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I also have a copy of Lucas Mason's book and tried the temperament several times. It seemed to me that starting the temperament with pure 5ths made it difficult, if not impossible, not to stretch the octaves too much trying to maintain the fifths outside the temperament. Major and minor seventh chords sounded awesome in the mid-range of the piano though. I gave up trying and now the book sits on my shelf as well.

I just did a simulation with pure 5th, too, using the iH curve I have for a Charles Walter console. The 4ths are busier peaking at about 2.2bps around A4-D5. The octaves look obnoxious with the 4:2 octave peaking at about 2.2bps also around A4-A5, which makes perfect sense when you consider the P4-P5 test for the 4:2 octave. A0 is -27 cents and C8 is +70 cents. In the F3-F4 temperament octave, the 4ths beat about 1/2bps faster than theoretical, the m3s 1/4bps slower, the M3s 1/4bps faster, but the M6s a whopping 3/4bps faster. The M6/M3 outside inside test would definitely need to be modified. F3-D4 beats about the same as G#3-C4, instead of G3-B3. The P12s also looks obnoxious with A3-D5 at 2bps and continually getting faster.


Don't think I'll bother giving this a try on my CW. But if I were to, I'd:

Tune a pure D4-A4 5th and a pure G3-D4 5th.

Then tune the G3-C4 and A3-D4 4ths, having them beat a bit faster than usual.

Next tune the G#3-C#4 4th to beat the same speed as the other two 4ths, adjusting both notes up or down until the G#3-C4, A3-C#3 M3s are properly progressive.

Last would be to tune A#3 for properly progressive Cm3s of G3-A#3-C#4 and the same for B3 with the Cm3s of G#3-B3 and D4. That completes the eight notes of the G3-D4 5th.


There are particular checks that can be used to refine this, but the important thing is to go back through the sequence of the three pairs of notes in the same order (C4 & A3, G#3 & C#4, A#3 & B3) refining each pair, then continuing with the next pair until no improvement can be made for the entire 3-part iteration. It shouldn't take more than two or three times around if good hammer technique is used. (Works the same regardless of the width of the 5th...)

For a large piano, continuing the beginning ladder of pure 5ths down to C3-G3 would be best, as the m3s would beat slower and you would still be wholly above the break.

(Please forgive the inevitable typos, but go ahead and mention them for posterity. )

Expanding the temperament would be by pure 5ths and any errors should be glaring, since the 4ths beat pretty fast.

I think for this temperament to sound acceptable, it would have to be very equal. If one 4th is too pure, one or more others would be even more busy. I think it would be very difficult to juggle an entire octave temperament to work out the proper speed of the 4ths. With tempered 5ths it is different. If one 4th is a bit busy, you could change it by compromising the tempering of the 5ths. Not so, if the 5ths are pure.

Anyhoo, I don't think I'll be trying it and will stick with pure 12ths. You folks do have me very curious about Mason's book, though. But before I get in too deep, is there a fee to join the "The New Tuning Dusty Book Club"?

Jeff,

You're free to contribute as much as you'd like as it's a non-profit club. :-)

Pwg

Cool! That means it's tax deductible!

Ha Ha Ha !!!

That's me laughing at myself for not realizing sooner the basic fatal flaw of Cordier's pure fifth tuning argument. He states that orchestras use pure 5ths and so for a piano to match the "intonation", it must also be tuned with pure 5ths. (Do you see the flaw, yet?) But any pure 5th tuned on a piano MUST be wider than one tuned on an orchestra instrument because of the combined effects of inharmonicity. Therefore, for the pitches on a piano to match those of an orchestra, tuned with pure 5ths, the 5ths on a piano must be tempered and not be tuned pure!

Now, sure, there are many other reasons to decide to tune a piano with pure 5ths, but the reason he gave is not one of them!

Again from this paper:

http://www.temperamentcordier.org/t...ce_cordier_chambonsurlac1991_english.pdf


"...

Because I believe that the tuning which corresponds to our present-day sense of well-in-tune stems from being accustomed to the orchestral scale that our ears have been hearing for a long time. Now this sense df "well-tuned" can only correspond to the PFET up to the violin mi4 : it necessarily registers as such from the bass to the top mi4 in the chain of perfect fifths that corresponds to the way stringed-instruments are tuned (excepting double bass) so there is no question of expanding any fifths below this upper limit : up to the chanterelle/top string of the violin, our “accoutumance” to the tuning of stringed-instruments has set a kind of fail-safe limit to intonation correctness : the cycle of pure fifths, - not tempered fifths!

