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Has somebody come up with a pure 12th aural tuning sequence?

I am interested in responses to this question as well.

I guess you could start with A4 and tune a pure 12th down to D3. Then infill that middle 12th by some means. It would have to be equal temperament if all 12ths are the same.

Expanding the middle 12th with pure 12ths should be more straight forward.

To my knowledge, no one has presented a certain way to subdivide the 12th from A4-D3. Bernhard Stopper presented a class at the PTG Annual last summer, but it was not really clear, and he did not complete the temperament. One problem is that the 12th may not be a pure 3/1 12th, it may require a compromise with the 3/2 12th, depending on the inharmonicities of D3 and A4!

The tuning also requires a very precise octave size, so you can't just tune the twelfth and fill in the octaves.

Years ago Bernhard presented a way to tune using a 12th bridge and playing 3 notes at once. There was a certain "taste" to hearing the correct interaction of 5th and octave. Even so, Bernhard said it would take cycles of refinement to get it right, and he didn't complete it in the class.

One interesting difference is the modified inner third/outer sixth test, different form the familiar test. The inner Major Third is a half-step higher, and beats the same as the outer sixth, for example not F-G-B-D, but F-G#-C-D!

I tune pure 12th aurally. What I do is start with a pure 12th and then place the note a fifth above the lower note and an octave below the upper note where it sounds best. A rule of thumb is the larger the piano, the larger "type" of octave. A spinet will be a little narrow of a 4:2 octave, a large grand a little wide of a 4:2 octave. I then tune a temperament within the 5th, just 6 more notes. That sequence is a reiterative one where each step is more accurate than the last. With good stability, it is unusual to need to go through it more than twice. It has been posted and discussed here before, but not much interest was shown. This 5th temperament is then expanded upward by tuning to the 4th below pure, then raising the pitch until the beatrate of the fifth below is about halved from when the 4th was pure. If they both tend to be too pure, the inferred M2 is too narrow. If too tempered, then too wide. (Think about it...) Like any aural tuning, you tweak a bit as you go. As you progress up, both the 4ths and 5ths beat faster, but you want the beat speed of the 5ths to increase more. They will beat about the same rate when you get to the upper note of the original 12th. This goes along with the octave "type" getting narrower as you go up.

Hope this helps.

Interesting discussion...
There seems to be some mystique around pure 12ths that has made some believe that it has some special musical qualities that are desirable. Maybe I'm missing something. I've experimented with various degrees of stretch in tunings, including trying to achieve perfect 12ths across the span of the keyboard. Its really not worth the effort, and it doesn't achieve anything special, and in fact, will decrease resonance in the top octave. Perfect 12ths in the top end will destroy the sympathetic resonance created by pure 2:1 octaves that enhances clarity and sustain in those highest notes that tend to lack substance.

That being said, pure 12ths can work great in the 5th and 6th octaves but I like to taper to 4:1 double octaves as I move up and then to 2:1 singles in the very top. I think 4:2 octaves work best in the midrange transitioning to 6:3 as you move towards the bass and slightly expanding the 6:3 as you move to the lowest octave. If I'm tuning for an organist, I keep the 6:3 pure all the way down. (I've learned that the hard way)

If I'm tuning 12ths aurally in the 5th and 6th octaves, I use the 6th/17th test (use a major 6th below the bottom of the 12th and compare that beat speed to the 17th created when using that same reference tone with the top note of the 12th. If the 6th and 17th beat the same it proves the 12th is pure. It will tend to make the double octave about a beat wide, and the single octave will tend to be wider still. I think of the pure 12th as the upper limit of the amount of stretch I feel comfortable with in that area of the piano. Beyond that, the 17ths start to sound 'edgy'.

If you take the "let the piano tell you" approach you can avoid wasting time on overly complex and theoretical solutions to the piano tuning problem.

Ryan, glad you joined in.

