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I quote a paragraph from BB's article:

It is really very simple but cuts right through to the very reason tempering is done in the first place. That is, the Pythagorean Comma, the "gap" between what 7 pure octaves and 12 pure 5ths would create: about 24 cents. It uses the piano's natural inharmonicity to fill that gap and goes beyond that to satisfy the ear's enigmatic desire to hear the upper registers sharper than they should be theoretically.

Starting with a temperament octave constructed with an appropriate amount of stretch (a compromise between a 4:2 and a 6:3 octave for Equal Temperament (ET)or just a plain 6:3 octave for any of the milder Well Tempered Tuning)[Note:the earlier HT's are more appropriately done with a minimum amount of stretch], begin stretching the octaves by comparing the octave itself, moving closer to a pure 5th but being careful not to make the resultant 4th sound too "busy".

Once you get into the 5th octave, you will notice that the beating in the 4th ceases to be of concern. That is because the coincident partials are high enough that they become so weak as to be inaudible. As the 4ths become less significant, you can concentrate more on the compromise between the octave and the 5th, trying to equalize the beating between them the best possible.


(1) First of all, what has the Pythagorean Comma to do with what is essentially Equal Temperament ?

(2) Secondly, could someone use examples to illustrate the words underlined ? I really couldn't catch what the author was trying to say.

Oh... everything, as ET is the way of distributing that discrepancy of 24 cents evenly over all the 12 fifths in the octave.

But, I might have misunderstood you - maybe rephrasing the question would help?



Yes... seeing it like this, the text presumes some practical piano tuning procedures that aren't really explained further. I'll try to rewrite Bill's text with extensive information:

begin stretching the octaves by first comparing the octave itself, then the fifth and fourth directly below the upper note (in stretching upwards) or above the lower note (in stretching downwards), moving closer to a pure 5th but being careful not to make the resultant 4th sound too "busy".

Does this help?

Thank you Patrick for trying to explain this. If your explanation was not sufficient, I can try to explain further. From what I understand, the original poster is not a native speaker of North American English. I am very sensitive to that kind of barrier in understanding and am willing to do whatever it takes to communicate properly.

The original poster is quoting from an article written several years ago. I have learned much in the meantime about writing. That is why I made note of the fact that the material needs to be edited (or really completely re-written) but the basic ideas are there and have not changed.

I fully understand how ET is the result of the 'comma' being evenly distributed across the keyboard.

But the author seems to suggest that the 'new' method of tempering octaves is achieved by some special method of distributing the 'comma'. Then again, the author is not formulating a 'new' temperament. The author is tempering octaves on what is essentially an ET. ET has already been tempered, and comma has already be evenly distributed. So my first question is what more can we 'further' temper the ET without distorting the ET ?

I assure you it has nothing to do with my understanding of the English language, be it North American or British English. It's all about 'gaps' that are missing in the chain where information is being presented. These gaps are not apparent to people who already have a good grasp of the theory of piano tuning.

For example, if you speak about a C key and refer to its 'close' keys as F and G, it's something that can be picked up easily by someone who has prior knowledge of the Circle of 5ths. But to someone who is new to such parlance, he will be dumbstruck because if one were to look at the keyboard it's D and E that are close to C, not so much F and G. If there was no previous reference to the Circle of 5ths, a newbie would not be able to relate what was being intended but conveniently omitted as a result of some calculated presumption.

Being systematic is not an easy feat, especially for experts who are over familiar in their field of work, as the more an expert a person is, the greater the temptation to presume.

I may be wrong here or misunderstanding the question, but once the temperament is tempered as ET, the octaves are simply the spreading out of that temperament, and if you slightly stretch them as you go, wouldn't that be considered 'further' tempering the octaves without distorting the ET?

Forgive me if I'm totally off point here. I may be chiming in on something that is beyond my way/style of analyzing tuning.

Just a little bit, I'm afraid.

What does the author mean by 'stretching' the octaves ? And without a ETD, how does one stretch the octave with precision ?

What is the purpose of comparing the 5th and 4th directly below the upper note ?

How do we compare ?

The above are questions that naturally spring out when one reads the paragraph. It's quite a common feature in most of BB's articles that I have read. A method is presented, but the purpose often not explained.

Looks like I'm not the only person around with the same question.

It is so difficult to even tune octaves, at least for me. I am a hobbist. Actually sound samples with videos would be very useful in understanding. Come to think of it, its also not easy for me to tune unisons

Cashley,

My understanding is that octave stretching has to be done in order to accommodate the inharmonicity of piano strings. It has nothing to do with the temperament or the Pythagorean Comma. The temperament is set in the mid-section. Once that is done, the Pythagorean Comma has been distributed (by which ever means) over the 12 semitones of the temperament octave.