..."

FWIW this is a simulation of pure 5ths tuning. In it I play consecutive notes, M3rds, 4th, 5ths, octaves and a combination. Note the degree of beating in the 4ths and octaves.

Sorry the video turned out a bit fuzzy

Cool! I take it these tone lack inharmonicity? (I guess that's how you would say it, rather than say harmonic tones...) The 2:1 and 4:2 octave beating sounded symmetrical and I could not hear any 6:4 beating in the 5ths is why I ask.

It sounds pretty dreadful to me...

Chris, can you tell me what led you to tuning a temperament within a 5th in the first place, even though you no longer do it? It's been a real game changer for me and am very thankful you mentioned it.

Of course anyone who plays a stringed instrument well (I play cello) knows that constant adjustment of pitch is necessary and obviously can be done on these instruments. Almost all other orchestral instruments also have a capacity to adjust pitch on the fly, some more than others. If left to themselves, they will play essentially in just intonation because they can, and because that is what sounds the best.

Obviously the piano has no capacity to equal this and therefore must make a compromise. When forced to play with a piano, the other musicians will instinctively adjust their pitches to closely match the piano because they can, and because that is what sounds best under those circumstances.

If the piano was tuned in a well temperament strongly favoring the key of C (such as Youngs) the other musicians would instinctively adjust to that because they can, and because that is what sounds best in those circumstances (assuming of course that the music being played is in fact in the key of C, F, or G). However they may have difficulties with their intonation if the music is in E Flat, F#, or some other key remote from C.

Just a few thoughts on the matter. Some form of equal temperament is doubtlessly best compromise when a piano is involved.

I personally am interested in Bernard Stopper's perfect 12th method. Have not tried it out yet but I am definitely interested.

Pwg

What led me was a desire to see if I could tune a piano by starting off from a tiny kernel and expanding outwards. Since expanding using balancing 4th and 5th is a very accurate way of expanding it seemed that all I need for a kernel is the minimum to enable such expansion, which is a 5th, and then taking advantage of the expansion method.

My method was a bit crude and started with tuning a best-effort tempered 5th down from A4 (D4) and a tempered 4th down from A4 (E4). Then I would make a best guess F4 M3rd down from A4. I would then do a mini 4th down M3rd up sequence from A4 using the existing notes as guides. There are only three chromatic 4ths involved. I tried to make the M3rds progressive. If I got it right then the last 4th (G4 down to D4) would match. If it did not match then I would readjust the F4-A4 M3rd and start over again. I then filled in the remaining F#4 so that the M3rd were all progressive.

At that point I am ready to start expanding. Most of the time the expansion resulted in reasonable progression of RBIs but sometime it did not. It may have been simply a sloppy kernel, a poorly scaled piano, or perhaps as Kees has indicated that it is not expected that ideal chromatic progression is possible for all types of intervals at the same time. If the RBI progression was not ideal but the SBIs all sounded proper and even then I was happy enough.

Since I started to get interested in unequal temperaments I went back to tuning a full 12 note temperament octave using methods I have discussed in other threads.

The opposite to starting from a single point and working outwards is to start from two wide points and infill. One is vulnerable to accumulative errors and the other is vulnerable to incorrect linearity of the infill, for example, CM3rds with M3rd ratios that distort the linearity of the remainder of the infilling.

I was a brass player and wondered at the time why keys with flats sounded better than keys with sharps and why a Bb instrument did not always sound good with a F or Eb instrument. I know now that the problem is with the 5th partial on brass instruments. The other problem was ear training...

Ah, the Stopper tuning. I have never heard of anyone that has been able to follow his tuning instructions and have them work. Neither have I heard of an actual theory behind it. It seems to be like a one-man horse that only Mr. Stopper can ride. Good Luck!

