51

Re: Well vs. Equal Temperament

A very interesting introduction and explanation of historical piano/harpsichord tunings prior to our modern equal-tempered tuning, including the Werckmeister III and Vallotti-Young "well tuned" systems, by Kyle Gann--

http://www.kylegann.com/histune.html

The Vallotti-Young is the author's preferred tuning for his own piano.
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Last edited by Stephen_Doonan (02-06-2016 19:32)

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Re: Well vs. Equal Temperament

AlexS wrote:

But what do you mean by "pure, just intonation"?

This has to do with musical notes whose frequencies are related by simple numeric ratios.

It all begins with Overtones, or Harmonics.  A "musical" tone (like a plucked string) sounds different than an "unmusical" tone (like hammering on an old drain-pipe).  The reason is that the different frequencies contained in the sound tend to be simple multiples of one another... a Fundamental (bass pitch), and a second harmonic at twice that frequency (which sounds an octave higher), and a third harmonic at three times the fundamental frequency (and which sounds an octave plus a fifth higher), and so forth.

The most important interval is the Octave... say from one "C" to the next "C" on the keyboard.  These notes share the same name, and in some way sound like the same note (except for one being higher than the other).

The next most important interval is the Fifth.  Take the third harmonic, and lower it by an octave (to bring it back into range).  Frequency was x3, and divide by two, so the Fifth is a frequency ratio of 3 to 2, or 3/2.  The interval from "C" to the next "G" on the keyboard.

Fourth harmonic is the octave of the octave.  Fifth harmonic sounds at two octaves and a third.  Lower by two octaves to bring it back into range.  Frequency was x5, and divide by four (two twice), so this ratio is 5 to 4, or 5/4.  This is called a Major Third, the interval from "C" to the next "E" on the keyboard.

From here the C-major scale can be defined... from C up a pure major third to E, and up a pure fifth to G.  Then from G up a pure major third to B (3/2 x 5/4 = 15/8), and up a pure fifth to D (which will thus be 3/2 x 3/2 = 9/4, but we can lower it by an octave to bring it back into range, getting 9/8 as the interval from "C" to the next "D" on the keyboard.  And from C down a pure fifth to low F (which will thus be 1 divided by 3/2 = 2/3, but we can raise it by an octave to bring it back into range, getting 2 x 2/3 = 4/3 as a Pure Fourth, the interval from "C" to the next "F" on the keyboard.  And from there up a pure major third to A.  4/3 x 5/4 = 5/3 as a pure major sixth, the interval from "C" to the next "A" on the keyboard.

The C-Major Scale in Just Intonation (all harmonically pure intervals, NO BEAT on any interval):
C 1
D 9/8
E 5/4
F 4/3
G 3/2
A 5/3
B 15/8
C 2

The Ratios:
D to C is 9/8 to 1, called Major Whole Tone
E to D is 5/4 div by 9/8 = 5/4 x 8/9 = 10/9, called Minor Whole Tone
F to E is 4/3 div by 5/4  = 4/3 x 4/5 = 16/15, called Diatonic Semitone
G to F is 3/2 div by 4/3 = 3/2 x 3/4 = 9/8, again Major Whole Tone
A to G is 5/3 div by 3/2 = 5/3 x 2/3 = 10/9, again Minor Whole Tone
B to A is 15/8 div by 5/3 = 15/8 x 3/5 = 9/8, again Major Whole Tone
C to B is 2 div by 15/8 = 2 x 8/15 = 16/15, again Diatonic Semitone.

None of these intervals match any interval in 12-tone Equal Temperament, except for the Octave.  And on a Piano with stretched octaves, that one fails too.

There are many other tunings and temperaments of interest... Pythagorean, the various Mean Tones, and the various Well-Tempered varieties.  But the Just Intonation is where they all take origin.

