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Generally, I think of the pure 5th as being a 3:2 5th, not a 6:4. However, on some scales, as you said, your suggestion may work better.

Also, the temperament is definitely F3-F4 but it could be tuned with C4 as the starting pitch. However, that would make the A4 end up at approximately A-439 rather than 440.

As you all may well know, the historically documented well temperaments all started from the note, C. My sequence simply arrives at what would normally be the first interval tuned, C4-E4 through the "side door", so to speak.

To tune from C4 as the starting pitch, tune C4 to the pitch standard; tune C4-E4 at 6 bps; tune F3-C4 as pure 5th, then tune F3-A3 equally beating with C4-E4 and the rest is the same.

Patrick,

I a commenting as I listen to your sound file.

• The F3-C4 5th sounded tempered to me, about like an ET 5th as you observed. Both RBI tests confirmed it as narrow.
  
  
• The A3-D4 4th was pure and the G3-D4 5th far too tempered.
  
  
• Both the F#3-C#4 5th and the F#3-B3 4th sound too tempered. The 5th does seem to beat faster than the 4th. This indicates to me that C#4 may be too flat and B3 may be too sharp.
  
• Both the G#3-D#4 5th and the A#3-D#4 4th sounded too tempered. Both of these should sound very nearly but not quite pure. This is as you noticed.
  
  
• The B3-D#4 sounds entirely too slow. Both this and the last test both indicate that D#4 is too flat.
• As you confirmed, both the A3-C#4 and B3-D#4 M3s sound too slow. This and the F#3-C#4 5th (and corresponding F#3-B3 4th comparison) indicate that both C#4 and D#4 are too flat.
  
  
• The G3-B3 M3 beats too fast. This and the B3-D#4 M3 test shows that D#4 is too flat and B3 is too sharp.
  
  
• The C4-E4 M3 sounds about right but the A3-E4 5th is too tempered (faster than the B3-E4 4th as you say). We had noted that the F3-C4 5th was tempered and not pure, the B3 was too sharp, so, this accounts for why the C4-E4 M3 sounds about right but E4 is too flat and so is C4.
  
  
• The F3-F4 octave is quite narrow. F4 is too flat. the 4th and 5th test also proved that F4 is too flat.
  
  
• The F#3-F#4 octave is too narrow as you confirmed.

It is past midnight now. I am sorry that I cannot complete this examination of the sound file now. However, I have confirmed each of Patrick's observations. None of the intervals are correct.

Sorry to have to say that.

Thanks for your comments Bill. Some of these observations I can see in the numbers too, but

makes no sense. If you look at Pat's F3F4 and the electronic one there is only 0.1 difference between the two. The F#3 F#4 octave which is supposedly too narrow is actually wider in my tuning than in Pat's.

Also in the temperament octave there were only 3 notes off by more than a cent (1.3 max) from Pat's tuning so how can all these intervals be so much off?

So I'm suspecting some data transmission error here.

Pat, are the numbers really being transferred correctly with the right partials for the offset? You will have to use the notes at precisely these partials too in your ETD as your software will have a different IH model. Can you try with a trial version of tunelab just to make sure?

Kees

Kees

Pat: Looking again at the offsets you measured and published here before I see your partials are different. The partials I use are the "tuning exam mode" values. This could explain some of the unexplicable things.

To make 100% sure we get our numbers right can you, next time you tune the midrange with my improved numbers, audio record each note for me? Then I can check here with tunelab that you did what was intended.

Below the full tunelab file which specifies the partial as well as the offset.