I don't think the problem with your 5th sequence was the scaling, especially in that part of the keyboard. I think the problem was there was no way to determine if the 4ths were correct for the starting 5th. Instead, the M3s were being adjusted sort of as a compromise between the starting 5th and the chosen 4ths. If the choice of 4ths were correct, great! If not, things wouldn't go right when expanding. As you say, and I couldn't agree more, expanding a temperament with 4ths and 5ths is extremely accurate and will show up even tiny flaws in what had already been tuned.

Doesn't look like my goal for this Topic will go very far, other tuning sequences for tuning within a 5th, but am very glad to have yours!

Maybe we can categorize these sequences. As you mention there is starting with a kernel and expanding, and there is starting with a skeleton and filling it in. But it could be argued that starting with a 5th IS a skeleton that is being filled in and it can be argued that once a skeleton is filled in, you expand it. So maybe the two aren't all that different. Maybe the difference is in size and not type.

Then there are the classic circle of 5ths methods. I am think of Fischer and White. Fischer used only 5ths and 8ths and the final test was closing an octave. White used 4ths and 5ths with the final test appearing to be closing an octave, but was actually progressive RBIs.

The more "modern" way is to set ladders of CM3s and fill those in. Each ladder is connected to the others with 4ths and 5ths and 6ths, of course, that are somehow adjusted to make the temperament as equal as possible.

Then there are various sequences such as Defebaugh's (as I understand it) that rely on the relationship of M3s and M6s, utilizing both the P4 test and the outside/inside test.

Something lacking from all these is how to determine the proper 4th and 5ths. Given any octave (or other interval for that matter) there is only one width of 4th and only one width of 5th that is correct. I don't see this directly addressed in any of these sequences (although there may be some that do...), but rather we seem to be required to futz around until we can't find any errors or call it a scaling problem, or fool ourselves into believing the RBIs are progressive, when they aren't, or decide that the SBIs don't have to be all that consistent, or look at the clock and decide it best to start expanding and work out the bugs later (rough it in, and polish as you go.)

So taking the idea of there being only one correct width for the 4th, and same with the 5th, for a given octave, how can these be correctly determined without trial and error? I kinda doubt they can be. But if we take the octave out of the equation, can't we start with a given 5th and from that determine the correct 4th (or visa versa)? That way there is for the one constant only one variable, instead of two. Of course the 5th we start with must result in an acceptable octave. But is that so hard, and isn't there room for the 4ths and 5ths to expand or contract going up and down the scale? I think so, and I am thinking it is a better option than what happens with other temperaments: a kind of mish-mash where whatever interval gets the most attention wins, typically the M3s.

Oh my, that's a lot of writing. Best give it a break!

Everyone's thoughts?

Getting back to the topic, the thing that I dislike the most from the simulation of pure 5th tuning is that the M3rds,M6th and M10ths all beat more rapidly than normal ET. The combination chords that I played all had a tense sound, and naturally it is the same in all keys. Some fast RBIs are tolerable if there are calmer ones to balance and contrast as with a traditional well temperament.

Out of curiosity I just tried a pure 4th tuning which would do the opposite, i.e. those RBIs are slowed and thus calmed across all keys. The unfortunate consequence is that octaves are contracted which would do the opposite to stretching a piano when expended. I don't think a whole piano would sound very good despite a short section in the middle of the piano sounding sort of kind of ok.

As usual the topic is morphing around a bit. I am thinking that setting the temperament within a 5th is another one-man horse - I'm the only one that rides it.

I got goofing around with inharmonicity numbers one time and came up with some figure for negative inharmonicity that allowed for both the 4ths and 5ths to be pure. It wasn't a curve, but the same value for all notes. Who knows what it would sound like!

I didn't care for the sound of the chords in your simulation either, and wondered if they were 2nd or 3rd inversion, having the widened 4th, rather than the pure 5th which would be found in the root position. Having a widened octave will of course made all wide intervals beat faster and all narrow intervals beat slower, but unless you have absolute pitch, I doubt it would be obvious when listening to individual intervals. But listening to entire chords, where the beat ratios between various intervals have changed, might be a different story. Perhaps this is some of the disgust I feel when playing shorter pianos...

I think a compromise between a slightly wide octave and not so narrow 5th works best and also results in nearly pure 12ths.

I just now saw this and did not have time to read everything. But since my name was mentioned, it was the book by Lucas Mason, "The New Tuning" that was published around 1985 that I have and know about. Cordier is apparently from France, so Isaac Oleg knew of his idea which is apparently an identical one.

