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Jim:

When using untempered fourths, there is less chance that the beat ratio between the fourths is not correct for the piano and cause an error in the chromatic M6 check of the CM3s.

My understanding of the workings of the Baldassin-Sanderson sequence is that there is only one possible set of frequencies (for a given set of initial CM3s) where the fourths will all beat the same and the other M3s will beat progressive. The beat rate of the fourths can be expected to be different depending on the scaling and the stretch, but for a given iH curve and initial set of CM3s there is only one beat rate for the fourths that will work. Of course how well a particular tuner can do this on a particular piano is where the rubber meets the road!

Really?!

Take this example of the contrary:

To set A4 exactly to 440 hz. we do not tune a pure, beatless, unison: A4-fork, but we tune instead an equal beating F2-Fork / F2-A4 M17th, which is a lot more accurate.

Another example is given by the tuning of octaves. We usually approach the octave by tuning it beatless, then refining it with tests by comparing beat rates of certain meaningful intervals, I.E. M3-M10, m3_M6, M3,-M17, etc. Here again beat rates are more accurate than purity.

In general, it is more accurate to tune an interval making it to beat at some given rate by comparison to another interval( i.e. equal beating, faster than..., slower than..., etc.) than making it beatless, because the margin left by this purity or beatless character is really very large.

Gadzar:

I agree that some speeds of beating are easier to detect than others. It also depends on the interval. Nobody suggests tuning the strings of a unison to be equal beating to another interval. Why? I would say because a unison is unambiguous. Unless there is a flaw, all the partials match. Other intervals like the fifth and especially the octave can be ambiguous. They can have more than one beat. The fourth is less likely to have more than one beat because of the faintness of the 8:6 partial match.

But what I am talking about here is using the fourths to produce a pair of chromatic M6s in order to refine the CM3s. It does not matter if the fourths are just or not, but they must be the same width. It is more likely to have them the same width by being beatless. It may not be know what beat ratio they should be to each other, so having them beat could cause an error.

Assuming that I have the two M6s then, what comes next? I appear to be missing a mental leap which lets me use these for CM3s refinement.

Jim:

If the lower M6 beats the same speed or faster or a lot slower than the upper M6, then there is room for improvement in the CM3s. Progressive RBIs that are 1 semi-tone apart require much closer tuning than RBIs that are 4 semi-tones apart.

Alright, it's a parallel interval progression check, I get it. And it's nice, for several reasons if I'm using it early on in setting my M3 ladder:

• I can use an M3-M6 check to see they're beatless, for accuracy.
• They involve two notes (E3, D4) that haven't been set yet (if you're using the "let the piano tell you" technique) and the other two notes in the test intervals have already been set. So the two M6ths are "high confidence" intervals.
• One of the M6ths is off of C#4, which while setting an F3-F4 CM3 ladder is the first note adjusted that sets a "good" M3rd width.

So if you're having trouble with that "let the piano tell you" technique, this would be one more indicator to check against.

Good one -- thanks, Tooner!

And you're right about this. When I mess up my CM3rds ladder, it's typically because I didn't estimate "middle" as well as I should have, between two other M3rd beat rates.

Jim:

I really should mention that I don't use this sequence, just happen to know about the check. I believe that an evenly progressive ladder of CM3s is not appropriate for challenging pianos, which is most of my business.


This same check (sometimes with m3s) can be used for the tempering of the fourths and fifths, once the CM3s are really set, especially when the fourths or fifths are tuned to all notes of the initial CM3.

I really don't see your point. How can you improve CM3 by introducing two more 4ths in the equation? If the beat rates of the M6s are not in a good progression it means nothing about CM3s, you may have misplaced your 4ths! The tests you make to set your fourths are the beat rates of M3s and M6ths; I don’t see why they would be more reliable than the tuning of CM3s. The fact that the M6 are chromatic doesn’t add precision at all. They may be equal beating or in an inverted progression without notice!

Isn't it easier to tune the CM3s progression directly?

The question here is:

Why don't you trust the setting of CM3's, all by itself?

All the tests you can imagine to check the CM3s make use of the same skills, estimations and appreciations needed to set the CM3s, and need the tuning of additional notes, subject to more errors, so how can it be more reliable than the direct estimation of the beat rate of M3s?

I have measured with my ETD and you can hear a 0.3 cents deviation in the tuning of F3 with easy. I’ve posted the procedure and results here in PW. And with the same accuracy you can then tune C#4. The tuning of A3 and F4 is by definition the width you select for the octaves in the temperament region. That gives the required precision to tune the CM3. No more tests or refinement are required. It wouldn’t be perfect, of course, but it would be accurate enough to set a good temperament. Trying to make corrections at this point of the sequence, in the setting of the CM3, is to waste time and efforts. There is the last step in the sequence: refine, which is best suited and placed to strive for perfection.

