI2 (mi3). 6:4 on aligne le 6eme partiel du LA1 (mi4) avec le 4eme partiel du MI2 (mi4). Et effectivement avec l'inharmonicité c'est différent, tout ceci est passionnant.

I think I have understood: To tune the fifth A1-E2 for example, 3:2 means that we align the 3rd partial of A1 (E3) with the 2nd partial of E2 (E3). 6:4 means that we align the 6th partial of the A1 (E4) with the 4th partial of the E2 (E4). And indeed with inharmonicity it is different, all this is exciting.

Thanks Philippe!

Maybe the text in the Flat temperament 'tuning description' could be altered to reflect those extra subtle details?. Currently it reads..

"Flat temperament
An equal temperament with harmonic stretching: setting the octave stretching to 1 will disable all stretching"

Perhaps instead of "... will disable all stretching".. to something like "... will provide normal stretching with no beats on octaves".. or other?

I could not find this quote in the manual. In the 2 links provided above you can read:

"Flat: octaves ratio is strictly 2, for use in certain circumstances, for example with synthesizers."

"It is quite usual to stretch octaves in a piano, but how much should they be stretched? Well… this might be a matter of taste! Adjust it to your own taste by modifying the octave stretching parameter. The main effect will be observed in the treble notes.

When the octave stretching parameter is set to 1, the stretching follows the natural inharmonicity of the strings (depending on the string length), so there still is a slight stretching. If you want no stretching at all, then use the flat temperament."

Ah, I understand now, thank you Qexl, we will update the UI text (something like "An equal temperament where the octave ratio is exactly 2 when the octave stretching is set to 1.")

Thanks Philippe, all of this is very interesting an perfectly timed with my experiments on tuning, both virtually in the computer and physically for my daughter's clavichord (not that there are so many octaves to stretch there, LOL). Two questions:

1. What if one wants less stretching than the natural inharmonicity but more than no stretching at all? The interface allows a minimum of 0.95 factor, but what if one wants less? Not sure why one would want less, I guess for the same reason why someone else would want more? I think I'd go for the "follow normal instrument inharmonicity", but still curious about options.

2. What is the difference between the Natural and the Harmonic stretching? I think I understand the harmonic, which is the one coming from the partials of the notes and it must be the one normally discussed, but how about the natural? My books do not mention that and Google knows nothing about it. Unless I'm also missing something, neither the manual, nor the embedded help mention anything about it -- so whatever you say here may be worth adding to the manual too.

Thanks!

PS: great work on the temperament (like everything else in PTQ!) -- the ability to do a full rebuild vs change string tension alone and everything else is really blowing my socks off! I initially resisted purchasing the full version (I thought Stage was sufficient and was ready to forgo the temperament) but I'm glad I've got Pro instead

Thanks a lot for your answer, sorry if I have yet another follow up.

I see, so if I understand you correctly, you are saying:

- the "normal" choices would be "no stretching at all" (octave ratio exactly 2), or "stretching like the default inharmonicity" (octave ratio variable and greater than 2)
- this parameter allows one to be "in between" these two conditions, in case one thinks/ears either as "too extreme"
- there is no reason to "anti-stretch"  (octave ratio less than 2) since that's non-physical and non-musical

Am I interpreting you right?

Oh, that makes lots of sense. Perhaps you should add these two sentences to the manual, to avoid other people like me from bothering you again or just keep wondering.

Thanks a lot!