It is not really something that anyone can claim is an idea exclusive to oneself, even if a certain person did discover and think it all out on their own. The fact is that an equal temperament can be created within virtually any size octave, from a 2:1 (or even narrower than that)to a 6:3 and beyond. Thia has not always been very widely understood, however. It was Kent Swafford RPT who told me about that many years ago.

I won't comment about Cordier because I have not studied any of that nor will I make any specific comments about Lucas Mason's book. Where I think the idea for an octave wide enough to cause the 5ths to be (or seem to be) pure comes from a specific market, namely New York City where there are lots of Steinway pianos and many of them in performance venues.

The Steinway scale design has high tension (any dealer will make a point of saying that) and high inharmonicity. The higher inharmonicity lends itself to a wider octave. The goal is often to please a concert artist who uses the whole keyboard and does not do much simple playing in the middle of it.

Therefore, it is more important to have the outer octaves sound more consonant and for triple and quadruple octaves sound in tune than it is for midrange octaves. Everybody likes the sound of a pure 5th. So, it is all a trade off of favoring one kind of interval over another. If the large piano has bright voicing and is tuned that way, the outer octaves will have a pleasing pitch to them and less beating between single octaves.

The trade of is that close harmony in the midrange sounds "tight" and "tart". The 5ths are pure but the 4ths beat 2-3 beats a second and all Major thirds, 19th and 17ths beat more rapidly. I went through my own period when that is what I did. When I went to study at Steinway in 1986, that is the way I tuned my piano for the instructor. He was not very impressed by it. He said it would produce a very brilliant sound and I said that is what I was after (being full of what I had read in Mason's book). My instructor said that some clients may want a mellower sound.

I have been through it all and I have to say that I would never tune a piano this way now, nor would I ever have for at least 30 years. There are better ways to achieve a more optimum compromise.

Thanks for chiming in, Bill. The title I chose for this Topic wasn't the best. What I am really interested in is the sequence used for tuning a pure 5th temperament, rather than actually tuning a piano with pure 5ths. See, I worked out a sequence for tuning ET within a 5th (tempered or otherwise), which is working very well, and was wondering what other sequences for tuning within a 5th might be known. I figured the most likely place to find them would be for tuning with pure 5ths, which I don't do either. Neither Cordier nor Mason seem to set ET within only a 5th.

So while I have your attention, Bill, do you know of any sequences for setting ET within a 5th? Btw, how 'ya doing?

Got Mason's book. It didn't come covered in dust, so I will have to take care of that detail myself, if need be.

I am mostly interested in how the temperament is set and came across some similarities to what I am doing. Starting with a D-A 5th, (his is D4-A4, mine an octave lower when possible), the 4ths of D-G and E-A are tuned in each of our sequences. His then tunes the 5ths of C-G and E-B in order to be able to compare the M6s of C-A and D-B and I guess have these in place for later in addition as an RBI speed check to help confirm the correct tempering of the 4ths. But they are not needed for the next step which is exactly the same as mine: Tune the D#-G# 4th so that the chromatic M3s of D#-G and E-G# are properly progressive.

Mason correctly mentions how D# and G# will tend to be more accurate than others in the sequence and that they are opposite in the circle of 5ths from the beginning notes of D and A. (D-G# and D#-A are tritones.) Since his sequence covers an octave, the circle of 5ths is a consideration. Mine is limited to a single 5th, and so there is no completed circle of 5ths. But he isn't clear as to why they tend to be more accurate and I'd like to mention this because it is a somewhat self-correcting feature of both of these sequences.

Consider that D and A are correct and fixed, and the 4ths of D-G and E-A are equal in size (beat the same) but are, say, 1 cent too wide. We would have the following errors:

D 0.0
D# ___
E -1.0
F ___
F# ___
G +1.0
G# ___
A 0.0

Now when tuning the 4th of D#-G#, the one cent wide error of the 4ths is repeated, but is symmetrical, as the error of the notes they are compared to (E and G) have symmetrical errors, one being sharp the same amount the other is flat. So after tuning D# and G# we have:

D 0.0
D# -0.5
E -1.0
F ___
F# ___
G +1.0
G# +0.5
A 0.0

As you can see, the 1 cent error in the tempering of the 4ths did not compound, which would make the error greater as when going around the circle of 5ths, but rather the error in tempering was halved for each note by tuning the 4th "symmetrically". (Kind of an awkward phrase...)

