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For the curious, here is a quick summary of the Eric Nikiforoff temperament sequence I talked about here:

• Set ladder of contiguous M3rds, C#3-A4
• Tune F#3 between C#3-C#4 (4th/5th)
• Tune G#3 between C#3-C#4 (5th/4th)
• Tune A#3 between F3-F4 (4th/5th)
• Tune C4 between F3-F4 (5th/4th)
• Tune D4 between A3-A4 (4th/5th)
• Tune E4 between A3-A4 (5th/4th)
• Tune D#4 to G#3 and A#3 (opposing 5th/4th)
• Tune B3 between F#3-E4 (contiguous 4ths)
• Tune G3 to D4 and E4 (opposing 4th/5th)
• Refine

And there's a video of Eric demonstrating it here. I note, as the original article does, that the first and last steps are big ones and deserving of separate study.

Enjoy!

Thank you Jim.

For the tuning of the first step, I guess it is a variant of the tuning of F3-A3-C#4-F4-A4 contiguous major thirds used in many sequences: Randy Potter, Jim Colemann, Sanderson-Baldassin, Bill Bremmer, and many others. So it will be relatively easy to find a way to tune the contiguous major thirds from C#3 to A4 in the order given here.

But for the remaing of the sequence, how do you tune:

4th/5th
5th/4th
opposing 5th/4th
opposing 4th/5th
contiguous 4ths ?

Equal beating?
2:3 ratio?
fifths slightly slower than fourths? How much?
fifths and fourths beating at certain rates: 0.6, 1 bps for example?

How do you tune them?

Excellent question Gadzar

I just watched the video on Eric's tuning sequence and was thinking about the tuning of the 4ths and 5ths. I look forward to Jim's response.

Gadzar’s question is excellent. It relates to the problem of the relationship between octaves and SBIs for the piano’s scaling. When any one is changed the others have to, also. And this problem is rarely seen as a factor when favoring CM3 tuning, but always as a criticism of fourths and fifths tuning. I have decided that with aural tuning, tuning the best possible ET always requires making estimates and then refining the tuning, regardless of the sequence. But by choosing the tuning sequence, we also choose where these estimates are made and how we deal with them.

Reading Gadzar and Tooner's posts, I get the impression that you are waiting "with baited breath" that this sequence is a breakthrough in tuning theory, or something, which it isn't at all. The title of the article is "One, Two, Three! A Temperament Sequence that is Easy to Learn." It is firmly on the CM3rds "side" of the debate vs. fourths/fifths sequences, if that's the way you want to look at it, since the approach to octave widths and setting the foundation of the CM3rds ladder is the same.


Yes, and I have written up my approach as well, which is a mashup of many of those techniques, particularly including the Stebbins "let the piano tell you" technique. See the first nine unnumbered bullet points here.


The short answer in the Nikiforoff article is "P4s are slightly expanded while P5s are slightly contracted." The first six steps after the ladder are straight-forward, I think. Choose an octave in the ladder that is already tuned, then tune the two notes in-between that have 4th/5th and 5th/4th relationships. For the two other notes depending on the P4 and P5 relationship, the notes are tuned are the "same-direction" interval test to notes that have already been tuned. The contguous fourths are tuned equal beating.


Well, "it depends," and I don't think there's anything particularly new here. After the ladder, I tune 4ths a tad under 1bps, and 5ths slightly narrow, maybe 0.2 bps. Then I immediately switch to refinement using combinations of parallel, contiguous, and other interval checks.

I try to be aware of the "confidence" with which I've tuned notes: at the beginning of refinement, the ladder notes have higher confidence than the ones I've just tuned with 4ths and 5ths, and so that weights my decisions as I refine and decide which notes to move. Occasionally I will adjust my ladder notes during refinement if it seems the best way to make the smoothest beat progressions.


I agree. The Nikiforoff article does not "solve" this problem, it takes another cut at a good combination of these techniques, arranged to favor novices who are just becoming acquainted with what it means to remember a sequence.


What I like about this sequence, is the insight that by extending the ladder down one more "rung," it discovers a means to stay within the CM3rds approach, yet rapidly rough-in the rest of the notes so I can get down to refinement.

I've written before that I'm still faster with my old sequence, but I've been practicing this one, trying to get to the point where I'm fluent. Then maybe I'll have some better thoughts about how it compares.

