Well, here is a demo video for what it's worth. It's not something I often do and I don't have very good equipment. This is a Church's Yamaha P-22:

https://youtu.be/_bwsPWGTBkw

Here is the sequence:

Tune C4 to pitch reference
Tune F3 to C4 as a P5 about 1/2 bps
Tune A#3 to F3 as a P4 about 1 bps
Tune G3 to C4 as a P4 about 1 bps
There is now the m3, G3-A#3 that will be used a reference later on.

Tune B3 to G3 as a M3 beating about 8 bps
Tune F#3 to B3 as a P4 about 1 bps
Check the progression of the resulting chromatic M3s of F#3-A#3 and G3-B3. Adjust both notes of the P4-F#3-B3 up and down together until the M3s are in proper progression, perhaps by "shimming" one against the other. DO NOT adjust G3 nor A#3. These will still be needed as a reference.

Tune G#3 as the middle note in the pair of Cm3s. Strive for a 5:6 progression between F3-G#3 and G#3-B3.
Tune A3 as the middle note in the pair of Cm3s. Strive for a 5:6 progression between F#3-A3 and A3-C4.

All 8 notes are now tuned. To determine if the chosen beatrate of the P4s was correct for the given beatrate of the P5, listen to the progression of the M3s and m3s. If the P4 beatrate was too fast, the m3,G3-A#3 will be too slow and the M3s F#3-A#3, G3-B3 will be too fast when listening to the progressions with all the other m3s and M3s.

Rinse and repeat.

In the video, I first made the 5ths and 4th pure to demonstrate how the sequence will show an error when the 4ths are not tempered for how the 5th was tempered.

I then tempered the 5th, but not as much as I really should have, and started the sequence again. At about 10:00 I realized the 4ths weren't right for the 5th by the progression of the RBIs, just as the sequence is supposed to show. So I retuned the 4ths with more tempering. After the 8-note temperament was completed, I then expanded the temperament upward with 4ths and 5ths. Then when I checked the F3-F4 octave, it was a little wide of 6:3, not all that great. I adjusted a couple intervals in the expansion to get the progression better, but never did get the D4 quite high enough.

The one thing I will need to work on is hearing the 5:6 cm3 ratio better. And, yeah, I should have tempered the 5ths more, but I guess that is part of doing a demo!

I like the idea of splitting an established tritone to get the middle note of two Cm3rds.

Doing the math, if the cm3s have the same beatrate, the error is 1.4 cents, so with a little practice we could expect a normal error to be no more than 0.7 cents, listening to just that one cm3. but when you start listening to the chromatic cm3s, then you can get much better accuracy.

Working with this sequence some more, I am seeing that if the P4s are wrong, but not TOO bad, you can fudge things to get either the m3 progression or the M3 progression, but not both. That is what, I believe, makes it bullet-proof. Of course you need to hear the progression...

Splitting the tritones into cm3s might work for aligning ladders of CM3s. Like if you have a ladder starting on F and another on G, you could tune the G# within the F-B tritone and the A# within the G-C# tritone. This would give you four bi-chromatic m3 to compare to each other in order to see how accurate the CM3 are.

This might be digressing a bit, but for splitting a given octave into two ET tritones can you think of a tritone test?

It's a 7:5 interval, 17.5 cents tempered. Even with a test, which I guess would have something to do with an m7, it wouldn't be very accurate. Probably be better to go with a beat ratio of 1:1.4 or about 2:3.

Very interesting approach, Jeff! I look forward to experimenting with this. I've never really tried using m3'ds much in temperament tuning - it's a compelling idea, especially used as contiguous intervals.

My last tuning today, I found myself using more 4ths and 5ths. It's always good to mix things up a bit and break out of old habits.

I've thought about this question. I'm not sure what you mean. Is there any test for splitting an octave evenly into any number of RBIs, e.g. is there a test for evenly progressing CM3s, other than trying to adjust the beat rates as evenly as possible?

In the case of F3-B3 vs. B3-F4, you'd have to compare the lower tritone's beat rate (at D#6) with the upper tritone's (at A6). Would those really be audible?

Conversely, in terms of test notes, the highest note (common fundamental?) with both tritone notes in its partial series lies two octaves and a m7 below the lower tritone note. In the case of F3-B3 this would be G0. Disregarding for a moment that this is out of keyboard range, I was thinking of the following test:

Compare G0-F3 (beat rate 1) to G0-B3 (beat rate 2) and ascertain the ratio between the two. Then compare C#1-B3 to C#1-F4 and again ascertain a ratio. Then compare the two ratios and make them equal to set B3 at ET pitch. (Hope this makes sense.)