Of course much depends on how close the 4ths are equally tempered to each other and how "properly progressive" the chromatic M3s really are. Still, it may be preferred by many to a typical 4th and 5ths sequence.

Another similarity is when tuning F and F#. Mason and I both fit these into the middle of a pair of contiguous intervals. The difference being his are CP4s (C-F-A# and C#-F#-B) while mine are cm3s (D-F-G# and D#-F#-A). This too will tend to "halve the errors", and when the resulting RBIs are compared to intervals where the errors were not halved, then the type and direction of the errors can be determined: symmetrical or unsymmetrical, too much or too little tempering of the 4ths.

Don't know how much interest any of you might have in this, so I'll wait for comments before continuing. Not sure where I might go with this discussion, anyway...

My eyes are beginning to glaze over! 😂

Pwg

Oooo...

Here is something to try. Take any already tuned minor 3rd, like A-C. Guestimate the A-C# major 3rd by putting C# where it seems about right. Now tune G# to C# as a tempered 4th. Next listen to the progression of the major thirds of G#-C and A-C#. If they progress too fast, both G# and C# are too sharp. If they progress too slow, or regress(?) then both G# and C# are too flat. You can shim one against the other for fine tuning.

Well, it's a nice sequence. However there are some points I don't like.

First of all: I miss the octaves which are absent in this sequence. In other less stretched ET tuning sequences octaves are a valuable tool used as tuning or check intervals. Here they are tempered and can not be used to tune or check a note.

Second: pure fifths. There is a range of pitches where a fifth can sound pure. One can inadvertently make random errors when tuning several "pure" fifths.

However I like this sequence because it's self-correcting. The tuning of C and B makes available the M3s CE and GB which can be compared with D#G and EG# and errors in the tempering of fourths become obvious.

In your example with fourths too wide by 1 cent, we will have CE and GB too narrow by 2 cents while D#G and EG# are too wide by 1.5 cents. The error of 1 cent in fourths is highlighted to a difference in the tempering of M3s of 3.5 cents.

The same self-correcting character is present in the Bremmer's sequence for tuning CM3s.

After you tune A4 and A3 let's suppose you make a bad estimation of F3 and you tune it 1 cent sharp. When tuning F4 it will be also 1 cent sharp. Now you are going to tune C#4. If you tune it, as Bremmer advises, from A3 looking for a 4:5 progression between F3A3 and A3C#4, C#4 will be flat by 1 cent.

Why? Because F3A3 is too narrow by 1 cent. So if you tune a A3C#4 to beat in the 4:5 ratio compared to F3A3, A3C#4 must be too narrow by 1 cent and C#4 will be flat by 1 cent.

F3 = +1
A3 = 0
C#4 = -1
F4 = +1
A4 = 0

Now, F4 is sharp by 1 cent. That means C#4F4 is too wide by 2 cents.

F3A3 = A3C#4 = F4A4 = +1 cent
C#4F4 = -2 cents

There is a 3 cents difference between the tempering of C#4F4 and the 3 other M3s.

The error of 1 cent in the tuning of F3 is "highlighted" to a 3 cents difference in tempering of M3s by this sequence.


When tuning C#4 in this sequence you must avoid hearing alternatively to A3C#4 and C#4F4. Because you will tune C#4 right in the middle to have these two M3s equally tempered, putting C#4 0.5 cents sharp. By doing so you will have disguised the error in the tuning of F3.

F3 = +1
A3 = 0
C#4 = +0.5
F4 = +1
A4 = 0

F3A3 = -1
A3C#4 = +0.5
C#4F4 = +0.5
F4A4 = -1

In this case you have less of a difference between the tempering of the M3s. In fact you have half the difference than in the first example. And the error in the tuning of F3 may be not so obvious.

You make some good points, Rafael. Some I'd like to question, others I'd like to challenge. I'll have more time to give a proper response tomorrow, then I'll be away for the weekend. (Taking some boys camping.)

Want to mention just one thing, or rather repeat one thing. I am interested in the sequence rather than actually tuning with pure 5ths.