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Hi,

I was just over at the rollongball.com site checking the comments re the Koval Variable Temperaments:

"Most temperaments don't play well when divided or multiplied much. This one does.... Starting with the KV1.5 (representing the offset for C) multiply all of the offsets by two, and the result is the KV3."

Does anyone know if there are other UTs that also play nice when all the offsets are multiplied/divided by some value?

This isn't life or death; I just have an idea or two I'd like to play with. If you don't know the answer, please don't put yourself out. This idea may go nowhere anyhow.

But if there is some sort of list of UTs with this property, I would appreciate learning about it.

Thanks,
-Joe

Hi Joe,

They will all multiply or divide... I think it had something to do with the m3/M3 ratios that we were playing with at that time - and my attempt to get the most change with the smallest offsets. You might just try a couple and see what you find!

Ron Koval

Jim Coleman Sr. divided his Coleman 11 offsets by half and called the resulting temperament Coleman 10.

David Bauguess

It was a judgement call when I was experimenting back then. Simply a preference I noted while searching for milder temperaments before I began designing my own. The variable temperament was something that I just stumbled on - it wasn't my intent at the time!

Ron Koval

Thanks Ron and David for the helpful info:)

I'll post more on this later tonight (though you've both probably thought of this yourselves).

Thanks,
-Joe

Greetings,
I tried this before, with a SAT it is simple to store a page with the temperament recorded on it, and then when you call that page up, you can apply the same set of offsets to this previously offset tuning which doubles the offsets. It is easy enough to see when you are going to widen the F#,B, or C# thirds beyond 21 cents,(if the added widths total more than that you are going to have a howler).

What is really wild is to use something like a Coleman 11 to spice up one of the Bach tunings. I never found a really useful reason to push the idea.
Regards,

Hi Ed and thanks for the input

Actually, I was sort of thinking of going the other way and narrowing the offsets....

I suspect that the majority of folks who are involved in the UT movement are primarily motivated by the musical possibilities presented in the relationships between the new harmonies. While this is of interest to me as well, it is not my primary interest.

As has been commented on here before, UTs can give the piano a "change of resonance." Actually, I consider this to be an understatement. In some cases, the piano seems to morph into another instrument. As an example, I have observed that when a piano is tuned to the "Representative Victorian (Moore)," the character... the tone changes to a clear sweet tone. Not just harmony, but the sound of the instrument itself. Yet when the same instrument is tuned to "the Factory Tuners of 1840," the piano now seems to project better. It seems to become louder and the individual sustain of each note seems longer. There is a night and day difference between the effects of these two temperaments on the piano. this is easily repeatable... tune a piano one way and play one it for a few days until you become familiar withn the new sound. Then, tune it the other way for afew days. Finally, tune it to ET.

Maybe the effects are real; maybe the effects are just subjective. But, I'm not the only one I know who has observed the difference after I used various UTs. And if the effects are observable by others, in some ways it doesn't matter if the effects are only subjective.

I want to be able to to take advantage of the variations in sound quality that UTs offer yet still remain in Equal Temperament.

I want to be able to offer this to my customers.

I want the option to subtly alter the resonance while remaining in ET if I find a poorly designed instrument. I want the option of subtly modifying the tuning to make the piano project better, or sound sweeter, or more celestial, or whatever seems to be called for.

I want to be able to do all this in a way that gives me predictable results. I want a palette of options to choose from that can reliably produce the results I choose.

We've all dealt with the question of "How accurate is accurate enough?" If we use a tuning fork at A=440 c.p.s., will a one-third cent variation produce a perceptible beat? No. One-third cycle will, but not one-third cent.

Back when I was a strictly aural tuner, I defined ET by good beat speed progression among the various intervals, and all the A's being beatless with each other across the keyboard, all the A#s being beatless with each other,all the Bs, and so on. And of course, A4 was to be beatless with the fork. Each tuning was custom designed to each unique instrument by tempering, expanding the temperament outward , correcting the temperament and even the type of octave that the initial temperament used, etc. A very involved process using much testing and retuning.

With advanced ETDs like the Verituner, much of this work is done in the "imagination" of the machine after it measures, rather than the aural trial and error method. (The Verituner can base its tuning on measuring 76 keys and is multi-partial.) Assuming the VT does its job correctly, a very good Equal Temperament can be laid out on the keyboard. Even with minute variations, if there is no perceptible beat between the slightly modified notes and the actual ET calculated by the machine, is the piano not still in ET?

