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Topic: wherein Pingu explains tuning to Bro D (split from Newton etc.) (Read 340 times) previous topic - next topic - Topic derived from Newton, Copernicus, G...

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wherein Pingu explains tuning to Bro D (split from Newton etc.)
Anyway, pianos don't use whole number ratios.
For years I've been meaning to ask you to explain tuning to me.

I know a fair bit about it already, but there's a huge amount I don't know, and I'm too lazy to research it.

Oh, you'd love it.  The first thing to know is that the Pythagorean comma isn't really a problem - it's very small when distributed amount 12 fifths.  The bigger problem is the syntonic comma, which is almost as big, and is the difference between a circle of only four fifths (e.g. from C to the E, a major third higher) and the "pure" major third given by the ratio 4:5.  In order to "purify" that third, you need to distribute all of it over just 4 fifths, and that means you are simultaneously OVER-compensating for the Pythagorean comma.  Which means that not only are your fifths too small (and much smaller than if you distributed the Pythagorean comma over 12 fifths), but you end up with a "wolf" at the end to make up the extra you've taken off.  Which you have to put somewhere where you hope you won't need it.

Systems in which you distribute the syntonic comma between four intervals of a fifth are called "quarter comma" mean tone. "Quarter comma" because you take 1/4 of a syntonic comma off each fifth, and "mean tone" because the resulting tones are the "mean" of the two "tones" given by the ratios of 9:8 and 10:9. The first is called the "major tone", and the second the "minor tone" (not a semitone, just a smaller "tone" than the "major tone".  Having all the tones the same size is convenient for Western music, but mean tone tunings have two sizes of semitone (because of the syntonic comma distribution". So on a fretted instrument, the frets are of alternating size (as well as decreasing over all as you go up the string of course).

I'll try to link a thing I wrote...

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Re: wherein Pingu explains tuning to Bro D (split from Newton etc.)
Reply #1
Trying to work through this post.
Oh, you'd love it.  The first thing to know is that the Pythagorean comma
OK so first of all, "Pythagorean comma" refers to the difference between 12 "pure" perfect fifths and 7 octaves, right?
Quote from: Pingu
isn't really a problem - it's very small when distributed amount 12 fifths.
Because a "pure" perfect fifth is 7.020 equal-tempered semitones, and 0.020 is pretty damn small, right?
(From 12 log2(3/2) = 7.020.)
Quote from: Pingu
The bigger problem is the syntonic comma, which is almost as big, and is the difference between a circle of only four fifths (e.g. from C to the E, a major third higher) and the "pure" major third given by the ratio 4:5.
Ok, (3/2)4 = 81/16, versus 80/16 (=5) for two octaves plus a "pure" major third.
Quote from: Pingu
In order to "purify" that third, you need to distribute all of it over just 4 fifths, and that means you are simultaneously OVER-compensating for the Pythagorean comma.
I'm guessing that you're referring to tuning systems in which you'd make a perfect fifth into a frequency ratio of 51/4:1, so that four of those perfect fifths would then give you two octaves plus a "pure" major third.
Quote from: Pingu
Which means that not only are your fifths too small (and much smaller than if you distributed the Pythagorean comma over 12 fifths),
12 log2(51/4) = 3 log2(5) = 6.966.  So if I'm reading you right, then this makes sense:  Such a fifth is smaller than the equal-tempered version, thus overcompensating for the Pythagorean comma.
Quote from: Pingu
but you end up with a "wolf" at the end to make up the extra you've taken off.  Which you have to put somewhere where you hope you won't need it.
I'm not sure I get this "wolf" thing, but I'm going to take a wild guess, and you can tell me whether I'm on the right track.

Suppose we're aiming to have a "pure" major third, and we're building an entire tuning system based on that.  Let's look at some of the frequencies we might get within one octave from C up to B.