Oh, and I forgot I wanted to make one more comment about this -- I think I would disagree with the italicized portion of your statement.

Go all the way back to the original Sanderson article, and I think you'll find that the two-octave temperament with a foundation in CM3rds is attempting to address that issue directly. You set the wide intervals first, compromising the octave widths as necessary to get good single and double beat rates, and then proceed from there.

Maybe someone like Bernhard Stopper would not agree that this is the best way, but I don't think you can say the problem is "rarely seen as a factor."

My contiguous fourths (and fifths) are not equal beating. I may not be able to get them to progress chromatically (due to the pinblock), but the contiguous SBIs should be progressive in the temperament. Otherwise, you may not have the fourth below a note beat faster than the fifth below a note.

The issue that I am addressing is not one of varying octave widths, and the need for a changing beat ratio for the ladder, but determining the correct beat rate for the fourths and fifths.

The Baldassin-Sanderson temperament sequence is the only CM3 sequence that I know of that does address it (and it takes nine notes to form the “mini” temperament) by tuning a series of fourths that all beat at the same speed. But when it gets to fourths that are contiguous, I believe they should be progressive and can be tuned that way. Chromatic fourths are much more difficult.

If it was not for this sequence, I would say never instead of rarely.

I think than both sequences, i.e. Sanderson-Baldassin's and Jim Moy's, address the setting of the fourths's (and 5ths) width.

I can not say the same of Nikiforoff's sequence as it tunes 4ths and 5ths without embedding 3rds simultaneously. Though, I've not seen the tests he uses to tune the 4ths and 5ths.


In the Sanderson-Baldassin's you first estimate A#3 as a fourth from F3, then M3 F#3-A#3 a bit (how much?) faster than F3-A3, then you estimate B3 as a fourth from F#3. This second fourth is supposed to be equal tempered than F3-A#3 but here the problem is what do we mean by that. Equal beating? A little, very little faster? How much? They are chromatic fourths and the difference IMO is not audible/tunable. Now you tune G#3 as a fourth from C#4. How do you temper this 4th? Supposedly same than F3-A#3. But again, does that mean equal beating? This time fourths are not chromatic, but 3 semitones apart. Maybe this time we can strive for a little faster beat rate!? Again, in my opinion the difference is negligible and untunable directly. Now we tune C4 as a M3 from G#3 a bit (how much?) slower than A3-C#4, then we tune G3 as a fourth from C4, same tempered than G#3-C#4 and again same questions: equal beating? smooth progression of fourths from F3-A#3/F#3-B3/G3-C4/G#3-C#4? Again I think this progression is not directly tunable. And now we finally have our test interval M3 G3-B3 wich must beat at an appropriate rate to have a smooth progression of all chromatic M3s from F3-A3 to A3-C#4, if this is not the case we have to adjust all the fourths and thirds consequently.

In this sequence the width of the fourths is determined by our decision of making them equal beating or gradually progressing from F3-A#3 to A#3-C#4 (if such a degree of refinement is possible).



In the Jim Moy's sequence we tune pure fifths from F3 and A3 giving temporary notes D3 and C4 and from these notes we place G3 to have to contiguous equal beating 4ths or we can tune 2 contiguous 4ths beating with a very little difference, it is up to us to decide the correct tempering of these contiguous fourths.

So in both sequences the width of the fourths is addressed, the tuning of the temperament octave is well solved taking into account iH as we check 8ves, 5ths, 4ths and 3rds.

Now, as I said, in Sanderson-Baldassin's we have tuned chromatic M3s with a bit faster rate, or in Jim Moy's bracketed hole step and chromatic M3rds. And in both sequences we have tuned equal beating or slightly progressing 4ths. These facts can be source of tiny imperfections in temperament. But in the final step, i.e. refine, we are supposed to correct any imperfection in the chromatic progressions. It is in this step where finally we will cope with all errors we may did previously in our sequence.

My personal preference is to tune a uniform progression of M3s and M10s and let the slow beating intervals fall into place. I am not inclined to tune a progression of fourths or fifths or octaves.

Maybe Sanderson-Baldassin's is more long to tune G3 and B3 (four 4ths and 2 M3s) but when it's done you have a nine note minitemperament to spread out to the hole piano. In the Jim Moy's sequence you go to tune G3 more directly (2 pure 5ths and a note forming 2 contiguous 4ths), but at the end you have only one note tuned. The advantage is that you make an enphasis in tuning bracketed CM3s.

