Piano World Home Page Forums Our Most Popular Forums Piano Tuner-Technicians Forum Pure 5ths Temperament?

Chris,

You are awesome!

Thanks for the offsets

Gadzar:

You are free to disagree, but I see no reason a pure 5ths temperament could not be tuned with Dr. White's sequence. Sure, the beatrates that are estimated would have to be altered along with some relationships, and the octave would have to be expected to be a bit wide, but no reason it would be inherently more difficult, or much different, than ET with a clean octave.

It's tempting to give it a try, just for fun.

Jeff,

It's not my test. I'm referring to one of the links I provided. The link was from a thread by Isaac Oleg:

If you are using offsets, you would be using and ETD, right? So why not just use an ETD to tune a pure 5th, divide the difference in the offsets of the two notes of the 5th by seven, and apply this semitone offset progressively to each note? This way inharmonicity would be taken into account.

There is one advantage I can think of for using the temperament within a fifth. It eliminates the risk of encountering the A4 to A3 dilemma.

When setting a traditional temperament, usually A4 is tuned to a reference. Then, A3 is tuned to A4. Then, whatever temperament sequence is used.

But, what happens as you begin to expand the temperament and discover the you should have chosen a different octave type for the A4 to A3 octave?

Do you completely retune the temperament? Or, do you fudge the A4 to A3 octave and leave A4 ever so slightly off?

By using these offsets, or by tuning the temperament within a fifth aurally, you could just set the temperament for D4 to A4. No more dilemma.

This is also an advantage of ETAs that set the temperament between A4 and A3, or C5 to C4.

If I'm remembering correctly, Bill Bremmer, RPT has a special procedure for expanding EBVT3 over the entire keyboard. First, he sets the EBVT3 temperament octave. Then, by using different octave types, he starts to finesse the tuning back towards ET as it expands from the temperament octave. If I'm wrong Bill, please let me know

This could even be an option for the One Pure Fifth tuning. You can continue to use pure fifths to expand the temperament. Or, you could try finessing things back towards a true ET.

This could be a tremendous time saver on smaller pianos that are going to be problematic when you cross over from plain wire to wound strings. Since many tuners probably aren't trying to continue ET at the transition anyhow, this could be an option.

My solution to the octave dilemma is to usually tune a 12th down from A4 or C5, and then a tempered fifth up and checking for an appropriate resulting octave. Although I like pure 12ths, there is no reason a tempered 12th wouldn't work the same. I then set the temperament within this tempered 5th. There can be an occasional adjustment during expanding, but then I think there always is regardless of the sequence. I expand with 4ths and 5ths and check with RBIs and 8ths and 12ths and 15ths as available up to around the treble break. Then I use a 12th spanner. I use RBI checks on the 8ths, 12th, and 15ths.

But even when guestimating an octave, and not having it dead on, I think there would almost always be some wriggle room that only a few notes would need to be retuned.

Using an ETA to tune the Pure fifths temperament is easy. I can think of several ways to accurately set the initial 8 notes within a fifth and then tune the rest by pute fifths using the ETA. The easiest way, I guess, is the one suggested by Jeff, that is tune A4 then tune D4 as a pure fifth and then tune the remaining notes in between dividing by 7 the offset of D4 and calculating the corresponding offset for each note.

There are other ways depending on the particular ETA. With Verituner we can choose the Large built-in Style and then adjust the Stretch setting, which can be set from -2 cents to +2 cents, I don't know if it will give enough stretch, in order to have a pure D4A4 fifth. Then directly tune the notes between D4 and A4. From there, use the Verituner in the measure mode to tune the rest of the piano by pure fifths.

Jeff, I agree that if you modify the tempering of octaves, fifths and fourths in the Braid's sequence then it is possible to use it to set the pure fifths temperament. The problem is you must know in advance how much you must temper octaves and fourths. And how much this temperings will be affected by iH from piano to piano. I guess this is possible, as it is perfectly possible to use it to tune ET with clean octaves and tempered fifths. But it should be modified from its original tuning intructions.

PS I guess this is true not only for Braid's sequence but for any known sequence. If you modify the tempering of the intervals it will work for Pure Fifths Temperament

Funny thing, that few realize, is that when using Dr. Whites sequence for an really great ET, you have to work out how much the 4ths are tempered kinda separate from the 5ths, and whether you need or ought to make either somewhat progressive. So I am thinking that with pure 5ths, it would be a bit easier.

Do you follow me?

Yes. This is right. This is so with CM3s setting also.

More on that in a few minutes...

I follow you. I have faced the same problem with my usual tuning sequence, so I have made some experiments using Verituner. I know that this is only a tuning aid and not a scientific instrument but supposing it is accurate enough I've found some interesting facts.