To split an octave into two tritones:
Tune A4-A3-A2 and insert C#3, F3, C#4 and F4 as rising M3s.
[Test A2-C#3, A2-C#4, C#3-F#, C##-F4, F3-A3, F3-A4, A3-C#4, C#4-F4, F4-A4 for progression.]
Tune F#3 as pure fourth to C#3 and E4 as pure fourth to A4. Check with 3rd/6th test.
Check the F#3-C3$ and A3-E4 fifths, they should be virtually pure, or extremely close. If not, you have a scaling problem.
Tune B3 as almost equal beating between F#3 and E4, a little faster on the top fourth. It will be "double beating" because of the un-tempered fourths.
Tune F#3 and E4 to produce a chain of equal-tempered fourths C#3-F#3-B3-E4-A4.
If all is accurate, you have placed B3 in the middle of octave F3-F4.

A simpler version is to tune progressive M3s, F3-A3-C#4-F4-A4, then pure fifths, F#3-C#4 and E4-A4, then tune B3 between F#3 and E4 as "double-beating" fourths, then temper the pure fifths and fast fourths to familiar beat speeds, then proceed within the octave.

Wandering off-topic, if you have tuned the extended two octave thirds and fourths, you can add B2, E3 and F#4. This allows testing for resonance in the slow beating intervals, comparing fourths and fifths across the midrange, and provides reference points to insert further notes, for example D3 can be tuned as a minor third between B2 and F3, testing as a P12 with A4a and comparing as m3 with C#3-E3 and M3 with C#3-F3. Add D4 as an octave, and with similar neighboring thirds tests.

This is kinda interesting, although I doubt they can be tuned accurately. Theoretical TTs are 600 cents and 17.5 cents wide of 7:5 just intonation. The beat is audible.

Why not tune 2 octaves and 4 stacked TTs like A2, D#3, A3, D#4 and A4? Adjust the D#s for a proper 2:3 progression of CTTs.

Or start with an A3-A4 octave. Tune the left string of D#4 as a beatless TT to A3 and the right string as a beatless TT to A4. Then tune the middle string to beat equally with the left and right string separately, about 3 bps?

Jeff-
I like your idea, I'm gonna try it!
Ed

What a great idea about taking advantage of beats between separate strings of a unison.

When trying this I find that going up a beatless tritone and down a beatless M3rd gets you very close to a correct tempered ET M2nd.

Isn't playing with tuning fun!

That's a great idea. I ran this with IH and even in the worst case of a spinet resulting D#4 is only 0.3 cent off, of course assuming you can accurately tune a pure 7:5 tritone.

Kees

I guess the test note would be the M2 below the lowest note, so G3A3 8:7 beats versus G3D#4 8:5 beats. I can't do it, but maybe it's possible?

Kees

Tune 2 perfect tritones AND two equal beating unisons. The killer is how much the tritone is tempered.

Maybe something could be done with ghosting, but what would it tell you? It is a very fast interval. The TT A3-D#4 is 15.5 bps. So you do a test and say, hmmm, this interval beats only 12 bps faster than that one. The TT is not tempered enough! I suppose a test could be used for initial pure TTs. Then you could set the test note to whatever speed you choose, and tune for a pure interval.

But I encourage others to try it and let me know.

I tried it using F3-F4 as the octave. My result on one try was not perfect, but showed the possibility. The beating is not easy to hear, but could be practiced In this case the upper beat is at the same level as F4-A4, but there is a lot of noise and a weak beat. My wife thought I'd gone crazy, and I'm not sure it's something customers would appreciate.

OK, back to the 8-note temperament. I really like it but want to try to be objective. Here are the pros and cons that I see and am asking for your input. Please, lets just talk about this temperament sequence, rather than get into how others work, let alone non-equal temperaments.

Pros:

Spans a small number of notes resulting in having chromatic intervals to judge the temperament very quickly.

Spans a small number of notes allowing the temperament to be set just above any break in scaling while avoiding higher RBIs that beat very rapidly.

Ensures that the proper ratio of the P4s and P5s beatrates is used.

The direction of error of the P4 to P5s beatrate ratio is indicated by which way the RBIs are non-progressive.

Any "cheating" to try to make the M3s progressive will result in the m3s being more unprogressive and visa versa. It is self checking.

Cons:

Is unsuitable for tuning across a break.

Requires the ability to tune a properly tempered P5 and consistent P4s.

Requires the ability to judge progressive m3s which are a bit faint and a bit fast.

It will take longer to complete unless you have the ability to tune cm3s at a ratio of 5:6.

Expanding the temperament must be initially done with P4s and P5s.

The octave type is not determined ahead of time, and may not be what is preferred after expanding the temperament.

Other Pros and Cons?