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Mark, you dropped a decimal place. To increase the beatspeed of G#3-C#4 (0.94bps) by 3% only requires raising C#4 by 0.05452073426

Cents = 1200 x ln[f2/f1] / ln2
f1 = 277.182631
f2 = 277.2699362
cents = 0.5452073426

Hmmm, I am not sure where you are losing me...

Try this yourself, instead. Determine how many cents the middle note of contiguous 4ths would be off for the 4ths to beat equally. Compare this to CM3s. Would this define the "window" you are looking for?

Maybe you are asking the wrong question. For some reason I am thinking about the Wright brothers. Instead of asking "How can birds fly?" They asked "How DO birds fly?" Are you asking how SBI tuners actually tune, or are you focused on how SBIs CANNOT be used to tune?

Jimminy Crickets!

You have raised the fundamental 0.09 hertz, right? The third partial is now 0.27 hz faster, right? The G#3-A#4 P4 started at 0.93bps, right? It is now is now beating at 1.20bps, right? The increase in beatspeed is 29 percent! ((0.27/.93) x 100)

OK, yes I am frustrated. How did you come up with f2 = 277.2699362? That must be where the "accumulating" error started.

OK Folks, I think I can explain this clearer:

We hear a beatrate change 6% and it now beats the same speed as the interval one semitone higher. Cool!

But why was there a beatrate and why did it change?
There was a beatrate because the interval was tempered and it changed because the tempering increased.

How much did the tempering increase?
Well, 6%, of course.

But what is that in cents?
6% of however much it was tempered in cents.

Can you give an example?
Sure! A P4 is tempered by 2 cents. In the temperament octave a typical P4 beats 1 bps. If you increase the beatspeed 6% it then beats 1.06bps. This is because the tempering increased, too. It is now tempered 6% more and is tempered 2.12 cents.

But what about RBIs?
Ah, RBIs! An M6 is tempered 16 cents. In the temperament octave a typical M6 beats 8 bps. If you increase the beatspeed 6% it then beats 8.48bps. This is because the tempering increased, too. It is now tempered 6% more and is tempered 16.96 cents.

Huh? Why wouldn't the same percent change in beatrates result in the same change in cents?" Hmmm, if the interval wasn't tempered at all, the beatrate would be zero. If you increased this beatrate by 6% how many cents would you expect the pitch to change?

Uh, it wouldn't change at all. Any percent of zero is still zero. But isn't an equal change in beatrate mean an equal change in cents?"
Only if the intervals are tempered equally. Like if our M6 (tempered by 16 cents) was an octave higher (beating at 16bps) and the beatrate was increased by 6% (to 15.09bps), this interval is tempered by 6% more, just like the one an octave lower (at 16.96 cents.)

So you mean for the same percent change in beatspeed an M6's tempering, in cents, will change 8 times as much as a P4's???
Yep!

And for the same amount of change in tempering, in cents, the beatrate of a P4 will change 8 times as much as an M6's???
You got it!

But then why do tuners use RBIs to tune with instead of just using them as checks?
Well, uhm, gee... Heck you think I know everything???

Hi Jeff,

I really want to spend my time exploring the best way to get accurate ET. If P4 have a superior method, I want to explore it.

I need to understand all methods if I am to teach them.

For example, personally I do not consider a P4/P5 sequence to be as accurate as a RBI sequence, but I realize that people need choices, so my basic tuning course teaches a sequence of P4/P5 that I call "Peanut Butter on Toast".

But my intermediate and advanced courses introduce and expand on RBI, because i "think" they are more accurate. But I want to be sure. If there is a superior, or even similar sequence that uses P4/P5, I want to explore and define it, and use it in my courses.

The P4 ladder you speak of, does it have two restrictions, like CM3?

i.e. F3 goes to F4 two ways
1) Octave: F3F4
2) CM3: F3A3-A3C#4-C#4F4

P4 ladder
F3 goes to ?
F3A#3
A#3D#4
D#4G#4
G#4C#5
...
There's no 2nd restriction within which to set the CP4

Here are my calculations:

Simple calculations from ET
A3C#4
A3 = 220Hz
A3(5) = 1100Hz
C#4 = 277.182631Hz
C#4(4) = 1108.730524Hz
A3C#4 = 8.730524bps

Then tune C#4 from A3 (M3) with 3% error

A3C#
Let C# of frequency, f, be the C#4 that has produces a BSD of 3% error
A3(5) = 1100
C# = f
C#(4) = 4f
A3C# = 4f - 1100

3% of 8.730524 = 0.26191572bps
A3C# = 8.730524 + 0.26191572 = 8.99243972bps

but
A3C# = 4f - 1100

so,
8.99243972 = 4f - 1100

solving for f,
4f = 8.99243972 + 1100
4f = 1108.99243972
f = 1108.99243972/4
f = 277.2481099Hz

Now tune C#4 from G#3 using P4 with a 3% error.