Once you have tuned all the notes between F3-A4, or even best and more suitable: the hole midrange C3-C5, then you can refine by playing chromatic runs of M3, P4ths, P5ths, M6ths, 8ves, M10ths and P12ths. If you find an uneven interval you can use tests of contiguous 3rds, 4ths and 5ths to detect the note(s) in fault and make corrections.

Gadzar:

When I tune an SBI, I can hear the slightest change in pitch because the beat rate is slow, or in the case of just SBIs, zero. I do not hear this when tuning RBIs.

Lets see, 0.3 cents at the fourth partial of the A3 is about 1 beat every 7 seconds. Do you think it is more likely to hear that sort of difference with an interval that is at or less than 1 bps or one that is at or more than 8 bps? Do think it is more likely to hear this difference between intervals with a ratio of 4:5, as in CM3s or between intervals with a ratio of 15:16 as in chromatic intervals?

But even so, is 0.3 cents enough accuracy to guarantee that all RBIs are progressive? As a proponent of CM3 tuning, you should be able to answer this.

But really, there are so many tests and checks available. Why not just let everyone mention what they prefer, and why? I see no reason to be dogmatic.

Tooner, I think how well each of us can discern changes in beat rates in both SBI's and RBI's has much to do with the time spent listening for them. Coming from a background more in line with a modified Braid White temperament I pretty well found a M3rd in the F4-A4 region almost useless, kind of like a revving chainsaw. Some people find it useful as the upper end of CM3rds and such but perhaps they spent a lot of time working in the +12 bps range. As a chromatic progression I think anything above 12 bps is not reliable....for me at least.

Well said!

Tooner,

Of course I can answer.

When tuning CM3s you can set these five notes (F3, A3, C#4, F4 and A4) in a way that every major third beats faster than its contiguous lower major third. And you can easily hear a deviation of 0.3 cents in the tuning of F3. Not because you can estimate its beat rate all by itself, but because of the way the progression of CM3 is affected by such a deviation. Remember that when you move F3 you have to readjust F4, so the beat rates of the three M3s: F3-A3, C#4-F4 and F4-A4 are all affected simultaneously and that changes the hole progression in a way that lets you detect the right spot for F3 with an acuracy of 0.3 cents.

That precision in the tuning of the CM3s makes unecesary any further testing. With these five notes tuned with that acuraty you can tune the rest of the temperament octave or tenth, and then you can refine the tuning in order to have an even chromatic progression of all the intervals (M10ths, 8ves, M6ths, P5ths, P4ths, M3rds).

BTW,

Do you think 0.3 cents is not enough precision?

Gadzar:

We can look at this both subjectively and objectively.

Subjectively, you seem to be able to hear a very slight difference in the progression of CM3s. Objectively, a change of +0.3 cents on C#4 of a perfect, theoretical set of CM3s will change the CM3 ratio from 3.97:5 to 4.15:5. That is not very much.

Subjectively, I do not hear small differences in the progression of CM3s. I do hear any slight difference in SBIs, though. And I can hear whether chromatic RBIs are in even progression easier than I can hear CM3s. Objectively, it requires greater accuracy to tune progressive chromatic M3s than it does to tune progressive CM3s.

No, 0.3 cents is not enough accuracy to guarantee progressive M3s and M6s, although progressive M3s and M6s can be tuned with errors greater than 0.3 even up to 0.9 cents. (I think this is part of the reason for the 0.9 cent allowable error on the PTG exam.)

I am pretty sure that you understand math enough to follow this. A M3 is 13.7 cents wide of just. If a sample M3 beats 8 bps at 13.7 cents, it will beat 16 bps at 27.6 cents. Twice as fast for twice as wide, same as if the M3 was an octave higher. This means that the 12th root of two can be used to determine how much a M3 (or any other interval) can be changed before it beats the same speed as the interval chromatically higher. So for the sample M3 to beat the same speed as the next higher M3 (8 bps * 1.059 = 8.472 bps) the interval would need to be widened to 14.5 cents, or 0.8 cents wider than theoretical (13.7 * 1.059 = 14.51 cents). But if we start with perfect theoretical ET and flatten the lower note of a M3 0.8 cents and sharpen the upper note 0.8 cents, then the interval will have a beat speed as fast as the interval two semi-tones higher. A tolerance of +/- 0.4 cents is not accurate enough either, because one M3 could be widened by 0.8 cents and the next narrowed by 0.8 again causing the beat rates to be unprogressive. Although a tolerance of 0.2 cents can allow some chromatic M3s to beat at the same speed, they would not be unprogressive. So I believe to guarantee that all M3s and M6s to be progressive requires a tolerance of +/- 0.2 cents, although +/- 0.4 cents will usually result in progressive M3s and M6s (but I have not used statistical analysis to work out a bell curve.) If you choose, you can work out the allowable tolerance for a progressive set of CM3s in the same way.