Let's say C = 1 arbitrary frequency unit (AFU).
G = 51/4 AFU.  D = 51/2/2 AFU.  A = 53/4/2 AFU.  E = 5/4 AFU.
B = 55/4/4 AFU.  F# = 53/2/8 AFU.  C# = 57/4/16 AFU.  G# = 25/16 AFU.
But working in the other direction, we get
F = 2/51/4 AFU.  B♭ = 4/51/2 AFU.  E♭ = 4/53/4 AFU.  A♭ = 8/5 AFU.
(I hope the flat symbol shows up properly.  It does on my screen, but different machines seem to treat special characters differently.)

Now comparing the G# to the A♭, they're apart by a difference of
12 log2((8/5)/(25/16)) = 12 log2(128/125) = 0.411 equal-tempered semitones
...which is big enough to be startling.  I'm guessing that your word "wolf" has something to do with this difference.

(And of course the fact that I put it at G# was arbitrary.)
Quote from: Pingu
Systems in which you distribute the syntonic comma between four intervals of a fifth are called "quarter comma" mean tone. "Quarter comma" because you take 1/4 of a syntonic comma off each fifth, and "mean tone" because the resulting tones are the "mean" of the two "tones" given by the ratios of 9:8 and 10:9. The first is called the "major tone", and the second the "minor tone" (not a semitone, just a smaller "tone" than the "major tone".
OK, I tried to do something like this above.  And yes, the tones are 51/2:2, which is exactly the geometric mean between 9:8 and 10:9.  (So you get an arithmetic mean when you work with the logarithms thereof.)
Quote from: Pingu
Having all the tones the same size is convenient for Western music, but mean tone tunings have two sizes of semitone (because of the syntonic comma distribution).
Chromatic semitone 57/4:16
Diatonic semitone 8:55/4
Right?
Quote from: Pingu
So on a fretted instrument, the frets are of alternating size (as well as decreasing over all as you go up the string of course).
But that's weird, isn't it?  If every second fret means going up a tone (51/2:2), then the 12th fret wouldn't give you an octave.  It'd give you six tones (125:64), which is an octave minus a "wolf", or something like that.  No?

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Re: wherein Pingu explains tuning to Bro D (split from Newton etc.)
Reply #2
Yikes. I use my piano to tune my guitar and a man to tune my piano. When someone plays a horn or harmonica, it's their job to figure out which holes to use.
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Re: wherein Pingu explains tuning to Bro D (split from Newton etc.)
Reply #3
You can Tune a piano, but you can't Tuna fish.

Spoiler (click to show/hide)

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Re: wherein Pingu explains tuning to Bro D (split from Newton etc.)
Reply #4
Trying to work through this post.
Oh, you'd love it.  The first thing to know is that the Pythagorean comma
OK so first of all, "Pythagorean comma" refers to the difference between 12 "pure" perfect fifths and 7 octaves, right?
Quote from: Pingu
isn't really a problem - it's very small when distributed amount 12 fifths.
Because a "pure" perfect fifth is 7.020 equal-tempered semitones, and 0.020 is pretty damn small, right?
(From 12 log2(3/2) = 7.020.)

Yes.
Quote
Quote from: Pingu
The bigger problem is the syntonic comma, which is almost as big, and is the difference between a circle of only four fifths (e.g. from C to the E, a major third higher) and the "pure" major third given by the ratio 4:5.
Ok, (3/2)4 = 81/16, versus 80/16 (=5) for two octaves plus a "pure" major third.
Quote from: Pingu
In order to "purify" that third, you need to distribute all of it over just 4 fifths, and that means you are simultaneously OVER-compensating for the Pythagorean comma.
I'm guessing that you're referring to tuning systems in which you'd make a perfect fifth into a frequency ratio of 51/4:1, so that four of those perfect fifths would then give you two octaves plus a "pure" major third.