A0 4 0.0 -14.52
A#0 4 0.0 -11.03
B0 4 0.0 -11.53
C1 4 0.0 -7.43
C#1 4 0.0 -7.93
D1 4 0.0 -7.14
D#1 4 0.0 -5.41
E1 4 0.0 -6.25
F1 4 0.0 -2.67
F#1 4 0.0 -3.84
G1 4 0.0 -1.43
G#1 4 0.0 -1.69
A1 4 0.0 -2.47
A#1 4 0.0 -0.19
B1 4 0.0 -1.46
C2 4 0.0 4.19
C#2 4 0.0 2.68
D2 4 0.0 3.15
D#2 4 0.0 3.93
E2 4 0.0 2.55
F2 4 0.0 5.48
F#2 4 0.0 3.91
G2 4 0.0 6.50
G#2 4 0.0 5.28
A2 4 0.0 4.90
A#2 4 0.0 6.27
B2 4 0.0 4.87
C3 4 0.0 7.53
C#3 4 0.0 5.79
D3 4 0.0 7.18
D#3 4 0.0 7.47
E3 4 0.0 6.24
F3 4 0.0 7.63
F#3 4 0.0 6.29
G3 4 0.0 10.26
G#3 4 0.0 8.33
A3 4 0.0 7.36
A#3 4 0.0 10.13
B3 4 0.0 8.90
C4 2 0.0 9.08
C#4 2 0.0 4.49
D4 2 0.0 6.39
D#4 2 0.0 7.14
E4 2 0.0 6.20
F4 2 0.0 7.63
F#4 2 0.0 6.40
G4 2 0.0 9.23
G#4 2 0.0 8.33
A4 2 0.0 8.02
A#4 2 0.0 9.24
B4 2 0.0 8.30
C5 1 0.0 9.21
C#5 1 0.0 7.94
D5 1 0.0 8.95
D#5 1 0.0 8.88
E5 1 0.0 8.26
F5 1 0.0 9.05
F#5 1 0.0 7.71
G5 1 0.0 11.52
G#5 1 0.0 8.33
A5 1 0.0 8.86
A#5 1 0.0 10.71
B5 1 0.0 9.71
C6 1 0.0 12.41
C#6 1 0.0 9.74
D6 1 0.0 12.36
D#6 1 0.0 12.56
E6 1 0.0 12.24
F6 1 0.0 13.89
F#6 1 0.0 13.16
G6 1 0.0 16.87
G#6 1 0.0 16.32
A6 1 0.0 17.24
A#6 1 0.0 18.44
B6 1 0.0 18.33
C7 1 0.0 21.37
C#7 1 0.0 20.96
D7 1 0.0 24.35
D#7 1 0.0 23.77
E7 1 0.0 24.87
F7 1 0.0 27.42
F#7 1 0.0 27.58
G7 1 0.0 32.28
G#7 1 0.0 30.90
A7 1 0.0 34.16
A#7 1 0.0 37.01
B7 1 0.0 38.31
C8 1 0.0 42.61

Kees

PS. I think if you try the first set of offsets I gave with the correct partials you may find it actually better than the 2nd "improved" set.

If I was right about the partials confusion Bill was right that the F3C4 you got would have been 2 cent narrow, i.e. a normal tempered one.

Kees

I can answer that for him I think. He (and I) starts at A3, then computes F3 such that F3A3 beats 6bps, then compute the rest of the temperament octave. The offsets from ET are then listed in the standard order starting from C (i.e. C4,..,E4,F3..B3). As all C's (for example) get the same offsets that's fine.

Kees

The correct partial selections are, of course, of prime importance. It does not matter which partials are chosen but if they are changed from one device to another, that would certainly account for the number of errors that were apparent in Patrick's trial.

Kees, Bill,

it was mismatching partials. I did a couple of mistakes - or actually, they can be boiled down to one huge mistake: not double-checking the basics.

I was under the impression that the choice of partials is set in stone for all PTG master tunings. I still don't know if that is the case, but my short experience of master tunings tell me otherwise (the one I witnessed in Vegas read some of the notes on the 3rd partial).

Then, as my RCT pocket version doesn't contain the tuning exam function of the "full" version, I dove into the RCT tuning styles library and grabbed a tuning from the PTG exam section, and used the partials from that.

Only when I read your response and got suspicious, I checked a few of the other tunings in the library. No consensus at all - a range of different partial readings.

In retro-perspective, I should of course have read the TuneLab or the RCT full version manual, and that way avoided wasting your time. I'm sorry - lesson learned

But now to the good news! ... Kees, your calculations are very close. I just got home from another session (this time with the right partials and all ) and I think I have a good representation of the tuning on the sound clip I am uploading as I write this.

To give a short summary, I'm very confident that ETD users soon can tune a very nice EBVT III! There are a few issues, but I'm sure that you (Kees) can solve them once you gets the aural feedback and a most educated and highly needed second opinion by you (Bill).