The experiment is as follows. Using Verituner in the measuring mode tune:

1. A4 to the desired pitch, for instance 440hz.
2. Tune A3 as a pure 4:2 octave from A4
3. Tune F3 as a tempered 5:4 third from A3, for instance 13 cents wide
4. Tune F4 as a pure 4:2 octave from F3

at this point the M3 F4A4 is expected to be wide by 13 cents, right?

Wrong!

Each and every piano I've measured gives a F4A4 M3 wider, most of the time, or narrower, rarely, than F3A3. No matter if I tune a less tempered or more tempered M3 F3A3, F4A4 is always wider (narrower rare times).


Let me repeat: In every piano I've measured, if I tune two octaves A3A4 and F3F4 the same size, then F4A4 is wider (narrower) than F3A3.

The only way to make F4A4 the same size than F3A3 is to stretch the F3F4 octave.

So when tuning CM3s you have to tune equally sized octaves with different sized M3s. Or you can tune equally sized M3s with different sizes for the octaves. Or the most rational: you can compromise octaves and M3s that fit in a smooth even progression.

There is no way to tune perfect sized octaves with perfect sized major thirds.

And problems are only arising... Once you have tuned an acceptable set of CM3s from F3 up to A4 you have to tune fourths and fifths that fit into these M3s.

I used to tune a variation of the Sanderson-Baldassin sequence, but now at the light of all this, I´ve abandoned it.

I tune the octave A3A4, with no tests, just the cleanest sound possible. Then I tune D4 and E4 as fourths and fifths from both A3 and A4. I've found than the relation between fifths and fourths A3D4, A3E4, D4A4 and E4A4 depends on the scaling of each piano. In some pianos fourths beat faster than fifths, in some pianos they beat the same and in some pianos fifths beat faster than fourths. Anyway, once I'm happy with A3, D4, E4, A4, that is: one octave, two fourths and two fifths, I divide the octave A3A4 into 3 CM3s A3C#4, C#4F4 and F4A4, with no testing. All I strive for is to have the same amount of tempering in each major third. I do not count or estimate beats. At the light of my little experiment, I know in advance that tuning perfectly sized thirds is not possible, so I only tune what sounds appropriate at this point of the sequence. Now I divide each one of the CM3s into fourths and fifths that build up that M3.

Namely for A3C#4: I tune A3E4 (already tuned), E4B3, B3F#4, F#4C#4 (C#4 is already tuned, so I do not move it). M3 D4F#4 may sound too tempered or not enough tempered compared to the other tuned M3s, so I try to refine my fourths and fifths to make it fit. Keeping an ear to the already tuned fourths and fifths. Also M6 A3F#4 is now tuned, so I keep an ear on it. At this point I may find that all fits nicely and I can continue, but most of the times I find something that doesn't fit and that I can not reconcile. Some fifths or fourths may sound too tempered or too pure and here I am forced to reconsider the tempering of A3C#4, it should be a little wider or narrower to produce better fourths and fifths. I even can retune my initial octave A3A4 to make it a little wider or narrower. I retune the CM3s in consequence and I re-do my fourths and fifths until I have acceptable and smooth fourths and fifths than fit with the prevously tuned fourths and fifths A3D4, A3E4, D4A4, E4A4 and also congruent M3s.

In the same way, I divide the M3 C#4F4 in fourths and fifths C#4G#4, G#4D#4, D#4A#3, A#3F4. I check RBIs available fit in the progression and retune whatever proves being wrong.

Same procedure to divide F4A4.

The iH of each note in the piano is different from one another. One can not think of tuning perfectly sized octaves, or perfectly progressive M3s, or an even progression of fourths, or equally tempered fifths. I fyou do so then the chosen interval will be good at the expense of others. The best we can tune is a compromise between all the intervals where none is perfect but none is unacceptable.

That is to say that there is no one fixed size for the semitone that fits all along the scale of the piano. The semitone is larger at the extremes of the scale. And it won't follow a continuous and smooth curve neither. The length and diameter of the strings do not follow a continuous smooth curve neither, as does not the iH.

Yes, I follow you, cont.

In my last post I deviated from the subject I wanted to treat.

Returning to my little experiment: tune two octaves A3A4 and F3F4 of the same size and finding out that M3s F3A3 and F4A4 are not the same size, let's repeat it but this time with 1.5 cents narrow fifths instead of pure octaves and a 2.5 cents wide fourth instead of the 13 cents wide M3.

This values of 1.5 cents narrow fifth and 2.5 cents wide fourth are tipical of a small grand or an uprigth piano and will build up a slightly wide (1 cent) 4:2 octave.

1. tune A4 to reference tone
2. tune D4 as a 1.5 cents narrow 3:2 fifth from A4
3. tune A3 as a 2.5 cents wide 4:3 fourth from D4
4. tune E4 as a 1.5 cents narrow 3:2 fifth from A3

As we know now, chances are that the fourth E4A4 won't be 2.5 cents wide. Depending on the particular piano the fourth E4A4 will be wider or narrower than the fourth A3D4. No matter how narrow or pure you tune your fifths, if they are the same size then your fourths won't be the same size. As in the case of the octaves, the only way to have equally tempered fourths is to make the two fifths of different sizes. So, once again, we have to make a compromise where the two fifths and the two fourths are not perfectly equalized but none of them is objectionable.