Yes.  Sorry I forget to mention the octave transpositions.  C-G-D-A-E gives you a very large major 3rd if all the fifths are pure.
Quote
Quote from: Pingu
Which means that not only are your fifths too small (and much smaller than if you distributed the Pythagorean comma over 12 fifths),
12 log2(51/4) = 3 log2(5) = 6.966.  So if I'm reading you right, then this makes sense:  Such a fifth is smaller than the equal-tempered version, thus overcompensating for the Pythagorean comma.

Yup.

Quote
Quote from: Pingu
but you end up with a "wolf" at the end to make up the extra you've taken off.  Which you have to put somewhere where you hope you won't need it.
I'm not sure I get this "wolf" thing, but I'm going to take a wild guess, and you can tell me whether I'm on the right track.

Suppose we're aiming to have a "pure" major third, and we're building an entire tuning system based on that.  Let's look at some of the frequencies we might get within one octave from C up to B.

Let's say C = 1 arbitrary frequency unit (AFU).
G = 51/4 AFU.  D = 51/2/2 AFU.  A = 53/4/2 AFU.  E = 5/4 AFU.
B = 55/4/4 AFU.  F# = 53/2/8 AFU.  C# = 57/4/16 AFU.  G# = 25/16 AFU.
But working in the other direction, we get
F = 2/51/4 AFU.  B♭ = 4/51/2 AFU.  E♭ = 4/53/4 AFU.  A♭ = 8/5 AFU.
(I hope the flat symbol shows up properly.  It does on my screen, but different machines seem to treat special characters differently.)

Now comparing the G# to the A♭, they're apart by a difference of
12 log2((8/5)/(25/16)) = 12 log2(128/125) = 0.411 equal-tempered semitones
...which is big enough to be startling.  I'm guessing that your word "wolf" has something to do with this difference.

Yes, exactly.  On keyboards, you try to "hide" the wolf by putting it between two notes a fifth apart that you will only rarely use.  On frets you have more options (but new problems).

Quote
(And of course the fact that I put it at G# was arbitrary.)

But it's where you typically put it.

Quote
Quote from: Pingu
Systems in which you distribute the syntonic comma between four intervals of a fifth are called "quarter comma" mean tone. "Quarter comma" because you take 1/4 of a syntonic comma off each fifth, and "mean tone" because the resulting tones are the "mean" of the two "tones" given by the ratios of 9:8 and 10:9. The first is called the "major tone", and the second the "minor tone" (not a semitone, just a smaller "tone" than the "major tone".
OK, I tried to do something like this above.  And yes, the tones are 51/2:2, which is exactly the geometric mean between 9:8 and 10:9.  (So you get an arithmetic mean when you work with the logarithms thereof.)
Quote from: Pingu
Having all the tones the same size is convenient for Western music, but mean tone tunings have two sizes of semitone (because of the syntonic comma distribution).
Chromatic semitone 57/4:16
Diatonic semitone 8:55/4
Right?

I think so, if I'm understanding your notation.  I have done the mathy part, but mostly it's in my ears these days.
Quote
Quote from: Pingu
So on a fretted instrument, the frets are of alternating size (as well as decreasing over all as you go up the string of course).
But that's weird, isn't it?  If every second fret means going up a tone (51/2:2), then the 12th fret wouldn't give you an octave.  It'd give you six tones (125:64), which is an octave minus a "wolf", or something like that.  No?

Ooh.  You might just be right.  Maybe the mean tone isn't actually the mean tone.  I'll have to think.

BTW, there are compromise "meantone" tuning in which you only take off, say 1/5th of a comma, or 1/6th, which can work well, although there is something a bit special about the pure thirds of1/4 comma meantone, and I've read that the beat frequencies harmonies in the triads, but I haven't checked that out.  It's rather unforgiving though, because it seems that we "sum" "out-of-tunedness", so by starting with a really quite out of tune fifth (an entire 1/4 too small), any deviation from absolute purity in the major third makes it horrible.  But well executed it is beautiful. 1/6 comma is much more forgiving, because deviations from accuracy in either the fifths or thirds (or even the octaves) don't sum to as much hell.