If A4 is beatless with the fork, is the piano at 440? If the piano is at 440.001, is it at 440? How close is close enough? The beat speed progression may not be nearly as smooth with these micro-offsets. Maybe some of the logic of the "reduced" or homeopathic UT still transfer to the temperament?

There would seem to be some UTs that are better candidates for this treatment than others.

For instance, the Equal-Beating temperaments would probably not make good candidates. If the offsets are changed, the Equal-Beating quality will disappear, which is what makes them special. There may be something interesting left. But the original logic of the temperament will be gone.

With other temperaments, rounding of the reduced offsets may cause unacceptable errors, or some offsets may disappear completely when rounded to zero.

With the Koval Variable temperament, even if carried to KV 0.25, none of the offsets round to zero, so they don't disappear.

There is probably some sweet spot where some of these temperaments can be reduced to and still have at least some effect on resonance.

According to Ron Koval's earlier reply in this thread:


Anyhow, sorry for the long-winded response. Does this seem to make sense to anyone, or am I just barking up the wrong tree?

Thanks,
-Joe

Joe,

The first thing you have to consider is that you can't get something for nothing. Each action has an equal and opposite reaction. For any interval that you may "improve", you will also cause "harm" to another. The WT I designed strikes a delicate balance in all of that.

However, it is clear that you are looking for something else and I believe I have the answer for you. I would suggest one of the Mild Meantones; specifically the 1/9 Comma Meantone.

First, you have to understand what a Meantone temperament is. In Meantone, all 5ths are tempered equally. The name of the Meantone is derived from taking the value of the Syntonic Comma which is 21.5 and applying a fraction to it. Therefore, in the classic 1/4 Comma Meantone, 21.5 is divided by 4, leaving the figure of 5.37. That is the number of cents narrow that each 5th would be tuned. Naturally, that leaves one 5th that cannot be tuned and will be very wide. It is known as the "wolf" 5th. The unresolvable 5th is usually left between G# and D#.

It is actually quite easy to program any ETD to tune virtually any gradation of Meantone imaginable. The ETD's 5ths already have 2 cent narrow 5ths. If you divide 21.5 by 11, the result is 1.95. Therefore, an 11th Comma Meantone is the virtual equivalent of ET. You may have heard about other Meantones, such as 1/5, 1/6, 1/7, etc. These are all just arbitrary numbers and theoretical ideas but Jean-Baptiste Romieu thought of all of those possibilities in the 18th Century. They are documented in Owen Jorgensen's first publication, Tuning the Historical Temperaments by Ear.

Unfortunately, there is no good way to tune any of them by ear but there is a very good way to tune them with an ETD and you don't have to stick to any particular fraction. You can vary the size of the 5ths to any degree you choose. I use a Sanderson device, so I am limited to 1/10 of a cent but I doubt that splitting that any smaller would make any difference.

Some of these ideas which were only theoretical have ended up having there own "magical" properties to them. It just took trying them to discover what they were. The 1/7 Comma Meantone with its (approximately) 3 cent narrow 5ths, for example, has many sets of equal beating M3's and M6's and each Major triad with the exception of the "wolf" triad has an exact 2:1 ratio of beating between the Major and minor third. Unfortunately, the 1/7 Comma Meantone, from what you say, would still be quite out of bounds for what you are after.

The 1/8 Comma Meantone is an ugly duckling. there is nothing appealing about it at all. However, the 1/9 Comma Meantone, from what you say may be what you are looking for. 21.5 divided by 9 yields 2.38. I round that off to 2.4. Since there is already -2 cents between each 5th on the ETD, drop the 2 and that leaves 0.4. That is the number you will use to program multiples of around the cycle of 5ths. I'll do that for you and show you exactly how to program the Temperament offset page in your ETD:

C: +1.2
C#: -1.6
D: +0.4
D#: +2.4
E: -0.4
F: +1.6
F#: -1.2
G: +0.8
G#: -2.0
A: 0.0
A#: +2.0
B: -0.8

What is special about the 1/9 Comma Meantone is that its 2.4 cent narrow 5ths sound hardly more tempered than ET 5ths and the one wide 5th between G# and D# is almost exactly as wide as the others are narrow. This means that if you play chromatic 4ths and 5ths, you would be hard pressed to distinguish it from ET. If you play chromatic M3's, there is a bit of unevenness but probably only a skilled piano technician would perceive it.