Alfredo Capurso, in his CHAS PREPARATORY TUNING thread, suggests to tune A3E4 faster than D4A4 and A3D4 slower than E4A4. He is a great aural tuner who makes wonderful tunings, he has posted some audios of his work.

He tunes progressive fourths. And fifths that are almost pure in the bass, then become more and more narrow tempered approaching the tenor and then become less and less tempered to almost pure in the treble.


Here is the link to Capurso's thread:


CHAS PREPARATORY TUNING


I hope all of you find it rewarding.

I am measuring the piano Inharmonicity aurally and that tells me what octave size sounds best. I have never had to retune the A3A4 octave, only when it drifts from the original size I set it to.

Search PW for "octave stretch" or "octave scale".

Hi Mark,

Not to drag this thread too far O.T., what checks do you use to verify A4 after you complete the temperament?

I usually start by tuning A3 to A4 where it sounds best. Then, 3rd/10th check.

D3 to D4 temperament. (Hopefully including some wound strings on a smaller piano.)

Expand temperament down to D2, verifying with checks and modifying even the original temperament octave to get good beat speed progression. Create a 2 octave temperament.

Expand upward to D5 using checks and checking for good beat speed progression. Also, use double octaves from the bass to make sure the D4 to D5 area doesn't go too sharp. Less sharp means slower 3rds, which gives a more chorus-like sound, rather than harshness. Create a 3 octave temperament, freely modifying and rethinking previous work if necessary.

Every note is provable.

And so on...

Sometimes the checks and progressions show that A4 should be at sightly different location than where I first started.

Of course, i won't know if I set A4 ideally until i have had a chance to explore the piano by tuning and getting more information.

You are right Joe.

If the first interval tuned in a tuning sequence is the octave A3A4 then all we have to decide its width are the partials of these two notes. We know nothing about how they align with the partials of the other notes, mainly a fourth, a fifth, a third or a sixth above and/or below. We have no means to know what these partials are until we try to tune those intervals.

So pretending to set the size of the A3A4 octave at the beginning of the sequence with no further information about the iH of other notes and treating this octave as if it was set in stone is an error. If you do so you are imposing your ideas to the piano instead of listening what the piano is telling you.

This is the same as tuning two equally sized octaves F3F4 and A3A4 and pretend having equally sized F3A3 F4A4 thirds: it's not possible, you will be fooling yourself if you try. You can not force the piano to tune like you want, it will tune like it was designed to. All depends on the actual scaling of the piano, on the individual iH of each string, there is no way to know in advance what this iH is, you have to tune the note to hear how it behaves with the other tuned notes.

When I was learning piano tuning I was told to tune A3A4 as a slightly wide 4:2 octave. Soon I discovered that this was fine only for some pianos but there were pianos who needed a pure 4:2 octave and there were some pianos who needed even a narrow 4:2 octave. Many piano tuners have refused to accept the idea but with time and insistence of my part and the fact than many ETD tuners were using such kind of narrow octaves, some of these skeptical tuners have finally accepted the need of a pure 4:2 octave though they still refuse to tune a narrow 4:2 A3A4 octave.

I guess this will be the same with my little experiment, if you tune a fixed size interval the other intervals won't be equally sized. You can not impose a fixed size of interval to a piano. All intervals used in piano tuning, namely octaves, fifths, fourths, thirds, sixths, tenths, etc... change along the scale of the piano, they must be compromised in order to be compatible with other intervals.

It is in that sense that you can not know the best width for the A3A4 octave until you have tuned the fifths, fourths, thirds and sixths, etc related to A3 and A4. These fifths, fourths, etc will tell you the most appropriate size of A3A4. As you say: every note is provable, nothing is set in stone, mainly if it was set at the beginning of the sequence with very little information available about the piano's iH.

Hey Folks, haven't been able to read everything carefully and won't be able to post much right now, but I have a thought to share. Besides the self-correcting and error-multiplying features of a sequence, there seems there can also be a note-arranging feature. I think this is different than the final-polishing feature that is mentioned as a step after roughing-in a sequence. Huh! I guess roughing-in is another feature...

Hi Joe,

Did you mean A3?

No Mark, I meant A4 might need to be retuned.

You could retune A3, but then you might have to retune all the work you based on A3. So, the dilemma is whether to retune A3 and all the work based on A3, or whether to just retune A4 slightly.

By using Jeff's one fifth idea, a way might be found to avoid this possible problem.

Raphael, I have also encountered pianos where A3 to A4 needed to be narrow.

Narrow 4:2 or 2:1?