Circular temperaments are also cool (but you have to fudge them on frets) - basically you distribute the Pythagorean comma unevenly round the whole circle, but take more of in the "near" keys (like C, G, F, D) to improve the thirds in those keys, and less off the rest.  For example, a really popular (these days) temperament is Valotti, in which you talke 1/6th comma off half the fifths and leave the rest pure.  That makes C, G and F triads identical to triads in 1/6 meantone, and makes others still better than equal temperament (which is really a very out of tune temperament because of the horrible thirds).  Even the remote keys in Valotti are rather eerily beautiful because of the pure fifths.  The worst major triad is E major, because it's still got a tempered fifth, but a horrible third as well.

So you usually fudge that slightly.

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Re: wherein Pingu explains tuning to Bro D (split from Newton etc.)
Reply #5
Thanks.  I'll have to ponder those last two paragraphs for a while, though.

I have to get the math of it, or else i don't really feel that I've understood it at all.

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Re: wherein Pingu explains tuning to Bro D (split from Newton etc.)
Reply #6
I like to get the gist of the math, and the sense that it would make even more sense if I got more of the gist. 

But then I don't push my luck after that.  Fortunately the ears can take over.

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Re: wherein Pingu explains tuning to Bro D (split from Newton etc.)
Reply #7
There's music and then there's music.
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Re: wherein Pingu explains tuning to Bro D (split from Newton etc.)
Reply #8
I like to get the gist of the math, and the sense that it would make even more sense if I got more of the gist. 

But then I don't push my luck after that.  Fortunately the ears can take over.
There's been a lot of work in defining the math behind musical instruments.  Turns out it's a 2nd level differential equation problem.

The Mathematics of Musical Instruments

Quote
This article highlights several applications of mathematics to the design of musical instruments.
In particular, we consider the physical properties of a Norwegian folk instrument called
the willow flute. The willow flute relies on harmonics, rather than finger holes, to produce a
scale which is related to a major scale. The pitches correspond to fundamental solutions of the
one-dimensional wave equation.

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Re: wherein Pingu explains tuning to Bro D (split from Newton etc.)
Reply #9
BTW, there are compromise "meantone" tuning in which you only take off, say 1/5th of a comma, or 1/6th, which can work well,
So as you take off a smaller fraction of the comma, your thirds get worse, your fifths get better, and your "wolf" gets less ferocious.
Quote from: Pingu
although there is something a bit special about the pure thirds of 1/4 comma meantone, and I've read that the beat frequencies harmonies in the triads, but I haven't checked that out.
"Beats" here refer to the differences between frequencies, right?

Given a "pure" major third (frequency ratio 5:4) and a perfect fifth from a 1/4-comma meantone system (frequency ratio 51/4:1), the "beat" between the first and fifth notes is something like 1.98 times the "beat" between the first and third.  So that's pretty close to harmonizing.  Compare with 1.96 for 1/5 comma meantone, 1.95 for 1/6 comma meantone, 1.88 for pure Pythagorean, and 1.92 for equal-tempered.

(Is this even what you're talking about?  I'm not sure.)
Quote from: Pingu
Circular temperaments are also cool (but you have to fudge them on frets) - basically you distribute the Pythagorean comma unevenly round the whole circle, but take more of in the "near" keys (like C, G, F, D) to improve the thirds in those keys, and less off the rest.  For example, a really popular (these days) temperament is Valotti, in which you talke 1/6th comma off half the fifths and leave the rest pure.  That makes C, G and F triads identical to triads in 1/6 meantone, and makes others still better than equal temperament (which is really a very out of tune temperament because of the horrible thirds).  Even the remote keys in Valotti are rather eerily beautiful because of the pure fifths.  The worst major triad is E major, because it's still got a tempered fifth, but a horrible third as well.
Heh.  Well, having a relatively crude ear and having been brought up on equal temperament, I don't even notice that the thirds are bad.