Yet, this temperament does clearly retain the characteristic Cycle of 5ths "color" that is so desirable to many. I use it for hotel piano bars, sometimes for Jazz pianists and one Church piano that I tune where the pianist likes color but prefers pastels.

I hit upon this simply through experimentation and found something I liked. If that is still a bit too much variation for you, however, there is nothing to stop you from going even milder by any degree you choose. I can only go milder by tenths but you could slit the differences in half or any other degree you choose if your ETD accepts hundredths. Ron Koval may be interested in that.

Here is a list for a Meantone with 2.3 cent narrow 5ths. Note that the largest deviation is 1.8 cents:

C: +0.9
C#: -1.2
D: +0.3
D#: +1.8
E: -0.3
F: +1.2
F#: -0.9
G: +0.6
G#: -1.5
A: 0.0
A#: +1.5
B: -0.6

Still too much? Try a 2.2 cent Meantone. Note that the largest deviation is 1.2 cents:

C: +0.6
C#: -0.8
D: +0.2
D#: +1.2
E: -0.2
F: +0.8
F#: -0.6
G: +0.4
G#: -1.0
A: 0.0
A#: +1.0
B: -0.4

Still too much? Try the 2.1 cent Meantone. Note that the largest deviation is 0.6. The "wolf" 5th ends up being 0.9 cents narrow instead of any amount wide.

C: +0.3
C#: -0.4
D: +0.1
D#: +0.6
E: -0.1
F: +0.4
F#: -0.3
G: +0.2
G#: -0.5
A: 0.0
A#: 0.5
B: -0.4

If you have an ETD that is programmable in hundredths, try the 1/10 Comma Meantone which is also a theoretical idea that Romieu pondered but for which tuning by ear would be virtually impossible. Note that the largest deviation is 0.40 cents. The "wolf" 5th is 1.25 cents narrow while the rest of the 5ths are 2.15 cents narrow.

C: +0.25
C#: -0.30
D: +0.15
D#: +0.40
E: -0.15
F: +0.30
F#: -0.25
G: +0.20
G#: -0.35
A: 0.0
A#: +0.35
B: -0.20

If you want to see any or all of these graphically depicted, copy and paste the figures in an e-mail to Jason Kanter at: jkanter@rollingball.com The results should be interesting as he may also point to something special about one or more of them.

Hi Joe,

In my experience with the variable series, going down to .8 wasn't worth it. 1.15 is the weakest that I use, with stops at 1.3, 1.5, 1.7 and 2.1 (those are the offset for C, the largest offset)

1.7 is surprisingly significant change from ET and the 2.1 is preferred by a few of my piano teachers - enough to really make a difference in the classical repetoire without causing pain in the arpeggio work.

1.3 is my most commonly used strength for 'regular' tunings.

Ron Koval

Bill,

I really have to thank you for your well thought out and detailed reply.

I had not even considered using mild meantone temperaments. I think you are right; this does seem to be what I am looking for.

Thank you!

Btw, I want you to know that I am busily entering values in my Verituner:



The The VT will allow me to enter offsets of up to 3 digits to the right of the decimal point. But after saving and reloading the temperament, the VT rounds the offsets to 2 digits to the right of the decimal point.

Thanks,
-Joe

Ron,

Thank you for the info I'll be entering those Koval Variable temperaments as well. The VT already has KV 2.0 and KV 3.0 built-in. Based on your post at the Verituner Forum, I have also entered the KV 1.15, 1.3, 1.7, 2.1, 2.5, and 2.9. I've also calculated and entered the KV 0.5. I will also enter the KV 0.25 as well.

It will be interesting to see if "homeopathic" versions of your well temperaments also affect the resonance of the piano.

So I now have multiple options for (possibly) affecting the resonance of a piano while technically remaining in ET:

• Equal Temperament
• Very mild Meantone temperaments as suggested by Bill Bremmer
• Very mild forms of Well tuning as based on the Koval Variable temperaments of Ron Koval
• And of course, variations in octave tuning.