Yes, an entirely valid point to bring up, though I don't think I've said anything regarding the correct sizing of the slow beating intervals. I've followed some of your discussions with Bill Bremmer, et.al., so I know you understand the approach the CM3 temperaments are taking. Given that, I think you'll agree that the Nikiforoff temperament doesn't speak toward that issue, it is simply another way of roughing in the notes between the ladder notes.

Perhaps because I have not reproduced the article in full, my summary unwittingly focuses on setting 4ths/5ths, in which case that's my fault, not Eric Nikiforoff's.

Getting off the thread topic, in a two-octave temperament, doesn't the selection of octave widths address your concern about the widths of fourths and fifths, though indirectly? The strategy, and major point of the two-octave temperament is to determine early, what the octave width tradeoffs should be given a particular piano's scaling. It follows that once you've done that, the 4ths/5ths widths fall out naturally, so long as you've correctly tempered the contained intervals (admittedly, no small task, which is why we all talk temperament so much). If you don't like what falls out, then on that piano perhaps different octave widths are appropriate.


The fact that it specifies equal width fourths in its intermediate temperament is not salient to the temperament's strategy. That part of the procedure is only a tactic to locate the note "in the middle" of one of the M3rds, splitting it into whole steps. In some descriptions, those fourths are only temporary reference points, to be re-tuned later. But it's a critical step, as anyone who has taken the PTG examination can tell you: perfectly smooth, parallel M3rds in no way guarantees your 4ths and 5ths aren't messed up entirely! (Which is how the piano is detuned before you take the exam.) I actually use a different technique for splitting that M3rd that I like better, which was summarized by Jim Coleman in the "Baldassin, Sanderson, Kimbell, Tremper" sequence.

One of the things I've noticed in different approaches to temperament is the speed with which people want to get to the refinement phase. Some want to just rough-in the first pass and then do multiple refinement passes with much tweaking and checking. Some want to make that CM3 ladder as "perfect" as possible, and take ensuing movements with equal care such that there is not much refinement to be done after the first pass.

To each their own. I think the same goes for whether we tune 4ths/5ths and check 3rds/6ths or vice-versa.

When I was learning the Sanderson-Baldassin sequence, I would read conflicting descriptions. Some would say 1 bps. Some equal. Some "the high fourth a wee bit faster."

I tried them all. And the resulting M3rd was never in a position with much more than a few cents accuracy, on average. Sometimes I'd hit it right on and jump up and down, and other times I was four or five cents off and the width of the third was wrong.

I probably just hadn't practiced it enough, but then I discovered the Kimbell-Tremper method of splitting that M3rd, my accuracy went waaay up. Within only a week's practice I was hitting it to within 1 cent consistently. As a result, I never went back to the "stacked fourths on both sides of a major third" method. (Then I discovered how to move the whole temperament down a M3rd and I was even happier.)

Right! Me too, I have had bad times tuning those 4ths and 3rds before geting a good G3-B3 M3.

So, I countinued my search of a more direct, less "try and error" sequence. Tough I've not tried it, the Kimbell-Tremper method seems to go straight-forward.

I am reluctant of what precision can be attained by tuning 2 pure fifths. There is a large margin left where a fifth is heard as "pure". You must try to tune both “pure” fifths exactly the same width.

I mean: to tune C4 from F3 you can tune a little sharp and then go down until it is pure. That will leave the 5th on the wide side of purity. When tuning D3 you must tune form the flat side up to purity in order to have the same width of the previos fifth tuned. Or visceversa: tune C4 from the flat side and D3 from the sharp side.

And you must also hear at the correct coincident partial and make the appropiate tests M6-M10 or m3-M3, in order to tune both fifths the same type: 3:2 or 6:4.

For me 5ths are not reliable enough.


That’s why I am using a sequence in which after tuning the ladder of contiguous M3s from F3 to A4 I tune a ladder of contiguous m3s: F3-G#3-B3-D4-F4-G#4, dividing the octave into 4 contiguous minor 3rds, just in the same way we tune the contiguous major thirds. This sequence is direct, has plenty of tests available from the beggining and there is no guessing and no try and error in it. Though in some pianos the tuning of m3s is not that obvious and you must use other intervals (which are often available) to solve some notes.