Anyway, I'm going to try to understand this Valotti thing.

If the C, G, and F triads are like 1/6 meantone, then I guess you must be talking about all the fifths F-to-C, C-to-G, G-to-D, D-to-A, A-to-E, and E-to-B being tuned like 1/6 meantone, because we need the B in there for the G triad.  This ratio is 1 : (45/4)1/6, or numerically about 1:1.497.

Then we'll use pure fifths to get B♭ from F, E♭ from B♭, A♭ from E♭, F# from B, and C# from F#.

(I'm thinking in terms of a keyboard, so I'll assume we have only one of G# and A♭ available, etc.  I'll make it the A♭, because the E:A♭ ratio actually makes a slightly better third than the E:G#, and you've said that E major is ugly.)

One might expect some sort of "wolf" at C#:A♭, but that ratio is 10935:16384, or about 1:1.498.  Which is actually a better fifth than the 1/6-comma meantone fifths (1:1.497) that we have in the "near" keys.  So no wolf to speak of.  Yay.

which major triadthird/first ratiocommentfifth/first ratiocomment
A♭1.263like 1/24 comma1.500pure
E♭1.260like 1/12 comma1.500pure
B♭1.258like 1/8 comma1.500pure
F1.255like 1/6 comma1.497like 1/6 comma
C1.255like 1/6 comma1.497like 1/6 comma
G1.255like 1/6 comma1.497like 1/6 comma
D1.258like 1/8 comma1.497like 1/6 comma
A1.260like 1/12 comma1.497like 1/6 comma
E1.262cheating with A♭ in place of G#1.497like 1/6 comma
B1.264cheating with E♭ in place of D#1.500pure
F#1.264cheating with B♭ in place of A#1.500pure
C#1.264cheating with F in place of E#1.498see "wolf" comment above

Does that look about right? :)

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Re: wherein Pingu explains tuning to Bro D (split from Newton etc.)
Reply #10
Incidentally -- since Harrison (Mr Longitude) has been mentioned in the Prisca Sapienta thread -- it seems that Harrison had some pretty strong opinions about tuning.

He felt that the ideal tuning would be based on pi.

In particular, he said that a tone should be 1/2π of an octave.  (That is, a frequency ratio of 1 : 21/2π.)  A diatonic semitone would be half of what's left over in an octave after taking out five tones.  (Frequency ratio 1 : 21/2 - 5/4π.)

So Harrison's major third is smaller than a "pure" major third.  (21/π = 1.247.)

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Re: wherein Pingu explains tuning to Bro D (split from Newton etc.)
Reply #11
BTW, there are compromise "meantone" tuning in which you only take off, say 1/5th of a comma, or 1/6th, which can work well,
So as you take off a smaller fraction of the comma, your thirds get worse, your fifths get better, and your "wolf" gets less ferocious.
Quote from: Pingu
although there is something a bit special about the pure thirds of 1/4 comma meantone, and I've read that the beat frequencies harmonies in the triads, but I haven't checked that out.
"Beats" here refer to the differences between frequencies, right?

Given a "pure" major third (frequency ratio 5:4) and a perfect fifth from a 1/4-comma meantone system (frequency ratio 51/4:1), the "beat" between the first and fifth notes is something like 1.98 times the "beat" between the first and third.  So that's pretty close to harmonizing.  Compare with 1.96 for 1/5 comma meantone, 1.95 for 1/6 comma meantone, 1.88 for pure Pythagorean, and 1.92 for equal-tempered.

