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OK, if everything is perfect then (assuming no inharmonicity)
P4g3d4 =-1.3280 P4bg3c4 = 1.7706 (beatrates).

If D4 was a cent flat then we'd have

P4g3d4 =-1.6671 P4bg3c4 = 2.2229

Is that difference really audible?


Resulting error in M2 C4D4 is 0.23 cent.

Kees

For me, the benchmark is plain: Progressive M3s and M6s, provided that 4ths and 5ths are tempered in the right direction. On some pianos you can't have both, so a jump in the RBI progression is needed. But this is a benchmark that I believe is rarely obtained, so perhaps it should be considered an ideal instead.

How do I judge the correct size of an M2 aurally? Well, not by playing it and listening, that's for sure! Myself, I listen to the 4ths and 5ths that are common to the two notes. Also two M3s or M6s that are two semitones apart (an M2) should be correctly progressive. If not, then the two M2s are tempered differently.

For example: If the G3-B3 M3 beats the same speed as the A3-C#4 M3, then the B3-C#4 M2 is tempered more (is narrower) than the F3-G3 M2. Listening to the SBIs that are common to the notes in each M2 (D4 and F#3) should point out where errors are.

If you are going to assume no iH, then assume 5ths that beat at -0.7bps, please. 5ths are affected much more than 4ths by iH.

Also, can you check your post, please? The P5-G3/D4 is -0.66bps...

As far as what is audible, I am afraid that without some sort of investigation we only have anecdotes. I can separately tune the three notes in a unison as 4ths and 5ths to an M2 and when sounded together it is beatless.

But I think I know what you are getting at. Let's look at it a little differently.

Let's say we are tuning WBW and each SBI is tempered 1.5 cents instead of 2. Would we notice? Maybe not, until G#3 is tuned! But let's say we are refining a temperament and an M2 is off by 1 cent (which also means that the 4th's and 5th's average error is 1/2 cent) would we be able to notice a difference between these 4ths and 5ths and the others? The "Beat Speed Difference Test" says yes.

As for the Beat Speed Difference test, I have found that the results are not practical. I.e. I can tune up to 3% on the test, but after I tune, record, and measure my temperaments, I'm finding much larger errors.

Possible reasons are:
1) Varying frequencies which produce varying beat speeds.
2) Varying measurements because it depends where I take the measurement.
3) Extraneous noise

I still have some more research to do to determine the effect of these conditions on error.

4) Physical limitations in the pinblock and rendering: 'ya can only get so close.

But have you determined whether there is less error in cents with SBIs compared to RBIs?

Hi Jeff,

For me, rendering is not a limitation, it is mearly a condition that can be dealt with. I will be talking about this in Denver.

What kind of physical limitations are you referring to?

Regards,

Mark C:

I said what kind of physical limitations: pinblock and rendering. You addressed one of them and then asked my what I was referring to. How strange...

Anyhoo, since you do not where the errors are coming from, how can you know where they are not coming from?

Also, since you have decided that the results of the Beat Speed Sensitivity Test do not reflect what really happens when you tune, why do you think another app that compares tuning an M2 with 4th and 5ths (or visa versa?) with tuning CM3 would be realistic either?

Those beatrates are correct. Remember G3C4 beats fast because (per your own instructions!) G3D4 is tuned pure and vice versa.

Redoing with 0.7 per your suggestion:
"tune C4 in G3-C4 4th at 1.4bps, then tune D4 G3-D4 5th at 0.7 bps."

Resulting error in M2 C4D4 is 1.24 cent.

Of course the correct P4 is closer to equal beating with the P5 than to beating twice as fast.

The "Beat Speed Difference Test" Mark put together puts an upper limit to our ability to disciminate beats by presenting interfering pure sine waves. Since what comes out of piano is less clean than that, accuracy will be less when tuning an actual piano. I suspect it is a lot less.

Someone will have to create a similar test with recorded real piano sounds to find out. It will of course depend on the piano, the range, and which particular interval you listen to. I find when checking M3 beatrate progressions (aurally or with a computer) there are usually some M3's with hard to hear (weak) beats.

Kees

Sorry, Kees, I misunderstood you:

I read:

"OK, if everything is perfect then (assuming no inharmonicity)
P4g3d4 =-1.3280 P4bg3c4 = 1.7706 (beatrates)."