Thank you!

I think I have enought to keep me busy for a while

Thanks,
-Joe

Not sure if that is a problem or not, Joe. I don't know a thing about the Verituner but Ron is an expert. I would be interested to know what you think about the 1/10 CMT. I wonder if you would be able to tell the difference from ET at all or whether it makes that tiny difference you were looking for. I have not tried anything smaller than the 1/9 CMT. It is about as mild as I would consider worth doing to make a key color difference between ET and anything else. Ron had a similar threshold.

I scanned the pages from Owen Jorgensen's first publication, Tuning the Historical Temperaments by Ear for Romieu's 1/9 and 1/10 Comma Meantone Temperaments, for what it's worth. I will e-mail Jason Kanter for some graphs and will also calculate an exact 1/9 CMT to the nearest hundreds to see if there is any significant difference between the exact 1/9 and the rounded off, 2.4 cent 5th Meantone that I use. I doubt that there would be.

Here are the scanned pages 200-206 from Jorgensen's book:

Here are the values to the nearest 100th cent for Romieu's 1/9 Comma Meantone Temperament. I will ask Jason Kanter to compare this and the 2.4 cent narrow 5th Meantone that I use:

The Romieu 1/9 Comma Meantone Temperament (rounded to the nearest 100th cent)
C: +1.16
C#: -1.55
D: +0.39
D#: +2.33
E: -0.39
F: +1.55
F#: -1.16
G: +0.78
G#: -1.94
A: 0.0
A#: +1.94
B: -0.78

While I like to be aiming for the "right" target, what's the real threshold for being able to achieve the target? Tenths of cents are probably a fine enough resolution, and hundreths are beyond what I can do, even though I included them in my spreadsheet stuff... Thousandths? Yikes!

Ron Koval

Ron,

I asked Jason Kanter to compare the 1/9 CMT (rounded to nearest 100th) with the 2.4 theoretical that I have used to find out if there is any significant difference. He did that for me for the 1/7 Comma Meantone and there was absolutely none! So, when I program that temperament or the variation that has one pure 5th inserted, I just use who cent numbers. Why bother with small change when it doesn't really matter?

Nevertheless, the first question I get from people is something like, "Why are these numbers so easy? Every other temperament I have seen has screwball numbers that I can't make hide nor hair of. These are just, 1-2-3-4-5-6. This can't be right or is it?"

Now you may be able to understand better why in my ultra mild WT, I still only used whole and half cents and all of the recordings that Grandpianoman put up are also only in whole or half cents.

The last three ultra mild meantones I posted would all "pass" the PTG Tuning exam with a "score" of 100 (like the ET via Marpurg would). I somehow doubt that anybody could tell one from the other at all when music is played but let's see how any and all of this plays out. There may be something in one of these ideas that has some true merit and possibly the small numbers would matter after all.

Related to this topic, previously, there was some interest expressed in the "Best Broadwood" temperament published in Jorgensen's big red book,Tuning. In my view, the term, "best" only meant that the temperament was closer to ET than any of the others. Whether this was any kind of improvement is a subject of debate because really, nothing can be improved as far as temperament goes, one can only search for a more personally satisfactory compromise.

If a Well Temperament that seems very nearly equal but still has some key color in it satisfies that desire, then perhaps this temperament serves the purpose but I can think of many possibilities, including my own WT that are far closer to ET than this. The two pure 5ths and the many equal beating intervals present in my WT make it a far more favorable choice. Don't be persuaded by a label is my advice. The Broadwood Temperaments were merely attempts at ET but old habits produced crude versions of mild WT's that had nothing special to offer.

In the interest of the question of the topic, I will take the published figures, cut them in half and ask Jason Kanter to create a graph for both.

Here are the published figures for the "Best Broadwood" temperament:

C: +5.0
C#: +1.0
D: +3.0
D#: +3.0
E: -2.0
F: +5.0
F#: 0.0
G: +5.0
G#: +2.0
A: 0.0
A#: +4.0
B: -1.0

It is easy for me to calculate whether any temperament would or would not "pass" the PTG Tuning exam for ET. Some people view that as a criterion or at least some kind of indicator. The "Best Broadwood" Temperament would theoretically "score" a disappointingly failing score of 48 on that exam. (Not quite "apprentice" level).