(Is this even what you're talking about?  I'm not sure.)
Quote from: Pingu
Circular temperaments are also cool (but you have to fudge them on frets) - basically you distribute the Pythagorean comma unevenly round the whole circle, but take more of in the "near" keys (like C, G, F, D) to improve the thirds in those keys, and less off the rest.  For example, a really popular (these days) temperament is Valotti, in which you talke 1/6th comma off half the fifths and leave the rest pure.  That makes C, G and F triads identical to triads in 1/6 meantone, and makes others still better than equal temperament (which is really a very out of tune temperament because of the horrible thirds).  Even the remote keys in Valotti are rather eerily beautiful because of the pure fifths.  The worst major triad is E major, because it's still got a tempered fifth, but a horrible third as well.
Heh.  Well, having a relatively crude ear and having been brought up on equal temperament, I don't even notice that the thirds are bad.

Anyway, I'm going to try to understand this Valotti thing.

If the C, G, and F triads are like 1/6 meantone, then I guess you must be talking about all the fifths F-to-C, C-to-G, G-to-D, D-to-A, A-to-E, and E-to-B being tuned like 1/6 meantone, because we need the B in there for the G triad.  This ratio is 1 : (45/4)1/6, or numerically about 1:1.497.

Then we'll use pure fifths to get B♭ from F, E♭ from B♭, A♭ from E♭, F# from B, and C# from F#.

(I'm thinking in terms of a keyboard, so I'll assume we have only one of G# and A♭ available, etc.  I'll make it the A♭, because the E:A♭ ratio actually makes a slightly better third than the E:G#, and you've said that E major is ugly.)

One might expect some sort of "wolf" at C#:A♭, but that ratio is 10935:16384, or about 1:1.498.  Which is actually a better fifth than the 1/6-comma meantone fifths (1:1.497) that we have in the "near" keys.  So no wolf to speak of.  Yay.

which major triadthird/first ratiocommentfifth/first ratiocomment
A♭1.263like 1/24 comma1.500pure
E♭1.260like 1/12 comma1.500pure
B♭1.258like 1/8 comma1.500pure
F1.255like 1/6 comma1.497like 1/6 comma
C1.255like 1/6 comma1.497like 1/6 comma
G1.255like 1/6 comma1.497like 1/6 comma
D1.258like 1/8 comma1.497like 1/6 comma
A1.260like 1/12 comma1.497like 1/6 comma
E1.262cheating with A♭ in place of G#1.497like 1/6 comma
B1.264cheating with E♭ in place of D#1.500pure
F#1.264cheating with B♭ in place of A#1.500pure
C#1.264cheating with F in place of E#1.498see "wolf" comment above

Does that look about right? :)


Yes!  I can't swear to the ratios, but the ranking is absolutely right.  You can see why the E triad is the worst even though it doesn't have the worst fifth.  When I'm tuning a keyboard in Vallotti, I think of E as the feeling you have when you are short of sleep and you are trying to compensate with too much coffee.  It must be the flat fifth and the matchstick supported eyelids of the major third.

And it's a nuisance, because you need it quite a lot for A minor, which is not a remote key (same key sig as C major of course i.e. no flarps).

So sometimes people transpose Vallotti round one.  Or, on a fretted instrument, sometimes you have a G# on on string and an A♭ on another, so you can play it there.  Or, on my viol, which has gut frets, double stranded, I split the first fret so I get a choice of sharp or flat.  There are some paintings of people doing that.  It works well to get C# on the C string but a proper F (not E# on the E string).

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Re: wherein Pingu explains tuning to Bro D (split from Newton etc.)
Reply #12
OK so I have at least a partial understanding of what's going on.  Yay.

Not sure I'm up to the point where I'd be able to follow some of the conversations you used to have with placebo messiah on this subject, but you have to start somewhere.

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Re: wherein Pingu explains tuning to Bro D (split from Newton etc.)
Reply #13
Have you ever dealt with tuning schemes in which more than 1/4 comma is taken off the fifths?

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Re: wherein Pingu explains tuning to Bro D (split from Newton etc.)
Reply #14
No.  Are there any?

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Re: wherein Pingu explains tuning to Bro D (split from Newton etc.)
Reply #15
Yes, but keep in mind that I don't know enough to tell the difference between the Real Serious Musicians and the Music Cranks.  So it may be only the latter that use the tuning schemes I'm talking about.