From ET:

G#3C#4
G#3 = 207.6523488
G#3(4) = 830.6093952
C#4 = 277.182631
C#4(3) = 831.547893
G#3C#4 = 0.9384978bps

G#3C#
Let C# of frequency, f, be the C#4 that has produces a BSD of 3% error

G#3(4) = 830.6093952
C# = f
C#(3) = 3f
G#3C# = 3f - 830.6093952


3% of 0.9384978 = 0.26191572
G#3C# = 0.9384978 + 0.26191572 = 1.20041352

G#3C# = 3f - 830.6093952

Substitute and solve for f:
3f - 830.6093952 = 1.20041352
3f = 831.8098087
f = 277.2699362

The rest was done using the formula for cents. But you can see that the f from M3 with 3% error is closer than the f from the P4 with 3% error.

I like that. I have a feeling that it is possible.

For example, after tuning the temperament using RBI, I look for chromatic P4 that do not increase in speed, and then check them with M3 to find out what needs correcting.

However, I have yet to find a P4/P5 method that produces accurate ET from the first interval.

Like I said before, my sequence results in little or no refining needed after the first pass. Even when checking P4 they often seem to fit.

The problem I have now is, after recording and measuring my temperaments, I find that the beat speeds do not progress evenly.

However, when checking back with the original test that set an incorrect note, I can hear the difference, or I can't.

If I can, then why didn't I set it correctly during the first setting? (I did but it drifted?)

If I can't, then the point is moot. It's as good as I can get it.

The real power of this sequence and method, for me, is that it is better than I am right now, meaning, I can continue to use it to tune more accurate ET by improving my BSDS.

Sorry, I'm Canadian. Was that Nixon? 30+ years ago?

Thanks Chris. Nice. Years tuning aurally?

Thanks Kees. Hope the pheasant came out great.

How many years have you been tuning aurally?

Since 44 years ago, but gone through a few decade long hiatus' without touching a tuning hammer, and only full time for 5 years.

I'll put you down for 30+ years. Your marks certainly show a refined ear.

No!

3% of 0.9384978 = 0.0281549
G#3C# = 0.9384978 + 0.0281549 = 0.9666527

You misunderstand why I brought CP4 up. It was not to demonstrate part of a sequence. I brought it up because you were unable to follow what I was saying and I didn't know where I was losing you. It was a mathematical exersize for you to better understand the beatrate sensitivity of P4s by comparing the maximum error of CP4s with CM3s.

Assume F3 and D#4 are correct. How many cents sharp will A#3 need to be to have the two P4s beat at the same rate?
Compare this to CM3s: F#3 A#3 and D4.

"However, I have yet to find a P4/P5 method that produces accurate ET from the first interval."

And from the rest of your post, you haven't found any other method that produces accurate ET from the first interval.

But you would only be satisfied with a 4ths and 5ths sequence that does???

But didn't you say you want to understand 4ths and 5ths sequences? Your actions do not show it. When there is a Topic about it you immediately spout off about how great your RBI sequence is. (Uh, the one that you don't seem to get to work...)

Mark, if you are really interested in SBI sequences, there are two fairly current Topics on it. Go there, ask respectful questions and try to understand the answers.

If you really want to know you will google it.

I appreciate your checking my work but not your sarcasm or talking down to me. You are usually more polite.

Consider a slightly different approach to what a cent is.

The fundamental frequency f(n), in Hz, of each note in the equal temperament system is defined by

f(n)=440r^(n-49)

where
r=2^(1/12)


and where n=1,…,88,with n=1 denoting the lowest note on the piano, A0, and n=88 denoting the highest note, C8.

By definition there are 100 cents per semitone or 200 cents per two semitones, therefore, the number of Hz per cent, HpC, can be written as

HpC(f(n))=(f(n+1)-f(n-1))/200

which is the difference in frequency in Hz for two keys a whole tone (two semitones) apart divided by the number of cents between f(n+1) and f(n), which is 200 cents.

Combining the two equations gives

HpC=( 440*r^(n-50) * (r^2-1) )/200

= 2.2*r^(n-50) * (r^2-1)

= f(n)/200 ((r^2-1))/r

Plotting HpC vs. n gives




So, according to the formula and the graph, at A0 there are 0.0159 Hz per cent while at C8 there are 2.42 Hz per cent.

Approximately. Cents are logarithmic, so the frequency of one cent less than a given frequency is less than the frequency of one cent more than the same frequency. That 6% difference comes back again!