The above, split in half would yield:

C: +2.5
C#: +0.5
D: +1.5
D#: +1.5
E: -1.0
F: +2.5
F#: 0.0
G: +2.5
G#: +1.0
A: 0.0
A#: +2.0
B: -1.5

Split in half, the "Best Broadwood" would theoretically "score" a disappointing 75 (apprentice level).

Quite some time ago, Mark from South Africa asked if I could scan the pages from Jorgensen's book for the "Best Broadwood" Temperament. It took me quite a while to get to it but experience in scanning other material has helped, so here you go for whatever it is worth:







We shall see if Jason Kanter's graphs show any redeeming qualities of either.

My own attempt at an ultra mild WT about 2 years ago was this:

C: +1.0
C#: -1.0
D: +0.5
D#: +0.5
E: -1.0
F: +1.0
F#: -1.0
G: +1.5
G#: 0.0
A: 0.0
A#: +1.0
B: -0.5

It would theoretically "score" a 90 (superior) on the PTG Tuning exam for ET although it is clearly still a mild WT. Jason Kanter did produce a graph of it at the time:

Bill Bremmer wrote:


and Ron Koval wrote:


This is just a guess. But my thinking is that the higher partials of each note will be affected more than the Fundamental. When a unison is tuned, “dead” or beatlesss actually a fairly large area. You can get the fundamental basically beatless and still deliberately create some beating on the higher partials by deliberately mistuning the unison.

The frequencies at higher partials are, of course, much higher. Also, the higher notes themselves are at higher frequencies at the fundamental. So, there are more “cycles to the cent” the higher you go. And vice versa.

There is a chart of notes and frequencies at http://en.wikipedia.org/wiki/Piano_key_frequencies which should be close enough for discussion here.

C1 (C4) has a listed frequency of 32.7032 Hz. C#1 (C#5) has a listed frequency of 34.6478 Hz. So there are 100 cents spread out over less than 2 Hz. However, the same would not be true of the higher partials of these notes.

C6 is listed at 1046.50 Hz and C#6 is listed at 1108.73 Hz. So approximately 50 Hz are distributed across 100 cents.

C7 is listed at 2093.00 Hz and C#7 is listed at 2217.46 Hz. Over 100 Hz are distributed across 100 cents.

(The non-technician reader should understand the the relationship between cents and Hz is more complicated than what I indicated above.)

The middle and higher notes also have their own higher partials, so we've got ourselves a real concoction of stuff happening here

So even if the piano is tuned beatless in the middle and the bass with ET while using these micro-offsets, I think it still might affect the resonance of the piano. And, it may have some small effect on the way the higher notes are tuned, which could also affect resonance. There could be cumulative change as the tuning works itself out.

The piano could be tuned in ET, and maybe still have some variations in resonance of a UT. I remember a term that was used years ago that I never seem to hear anymore: “Concert Tuning.” Years ago, it was common practice for some tuners to tune in Equal Temperament using only 4ths, 5ths, and octaves. (I'm sure you remember them )

But, the placement of the notes was refined in “Concert Tuning.” All the checks were used to determine the ideal placement of each note... 4ths, 5ths, M3rds, m3rds, 6ths, 10ths, and so on. Yet, the “4ths and 5ths” temperament was accepted as tuning in Equal Temperament.

Bill, maybe there won't be any noticeable difference in the sound of the piano when using these micro-offsets. I have to start experimenting. Or, maybe there will be. Or, maybe the difference will to considered too insignificant to bother with.

I guess my posting here was premature because I only posted an idea, not reported observations. But, I am glad that I did. I have a better understanding of how to go about using these reducible UTs . And you have made me aware of the possibilities of various Meantone Temperaments. I think I have avoided some cul-de-sacs because of your and Ron's and Ed's advice.

And thank you so much again for all the info and the offsets.

And thanks to everyone who participated here.

-Joe

Indeed Joe, that is the experience I had when tuning the ET via Marpurg. It was originally intended to be a different way to tune ET but like most Quasi Equal Temperaments, there was a flaw in the logic. Although 4ths are 2 cents wide and 5ths are 2 cents narrow in ET, they do not beat exactly alike. As you can see from Jorgensen's 1/10th CMT which is very nearly ET. the 4ths beat 1/3 faster than the 5ths in true ET.