The reason I chose M3s instead of M6s for calculating the allowable tolerance is because M6s are 2 cents further from just intonation than M3s, and do not require as close a tolerance for progressiveness. The closer to just intonation, the closer the tolerance that is required. This is an objective reason for using SBIs. Since they are 7 times more just than M3s, when they are out of progression, they show an error seven times smaller than M3s would. In a typical piano, this is a smaller tolerance than the pin and string can be reliably set.

Once I understood all this, including my personal abilities and limitations to hear RBIs and SBIs, the choice for me was obvious. I choose to tune with intervals that have a tolerance greater than I can actually tune. It is like doing carpentry work that requires +/- 1/8 inch tolerances. I want a tape measure with 1/16 inch marks, not ¼ inch marks.

You understand nothing!

You seem to ignore what I say, as always.

I'll put it in numbers to show you how a deviation of 0.3 cents in the tuning of F3 can alter the progression of the CM3s.

I'll make the calculations without iH. I leave to you the work to put iH into play with your brand new database.

The frequences in hz. are:

F3=174.61
A3=220
C#4=277.18
F4=349.23
A4=440

The beat rates in bps of CM3s are:

F3-A3 = 6.93 (4*220-5*174.61)
A3-C#4 = 8.73
C#4-F4 = 11.00
F4-A4 = 13.86

The ratios of the progression:

A3-C#4/F3-A3 = 8.73/6.93 = 1.26 aprox 5:4
C#4-F4/A3-C#4 = 11.00/8.73 = 1.26 aprox 5:4
F4-A4/C#4-F4 = 13.86/11.00 = 1.26 aprox 5:4

Now if we detune F3 by an amount of 0.3 cents flat we have:

F3=174.58
A3=220
C#4=277.18
F4=349.17 (F4 is adjusted to 2*F3)
A4=440

F3-A3 = 7.08
A3-C#4 = 8.73
C#4-F4 = 10.76
F4-A4 = 13.86

A3-C#4/F3-A3 = 1.23 between 6:5 and 5:4
C#4-F4/A3-C#4 = 1.23 between 6:5 and 5:4
F4-A4/C#4-F4 = 1.32 almost 4:3


And now you see that the ratio of the latest is 4:3 instead of 6:5-5:4 of the two preceding ratios. That is a very easy to hear difference: F4-A4 beats 4 times for 3 beats of C#4-F4!

In simple words F4-A4 beats too fast and you can distinctly hear the uneveness in the progression!

You need no much training in apreciating the difference between 5:4 and 4:3 ratios. Remember that you are hearing and comparing these ratios, you are not trying to estimate a single 5:4 ratio, you are evaluating the whole progression.

Of course it is not possible to distiguish the change in the beat rate of F3-A3 from 6.93 to 7.08. But after you adjust F4 and you play the four CM3s, you can easily detect the uneveness at F4-A4 which will beat too fast compared to the other CM3s.

In the same way, if F3 was tuned 0.3 cents sharp then you will find that F4-A4 will beat too slow in the progression.

Could you please borrow an ETD, or install the trial version of Tunelab for free in your computer and give it a try?

You will change your mind if you do!

You'll see that your ear is accurate enough to detect that 4:3 to 6:5-5:4 ratio difference and you can tune F3 with an accuraty of 0.3 cents.

About 0.3 cents being enough acuraty, please read my words: 0.3 cents in the tuning of F3 from A3. That means just that. A3 is tuned to A4 and F4 is tuned to F3 with the precision you can achieve in tuning your octaves.

That doesn't mean you can have F3 0.3 cents flat with A3 0.3 cents sharp, that would be a "mathematical" conception of acuraty, based on random errors, which is not the case.

And you seem to ignore consistently that this is only the beggining part of the sequence, after that you will refine the tuning of your temperament and correct notes testing all other kind of intervals.

Gadzar:

I posted: “Objectively, a change of +0.3 cents on C#4 of a perfect, theoretical set of CM3s will change the CM3 ratio from 3.97:5 to 4.15:5. That is not very much.”

Objectively, our math agrees: 4.15:5 is the same ratio as 4.98:6.

Subjectively, our opinions differ. I do not think this is much of a difference. And btw 3.97:5 is the same ratio as 4.76:6, not far at all from 5:6 and also the same ratio as 3.18:4 not far at all from 3:4.

Although I used to do things a little differently, I now prefer to use each new note that I tune in order to refine the others already tuned, and not wait until the end for refinement. A fourth and fifth sequence is great for this.

I consider much of this to be in the realm of personal abilities and preferences. I do not feel a need for us to agree. Do you?

No, I don't feel the need to agree.

But, if you can not hear the difference between CM3s with a beat ratio of 4:3 instead of 5:4, then I can not see how you can tune a piano by ear!

I wish I could hear a piano tuned by you in ET.

Gadzar:

Progressive, chromatic RBIs is the proof of ET, not CM3 ratios. How they are achieved does not matter.