Of course, one of the consequences of taking more than 1/4 comma off the fifths is that the major third is smaller than a "pure" major third (while most tuning schemes make the major third bigger than a "pure" major third).

[eta:  Another consequence is that the wolves are huge.]

Take Harrison's idea (that I mentioned a few posts ago), for example.  It's nearly equivalent to taking 3/10 comma off the fifths.  It has proponents (probably Cranks) who swear it sounds the nicest.  While I don't think there's anything specially wonderful about Harrison's use of pi -- it was probably a case of numerowanking -- he seems to have accidentally found a pretty good approximation to a case where the beat frequencies harmonize in the major triads (closer than 1/4 comma in that regard).

And then there's Kornerup.  The idea here is that (in the logarithmic "space") the ratio of a whole tone to a diatonic semitone is the golden ratio.  It works out to be pretty much the same as taking 0.267 of a comma off the fifths.

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Re: wherein Pingu explains tuning to Bro D (split from Newton etc.)
Reply #16
Oh, wait, I do know of some. Some circular tunings (i.e. with no "wolf" because the pythagorean comma is distributed usably) have over-large fifths in places.

I think.

I don't know about the ones you mentioned.  I'm not all that well-up on weird and wonderful temperaments - I was very much driven by the search for something that would work on both keyboards and viols, or viols and lutes, that I was capable of tuning myself. Because then I'd never have to play with a harpsichord that was more out of tune with me than I could do myself.  But I did enjoy learning to tune in various meantone temperaments, and Vallotti, with and without tweaks to improve A minor.

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Re: wherein Pingu explains tuning to Bro D (split from Newton etc.)
Reply #17
you tuned a harpsichord?
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Re: wherein Pingu explains tuning to Bro D (split from Newton etc.)
Reply #18
And yes, a minor is a pretty important key to be able to play in tune.
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Re: wherein Pingu explains tuning to Bro D (split from Newton etc.)
Reply #19

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Re: wherein Pingu explains tuning to Bro D (split from Newton etc.)
Reply #20
Yikes. That sounds like a job for an afternoon.
Love is like a magic penny
 if you hold it tight you won't have any
if you give it away you'll have so many
they'll be rolling all over the floor

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Re: wherein Pingu explains tuning to Bro D (split from Newton etc.)
Reply #21
And yes, a minor is a pretty important key to be able to play in tune.

I stopped thinking of the triads as being "in" or "out" of tune - more as having colours.  As I haven't got perfect pitch, that was neat - it meant I could hear the keys as different qualities.  Being synaesthetic helped - I have very definite "feels" for each major triad in Vallotti - not their absolute pitch but their relative temperedness.  Some are the same as others, of course, but that still makes the actual keys sound different, because chords within those keys have different colours in different places.

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Re: wherein Pingu explains tuning to Bro D (split from Newton etc.)
Reply #22
Yikes. That sounds like a job for an afternoon.

Not really.  I can do two registers in under half an hour.  The tricky part is the first octave, when you are setting the temperament.  Then you just have to tune everything to that octave, and the temperament actually helps you set the octaves, because you have "checks".  It's actually harder to tune an octave than other intervals because they are very "tolerant" of slight errors (not a big increase in beat rate for a quite large error) and it accumulates if you don't check them against the thirds.  But the thirds have a big increase in beat rate for a small amount of error, so they are great for checking the octaves.

I have a nice story about the first time I had to tune a harpsichord for reals (a concert).

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Re: wherein Pingu explains tuning to Bro D (split from Newton etc.)
Reply #23
It's a lot easier than a piano btw.

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Re: wherein Pingu explains tuning to Bro D (split from Newton etc.)
Reply #24
Sure but it's a lot harder than a cool.
Love is like a magic penny
 if you hold it tight you won't have any
if you give it away you'll have so many
they'll be rolling all over the floor