Nevertheless, the ET via Marpurg remains a very close approximation of ET, also falling "under the radar" of the PTG Tuning exam with all notes less than 0.9 cents from exact ET (three of them, F, A and C# being the same as ET), so if perfectly executed, it would "pass" the exam with a theoretical "score" of 100. It has no Cycle of 5ths color and therefore is perceived as ET.

What was interesting to me during previous discussions of the ET via Marpurg was a post from Gadzar (to whom I have sent a private message asking him to re-post it if he can find it, for I could not). A British technician in 1949 (if I remember correctly) advocated tuning ET with equalized 4ths & 5ths because he found that it made an overall and favorable difference in the final outcome of the way the piano sounds compared to true ET.

I was intrigued by this and found it to be true. So, what turned out to be initially somewhat of a disappointment, it has now become the only way I will tune ET. Not only did it improve the sound of ET for me on the piano but it actually proved to be a shortcut.

When I now choose to tune ET, I do it aurally and use only 4ths & 5ths once I have tuned the initial set of Contiguous Major Thirds (CMT). When expanding the temperament octave in either direction, I use only 4ths & 5ths and make the octave, 4th & 5th all have the same tempered sound which leaves each sounding very nearly but not quite pure. Once I have a double octave tuned, I play the double octave plus the octave-fifth and single octave below it and find the point where the double octave note being tuned resonates in seemingly perfect purity with the three notes below it. In reality, there is an imperfect coincident partial match for all but the slight beat there is in each tends to cancel itself which leaves the piano sounding amazingly "in tune" with itself. There is absolutely no need to use any Rapidly Beating Interval (RBI) checks!

To use an ETD to tune the ET via Marpurg, you do need hundredths of cent capability and this is definitely an example of where those would matter. Here are the offsets for the ET via Marpurg:

C: -0.05
C#: 0.00
D: -0.16
D#: -0.78
E: -0.50
F: 0.00
F#: +0.60
G: +0.56
G#: -0.05
A: 0.00
A#: +0.16
B: +0.56

Here is one of Jason Kanter's graphs of the ET via Marpurg. It looks more irregular on paper than it sounds:



Ron Koval might be able to tell you what kind of stretch to use with the Verituner to insure that the 4ths, 5ths and octaves all continue to agree but my guess is that it would be nothing special at all, just the default stretch but maybe some kind of departure from that in the highest and lowest ranges.

You have a lot of options to try and indeed, the sound of the whole piano when completed is what will matter. I would be very interested to know which idea ultimately suited you the best.

Bill,

Again, thanks for your help

I do like ET much of the time. The effect can be quite stunning. But, I have noticed that there have been times when ET, even though properly applied, leaves the piano sounding a bit blah. So, I am looking for options.

For me personally, I want to explore remaining within ET and yet finessing it to try to alter the sound of the piano, but only in those cases where standard ET seems limiting to the sound. I'm not so much interested in exploring key color as I am in subtly altering the resonance.

I don't think I will settle on any one specific variation. It would probably be best if there were multiple options available. If I have to tune an anemic spinet for use in a large room, maybe I'll use approach A. For a specific make of piano that gets weird sounding at the break, maybe approach B. And, so on. And if the piano seems fine in standard ET and the customer is happy, I'll just continue tuning it as before.

Just as a quick aside, I have had some health issues recently. I am recovering, but slowly. I am physically only able to handle a very light schedule at this time. So, I've had time to ponder on these things I am going to be testing these things. But it may take me a while before I have anything definitive to report. I am sorry about that. But, I will report.

Thanks,
-Joe

That's a very interesting and profound concept.

Clearly any standard analysis of UT by looking at the quality of M3's in various keys is irrelevant. As fat as I know there is no theory for this whatsoever.

I came up with an argument why ET might sound worse than approximate ET:

Let's assume beats are "bad" and they should not be heard as much as possible. With 12 keys per octave we can't make all intervals beatless so we have to distribute the beats over all intervals in some way.

When a piano is tuned in a clinically perfect ET the beat rates are logically organized. The human mind is a good pattern recognizer, so if the beats are logically organized we will notice them more than if they were randomly organized.

So if we offset ET by a small random amount we randomize the beat rates and make them less audible perceptually.

Any thoughts? Perhaps I should have started a separate thread on this?

Kees