Some notes inside the temperament, and a general need for more stretch just above and below (to avoid noticeably beating 5ths). But, as I said before, very close - nice numbers, Kees

Here is the link to the sound file:

Nitpicking Kees EBVT III (March 29, 2011)

I didn't see this post until now, sorry...

the three challenges to work out math-wise i think would be

1) the placement of F3 (and thus F4 and F5)
2) expanding the stretch slightly directly beneath the temperament, and across the tenor break
3) expanding the stretch slightly directly above the extended temperament (F5 ->) to get a smooth transition into mindless octaves.

Both 2 and 3 should be able to solve mathematically by giving priority to the 5th, I think?

Let's go mainly with the Yamaha C5 for now, but once we get it confirmed I'd think the next logical step would be to try it on the Hellas piano (which has a challenging scale, to say the least!)

Cool. Now the real work can begin.

Your F3A3 beats at 6.38 bps, mine at 6.15 bps. This is from looking at the spectral band around 880Hz of you playing the interval and measuring the time between 10 beats.
For your master tuning I mixed your F3 and A3 and then measured it.

Unless I'm making some strange mistake, I think your internal clock is a bit fast. Maybe too much coffee?

Regarding 2 and 3: I have some difficulties with that. Even if I tune pure 5ths I don't get the kind of stretch you get in the C3-E3 range. Let me ponder that some more.

Kees

Pat, here's the Hellish Helsinki. F3A3 is supposed to beat at 5.9 Hz using the formula bps = 6/6.9 * ET rate. A=440.

Have fun,
Kees

A0 4 0.0 -42.83
A#0 4 0.0 -37.33
B0 4 0.0 -36.82
C1 4 0.0 -30.36
C#1 4 0.0 -29.57
D1 4 0.0 -27.76
D#1 4 0.0 -24.83
E1 4 0.0 -25.13
F1 4 0.0 -19.20
F#1 4 0.0 -19.65
G1 4 0.0 -16.83
G#1 4 0.0 -16.17
A1 4 0.0 -16.32
A#1 4 0.0 -13.18
B1 4 0.0 -14.37
C2 4 0.0 -3.43
C#2 4 0.0 -4.59
D2 4 0.0 -3.98
D#2 4 0.0 -2.83
E2 4 0.0 -4.43
F2 4 0.0 -0.91
F#2 4 0.0 -2.55
G2 4 0.0 -0.05
G#2 4 0.0 -0.96
A2 4 0.0 -1.19
A#2 4 0.0 0.46
B2 4 0.0 -1.37
C3 4 0.0 1.67
C#3 4 0.0 0.05
D3 4 0.0 1.33
D#3 4 0.0 1.85
E3 4 0.0 0.37
F3 4 0.0 2.11
F#3 4 0.0 0.58
G3 4 0.0 3.72
G#3 4 0.0 2.13
A3 4 0.0 1.67
A#3 4 0.0 4.45
B3 4 0.0 2.11
C4 2 0.0 2.88
C#4 2 0.0 -1.37
D4 2 0.0 0.63
D#4 2 0.0 1.58
E4 2 0.0 -0.09
F4 2 0.0 2.11
F#4 2 0.0 0.18
G4 2 0.0 3.06
G#4 2 0.0 2.13
A4 2 0.0 1.70
A#4 2 0.0 3.36
B4 2 0.0 1.65
C5 1 0.0 3.91
C#5 1 0.0 1.96
D5 1 0.0 3.19
D#5 1 0.0 3.09
E5 1 0.0 2.14
F5 1 0.0 3.55
F#5 1 0.0 1.58
G5 1 0.0 5.01
G#5 1 0.0 2.13
A5 1 0.0 2.94
A#5 1 0.0 4.86
B5 1 0.0 2.90
C6 1 0.0 5.95
C#6 1 0.0 3.01
D6 1 0.0 5.60
D#6 1 0.0 5.79
E6 1 0.0 4.93
F6 1 0.0 7.07
F#6 1 0.0 5.48
G6 1 0.0 9.34
G#6 1 0.0 8.25
A6 1 0.0 9.03
A#6 1 0.0 10.21
B6 1 0.0 9.32
C7 1 0.0 12.68
C#7 1 0.0 11.32
D7 1 0.0 14.28
D#7 1 0.0 13.48
E7 1 0.0 14.16
F7 1 0.0 16.63
F#7 1 0.0 15.54
G7 1 0.0 19.71
G#7 1 0.0 17.82
A7 1 0.0 20.62
A#7 1 0.0 22.85
B7 1 0.0 22.72
C8 1 0.0 26.68

hmm, before we blame it all on my java consumption, what do you get for the faster G3-B3?