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Re: Well vs. Equal Temperament

Stephen_Doonan wrote:

A very interesting introduction and explanation of historical piano/harpsichord tunings prior to our modern equal-tempered tuning, including the Werckmeister III and Vallotti-Young "well tuned" systems, by Kyle Gann--

http://www.kylegann.com/histune.html

The Vallotti-Young is the author's preferred tuning for his own piano.
---

'nice link!  smile

Here's another experiment in a similar vein to the last one, but now with the 1796 Broadwood using the 1799 Young temperament (young2.scl) that Gann mentions in the link Stephen provides (here at A=424), in juxtaposition with the same Bluethner-based Steinway D as above.   The renders are of Beethoven's Op. 27 n. 2 3rd mvmnt, but sadly the midi file is unattributed in my collection (if anyone recognizes the performance then please let me know).  Reverb waveform for both renders is Archduke Hall using the same custom mic setup.

http://www.forum-pianoteq.com/uploads.p … einway.mp3

http://www.forum-pianoteq.com/uploads.p … adwood.mp3

which, while admittedly having little to do with historic tunings (other than perhaps issues of material stress vs string tension), reminds me of one of the many reasons why I love to play Beethoven on period instruments so much: all nuance and color aside, in big passage stuff like this the instrument literally shakes!  B's music pushes the old pianos right up to their limit and no small part of the excitement of such performances is the fear that something catastrophic might happen!  big_smile

Last edited by DaveyJones (03-06-2016 10:18)
Wahre Kunst bleibt unvergänglich.

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Re: Well vs. Equal Temperament

Thanks, OrganoPleno, and others too, for interesting historical lessons smile

Here goes just another layman's question: is it possible to try this "just intonation" thingy in Pianoteq, and in what particular version of it? Do I need "Pro" to tune my piano like that? Or mb I could use Standard (perhaps with some presets created by community members owning a "Pro" versions)?

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Re: Well vs. Equal Temperament

AlexS wrote:

Here goes just another layman's question: is it possible to try this "just intonation" thingy in Pianoteq, and in what particular version of it? Do I need "Pro" to tune my piano like that? Or mb I could use Standard (perhaps with some presets created by community members owning a "Pro" versions)?

Yes, it can be done in PianoTeq, and the Standard Version is sufficient.

Opening PianoTeq Standard, click on the area for "Tuning".

It should say (as default): "diapason 440 Hz, temperament equal".

Press the "mu" key to the right of diapason...
now it should say "scale default, keymap default".

Click on Scale Default, and it gives two choices...

Tuning presets:   many to choose from.  1/4 Comma Meantone (Pietro Aaron 1523) is a classic, with Eight pure thirds (5/4), and all fifths scrunched a little small (except for a single too-wide "wolf fifth" to close the cycle.

Pythagore (Pythagorean) is all pure fifths (3/2), except for a single narrow "wolf fifth" to close the cycle.

Or choose "load tuning from file"... this lets you load a Scala-File for some particular tuning.

The website http://www.huygens-fokker.org/scala/scl_format.html lets you download the Scale Archive, containing over 4500 tuning files for use with PianoTeq.
http://www.huygens-fokker.org/docs/scales.zip

Unfortunately, the "Just Intonation" is not included either way.  Probably because its appeal is strictly rational, but for playing any kind of music it actually sounds rather horrid.  In principal, you could program it into a Scala File on your own.  The difficulty is what to do with the black keys.  Are they related to a white key (above or below?) by a fifth (or a third?), or do they attempt to "split the difference" in one fashion or another?  In this way, we see that "Just Intonation" is not completely well-defined, and actually has numerous interpretations... none of them very satisfactory for actual music.

One interpretation is found at:
http://xenharmonic.wikispaces.com/12highschool1
We see that, for the white keys, all the fractions are the same as in my previous post.
By copying this little table (including the lines with "!", which are comments), saving it as a Text File, and then renaming it to be a Scala File (*.scl), it should actually play when loaded into PianoTeq as above.  Elsewhere on that Website are many other such interpretations.

The best Scales for the study of Early Music are the Pythagorean (for Medieval and early Renaissance music) and the 1/4 Comma Meantone (for late Renaissance through early Baroque music).

For late Baroque and early Classical music, you can explore the tunings of Neidhardt, Vallotti, Kirnberger, and Werckmeister (all included in PianoTeq Standard), which are basically modifications of Mean-Tone to soften the Wolf Fifth(s) and allow playing in all Keys.

Enjoy!

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Re: Well vs. Equal Temperament

'just a shout-out here for Brad Lehman's Bach temperament: http://www.larips.com/

http://www.forum-pianoteq.com/viewtopic … 37#p949137

smile

Wahre Kunst bleibt unvergänglich.