Here's an article by Jim Coleman, one in a series on #pianotech where he summarizes it. It doesn't match mine step-by-step because I've mirrored the ladder around F3, but the concept is the same:

http://www.ptg.org/pipermail/pianotech/1996-October/010308.html

And this is before he started calling the temperament sequence "Baldassin, Sanderson, Kimbell, Tremper." Maybe mercifully so


I agree, if you're just listening to the 3:2 directly. But I think I've had good accuracy with the M6-M10th check. Maybe I should go measure exactly how closely I'm getting though. Time for Tunelab.


Well, are you going to make us beg? Post it! I posted mine, so you post yours (But in a separate thread, please...)

I'm going to quote the money paragraphs from Jim Coleman's article here, just so it shows up on searches:


And that changed my life for the month after I read it :-)

Jim,

I have already posted my "Contiguous Major/Minor Thirds Temperament" here

I see it! And I'll try it. My fault that I missed it earlier, I unfortunately sometimes start ignoring threads depending on the level of bickering :-)

I find very constuctive and interesting to read and try others efforts and sequences to tune E.T. Even if I'll have to re-start from zero and quit what I was doing until now.

The alternative is to stick to a sequence, and remain sticked to it forever. We still see tuners that use the more than 100 years old Braid-White sequence or a variant of it and are affraid to switch to a more modern, actual, accurate and reliable method, like the ones we are talking about here.

If I ever find a better sequence than the one I use now, I will switch volontiers!

OK, I tried it, posted some fixes here, thanks Rafael!

After trying this, I can see the advantage of having the Cm3 ladder in parallel to the CM3 one. With the sequence I have been using for a while now, and posted earlier, even a small mistake in the "two-5ths to make contiguous 4ths procedure" can cause the "other" CM3rds ladder that is a whole step off from the first one to be "shifted" in the wrong direction, even though it progresses well chromatically. Same thing goes for the original Baldassin-Sanderson technique. For that reason, I have always looked to the earliest 4th/5th check I can make in order to make corrections.

But now after some reflection on the Nikiforoff temperament sequence, I am re-considering something I read in Kent Swafford's Every Which Way Temperament:


The simplicity of the Nikiforoff sequence gets me to the refinement steps faster, while still maintaining the robustness of the CM3rds approach. I think the combination of the two is very powerful. So it has caused me to refocus some of my study efforts on making my refinement techniques more efficient, rather than concentrating so hard on the temperament sequence.

The Cm3 technique appears to be very powerful, but it also looks like a lot of practice to make it efficient, and even then I wonder if it can be done "mindlessly."

Jim:

Excuse me if this has been mentioned before. After a set of CM3s is tuned, and then the fourths and/or fifths are tuned to the starting note, there is a chromatic pair of either M6s or m3s that can be used to check the accuracy of the ladder and/or the fourths and fifths.

Example: F3 A3 C#4 are tuned as part of a ladder. E3 and D4 are tuned to A3. The beat rate of M6 E3-C#4 can be compared to M6 F3-D4. If the E3 and D4 are tuned beatless this can be used solely as a way to refine the ladder of CM3s. And if tempered, can be used as a way to refine the fourths. When these test are continued for all the fourths and fifths for the F3’s and C#’s., there is a nine note mini-temperament that has sufficient cross checks to ensure that the fourths and fifths have the proper beatrate. I wonder if there is a number theory significance to the nine note mini-temperament. Baldassin-Sanderson produces one, and it is even significant with 4ths and 5ths sequences where a ladder of CM3s is produced with the ninth note that is tuned.

Sorry that I don’t have time to read all the posts carefully, just wanted to toss this in.

Yes, I follow this.


Could you explain this in further detail? To what are you comparing E3 and/or D4 if they are both tuned beatless to A3? I follow using the tempered 4ths, but I'm not sure what you are doing with the beatless ones.


I don't know. My impression has always been that the size of the 4ths in that sequence was not particularly important, except that they all needed to be the same so that both sides of the the resulting G3-B3 M3rd are positioned symmetrically. If it was not, then you compensated by adjusting your 4ths again. Which was a lot of work since there were four of them. And when you've got G3-B3, then the sequence goes back and re-tunes those 4ths anyway by bracketing M3rds.