6.16 Hz.

Bi[EDIT: add 'e']ng terse is considered rude so I add this sentence.

Kees
EDIT2: I'm really excited by these results. This week has been nothing but bad news on other (unrelated) matters, this really makes my day (week?)!

haha

I listened to the mp3 again and I still hear the F3-A3 as beating slower! Maybe my internal clock is reversed?

The tapping i did was at 6 bps, and what I heard from the G3-B3. (I keep a small alarm clock on the floor, and I tapped 16th triplets to the tick). I don't know, I hope Bill can listen and report on what he hears?




I'm sad to hear about the other stuff, but I'm glad that this ground we are gaining can restore some of the overall balance! I'm really excited about this, too.

Now I have to go to a concert - I tuned for an interesting Danish pianist that is playing a solo concert tonight: Carsten Dahl

Re F3-A3: Maybe you've mutated EBVT unconsciously to a slightly milder version, your comments about C#E# being wider than you were used to being consistent with that.

Now the requirement to set F3A3 to some magic number 6 is I think a weakness of the EBVT aural recipe. I can't think of an easy way to set it without ever counting absolute beats per second. A complicated way would be to first tune ET with a 4:2 F3F4/A3A4 octave using for example contiguous thirds to set F3A3 to whatever the piano tells you, 6.9 on nice grands. Then retune F3 so that F3A3 beats between D3F#3 and D#3G3. EBVT via ET!

Kees

That sounds rather plausible... Wouldn't these numbers you posted earlier hint at that, too?

PS I like the verb "mutate" here...

C 3.8
C# -1.3
D 0.9
D# 1.6
E -0.4
F 1.8
F# -0.3
G 3.1
G# 0.7
A 0.0
A# 2.9
B 0.0

Yamaha C5 (verified by Pat)

C 3.6
C# -1.0
D 0.8
D# 1.3
E -0.6
F 1.4
F# -0.3
G 3.0
G# 0.9
A 0.0
A# 2.5
B 0.2

Kees, Bill,

I'm busy until next week (concerts and such), and I know at least Bill is occupied as well, but I just want to make sure that the next logical steps are:

1) Bill giving his feedback on my last talk-through of Kees' EBVT III calculations transferred to the Yamaha C5,

2) me tuning the Hellas "Hellish" Helsinki to Kee's numbers, and reporting back aurally (and in writing)?

Re1: sure, but keep in mind that I have a firm opinion that your F3 is too low. If Bill judged is to be right, he will have to redefine EBVT

Re2: I would expect for poorly scaled pianos that aural tuning and computer tuning would diverge. Aural tuning takes every particularity of the stings into account. Computer tuning fits the irregularly scaled piano to an ideal model without irregularities.

One could argue
a) that aural tuning is superior in this case as it takes deviations from ideal inharmonicity models in to account, or
b) computer tuning is superior as it abstracts an ideal piano like the one under consideration with all those irregularities smoothed out, and thus produces a tuning accommodating its flaws.

I have done some experiments with say a 4:2 octave tuning in the midrange based on 1) inharmonicity data on all strings and 2) an abstracted inharmonicity model smoothing out all irregularities and found 2) to be superior aurally. So it may be better to ignore beat deviations in favor of a smooth tuning.

Getting off topic here, sorry.

Kees

The more I think about it, this might very well be the case. That would make for a slight tilt towards ET (C#4-F4 and G#3-C4), right?

Anyways, there is no question you should go by Bill's EBVT III and his 2nd opinion on my comments. We are of course trying to rule out personal preferences (and errors) and get as true an ETD representation of the temperament as possible.

But if we leave that open for a while, we still need to avoid beating 5ths outside the temperament. For that I suspect we'll need the aural output of different pianos. That is why I'll try to get the Hellas numbers aurally verified.

One could argue that it's a bad instrument, and one would be perfectly right Still, although this piano is a national "treasure", similar instruments are found all over the world. We, for example, do not have spinets at all in my country, but I'd think their numbers will differ noticeably from a Steinway D