I see using 12ths, whether narrow, pure or wide, as a great way to "let the piano tell you". If there is a general sense of stretch that you prefer, you can define it by the width of the 12th and transfer that sense of stretch to shorter or longer pianos by striving for the same type of twelfth. As I mentioned, a pure 12th produces wider octaves octave types in large pianos and smaller octave types in shorter ones. And if the sense of stretch of pure 12th is too much for your taste, then tune with slightly narrow 12ths. To me THAT is the value of tuning with 12ths. Tune the type of 12ths you like, and "the piano tells you" what the octaves should be. From the few other techs that I follow (they use a machine) wide 12ths seems to be the standard.

But we are getting off the subject of the OP's question. Ryan, you seem familiar with 12ths tunings. Is there a way you do it aurally?

Really? Think about it.

Well, I'd say the F-D 6th beats closer to G#-C than the G-B, but not the same. Unless its a smaller piano where the break is E-F when it really should be like G#-A. In those cases the M3s really slow down toward the break. Then F-D might beat like A-C# even. I wouldn't bother trying the test across a break, of course

I reason that because, unlike expanding with octaves, expanding with 12ths will put any unequal semitones on a different named note. For example, lets suppose that you make middle C slightly sharp to make some weird kind of "temperament" then when expanding upwards the G would be sharp instead. The "temperament" will be different for every 12th block.

Or am I missing something?

OK, but would having A#0, F2, C4, G5, D7 all sharp and another series, like D1, A2, E4, B5, F#7 all flat make an equal temperament?

I think we can easily get locked into one interval and think everything is OK. It is a major flaw, in my opinion, with setting a temperament with contiguous major 3rds. The same thing with an octave based temperament. But when including the 5th and octave within a 12th, you have "cross checks" with Slow Beating Intervals. And SBIs, being less tempered than RBIs, are inherently more accurate.

Ooops, I better get off the bandwagon.

Me too! I've missed you guys - It's interesting to check in every once in awhile and see what's happening!




As I said, I usually only pay attention to 12ths as I'm expanding the temperament into the treble. getting the 6ths and 17ths to beat equally is a convenient way to keep the stretch in a nice place. I don't really see the point in focusing on the 12ths through the rest of the piano.

Since I mostly tune with open strings (instead of using temperament strips), the slow beating intervals are harder to nail down. I find the fast beating intervals very useful, so I use a lot of aural checks involving 3rds 6ths 10ths and 17ths. These tell you everything you need to know about your octaves, double octaves, and 12ths.

I have used the following pattern for many years and find it to be highly accurate and controllable (not to mention pretty quick too). By making the 4ths beat about 1.25-1.5 bps, and making the 3rds slightly faster than "usual", the 5ths will be nearly pure, the octaves will be slightly expanded, and the 12ths will be VERY close to pure.

A4-A3
A3-F3
F3-A#3
A#3-F#3
F#3-B3
B3-G3
G3-C4
C4-G#3
G#3-C#4
A#3-D4
B3-D#4
C4-E4
C#4-F4...extrapolate as needed

Some others have tried this and like it. Doesn't work on a POJ piano, but works great on a decently scaled one.

Pwg

What is that?

How do you do that?

POJ : Piece Of Junk

PSO: Piano Shaped Object

I'll get back to you.

For those of you who focus on pure 12ths, do you carry these all the way to C8?

Not really possible to carry much of any partial match to the top octave on the hand-me-down spinets and rusty-musty church basement antiques that I usually tune.

But I do like the "edgy" sound that a slightly overstretched top octave makes. I do understand what you say about the increased resonance that pure 2:1 octave might give in the top octave, but I don't tune pianos with such resonance.

Another practical thing I like about using the 12th spanner when tuning the higher treble is, for example, the 3rd partial of C6 is heard stronger than the 1st partial of G6 when tuning G7. But after tuning the lowest note in the temperament I don't use the spanner until somewhere around the treble break. I do better expanding the temperament upward with 4ths and 5ths and checking other intervals, including the 12ths,also using RBI checks. You can fool yourself, or and least I can, into thinking you have tuned a pure interval when you really haven't.

I find this very laborious and time-wasting:


instead of this hearing a pure 12th is not difficult and much quicker...

No

Pwg