To mean: theoretically the beatrates are... So let me re-respondt to:

"OK, if everything is perfect then (assuming no inharmonicity)
P4g3d4 =-1.3280 P4bg3c4 = 1.7706 (beatrates).

If D4 was a cent flat then we'd have

P4g3d4 =-1.6671 P4bg3c4 = 2.2229

Is that difference really audible?"

Those beatrates aren't what I would be listening for. I would be listening for the beatrates when the 5th beats half the speed of the 4th, but since we are talking theoretical the ratio would be 1:1.35 :

Theoretical: 0.89bps/-0.66bps
With D4 flattened 1.0 cent: 1.12bps/-0.83bps

A tuner might not hear it being wrong all on it's own, but compared to others, yes, I think they would, especially while actually tuning.




And maybe I still haven't had enough coffee this fine Monday:

'Redoing with 0.7 per your suggestion:
"tune C4 in G3-C4 4th at 1.4bps, then tune D4 G3-D4 5th at 0.7 bps."

Resulting error in M2 C4D4 is 1.24 cent.'

Are you referring to where I said:

"If the C4-D4 M2 is out of whack, tune G3-C4 4th at 1bps, then G3-D4 5th at 1/2 bps. Otherwise just tune G3 so the 4th is about twice as fast as the 5th."

If so, the intent is to leave C4 where it is ala WBW and tune the 5th to 1 bps and the 5th to 1/2bps (or 0.7bps with no iH).

I get an error of -0.35 cents on the C4-D4 M2.



But the real intent of the exercise is to see how SBIs can be used to tune and evaluate an M2, not what beatrates would be theoretically perfect. I suspect that change in the 5th's beatrate from 0.7 to 0.5 makes things easier to hear.

OK, I read your sentence in another way.

"Tune G3-C4" is ambiguous and I still don't know which note you want to move when you write "then G3-D4 5th at 1/2 bps".

Anyways, theoretically G3C4 is closer to equal beating with G3D4 than it is to beating twice as fast so would it not be better to make them equal beating in your example? I guess you say inharmonicity changes this but I'm not sure.

For example on a simulated Steinway D tuning, using 4:2/6:3 equal beating octaves I get beat rates

On Heintzmann upright with 4:2 octaves I get

With 4:2/6:3 octaves I get

where now the P4 is more than twice as fast as the P5 but this is not a good octave. Something in-between 4:2 and 4:2/6:3 (which sounds good) produces

with P4 close to twice as fast as P5. So it (get P4 to beat twice as fast as P5) seems to work for the Heintzman but not for the Steinway.

Kees

Kees:

Someday I may tune an SSD. I'll let you know then how it works out.

In the meantime, if you give this exercise a try let me know what you think.

Prout and others:

Have you had the chance to experiment with Spear? Have you come to any conclusions after seeing the notes in Spear?

More generally, has anyone else been experimenting with the Spear program to look at notes? I know that Olek used it a bit, but I haven't seen much discussion. Seems valuable, assuming that it's accurate. Does it raise some questions about things that have been discussed in this thread about tuning to SBI versus RBI? Ie:

1. Is the bandwidth of a partial a bandwidth of simultaneous freqs, or does the fundamental drop out earlier than we might expect on some notes, making other "near-partials" more audible, so that they show up in other spectral analysis programs as having more amplitude than they actually have? These programs take the entire time of the recording of a note, and then graph the relative amplitude of all of the freqs over that time. In other words, in a ten second recording, if one freq was loud at the start, but vanished midway through to be replaced suddenly by another single freq at the same amplitude as the first, the graphed result would show that the recording consisted of two freqs of equal amplitude. The situation with a piano note is of course much more complex, with beats causing some partials to rise and fall in amplitude, etc, but the same averaging problem appears.

2. Does it offer the opportunity for greater precision in choosing whether to tune to the start or the end of a note? I avoid the phrase "tune to the attack," since Spear is sometimes showing a distinct change in partial amplitude well after the attack transients die. I'm not sure if these changes correspond with a given decay stage--the decay of the amplitude of the entire note.

Should I start a separate thread for using Spear? Jeff and others--I don't mean to interrupt your discussion. Analyzing notes in Spear is relevant, but it takes things in a different direction.