Now, after setting the temperament...

1) the bass of the piano has to be tuned so that
a) the octaves, fourths, fifths and other intervals are as pure as possible, and
b) the harmonics (partials) of the bass strings do not beat too harshly with the mid-section and treble section of the piano.

and...

2) the treble of the piano has to be tuned so that
a) the octaves, fourths, fifths and other intervals are as pure as possible, and
b) the harmonics (partials) of the bass and mid-section do not beat too harshly with the treble.

Because of inharmonicity, in both cases (bass and treble), the requirements of (a) and (b) are mutually exclusive, and a compromise has to be found between the two. If there were little or no inharmonicity (e.g. thin, low-tension harpsichord strings), there would be little or no need for a compromise. But in a piano, there is significant inharmonicity. So, what to do?

Both when tuning the bass and the treble, octaves must be stretched in order to achieve a compromise between (a) and (b).

If no stretch is applied, then the bass sounds too sharp (even tough octaves are pure) and the treble sound flat (even though octaves are pure). Again: this has little to do with the temperament. (Although Bill writes that different octave sizes are suited to different temperaments - I'm not sure why he writes this.)

That is my understanding, as a layman - perhaps a pro can confirm this - I'm open to correction.

Cashley:

The article you quoted brought up some questions in your mind. Rather than try to explain the article, I will try to answer your questions.

” So my first question is what more can we 'further' temper the ET without distorting the ET ?”

The ET is not what is tempered. The intervals are tempered from just intonation to produce ET.

The Pythagorean comma is based on a 2:1 octave ratio. But what if the octave ratio is 2.2:1? Then the octave would be “stretched” or “tempered” and the same would happen to the comma. And so each interval would be tempered slightly differently to produce ET.

With harmonic tones, (no inharmonicity) the values of all notes can easily be computed by using the 12th root of the octave ratio. So we can have an ET based the 12th root of any number.

I will stop here and see how well Mark answered your other question.

I think you have a sharp mind and will be able to understand all of this if it is presented to you in order.

Mark, your understanding is good and thorough.

I suggest reading this article from Wikipedia on the subject of inharmonicity first: http://en.wikipedia.org/wiki/Inharmonicity

Theoretically, an octave on a piano has the upper note's string vibrating at twice the frequency of the lower. Because of inharmonicity however, the upper note must vibrate at a slightly greater speed in order for the octave to have no beat. That means that when a piano technician tunes an octave so that it sounds as it is typically expected to sound, a "pure", beatless or "just" sound, (all three terms are synonymous), the upper note is naturally tuned slightly sharper than it would be theoretically. The octave is therefore, "stretched".

All strings have more than one frequency that is heard when they vibrate. There is the obvious, the fundamental frequency, the one we identify the string as having, such as the 440 cycles per second of the note A4 (the A above middle C).

All strings have overtones (also called harmonics). North American piano technicians use the adjective "partial" as a noun and then pluralize it to be synonymous with overtones and harmonics except that the fundamental is identified as the "first partial". ("Partial" may simply be short for "partial tone".) The words overtone and harmonic could only imply a higher frequency than the fundamental.

The partial series is like a large dominant 7th chord in music: Fundamental (1st partial), Octave (2nd partial), octave-5th (3rd partial), Double Octave (4th partial), Double Octave-Major 3rd (M3) (5th partial), Double Octave-5th (6th partial). Double Octave-minor 7th (7th partial) and Triple Octave (8th partial). The partial series is infinite but the higher the partial, the fainter it is, so generally, tuning usually only involves listening to and arranging the first 8 partials because beyond that, they are too faint to be of any consequence. The exception to that is in the very low Bass, where partials beyond the 8th can be clearly heard.

It is also important to understand that because of inharmonicity, the higher a partial is, the sharper it will be. Between the first and second partial, there is only a small amount of inharmonicity. In the midrange, that is usually between 1 and 2 cents. A cent is defined as 1/100th of a theoretical half step. A half step is the difference in frequency from one note to the next.

Between the 1st and 4th partial, there is usually 2-4 cents difference. Between the 1st and 8th partial, there is usually 18-20 cents difference. This creates a dilemma when tuning octaves.

Take the example of tuning C4, C5, C6 and C7. If C5 is tuned from C4 so that the first partial of C5 is exactly in tune with the second partial of C4, the octave will sound perfectly "pure". So far, so good. Now, if C6 is tuned similarly from C5, the C5-C6 octave will also sound pure but the double octave, C4-C6 will be slightly narrow and exhibit a slight beat. If C7 is similarly tuned from C6, the C6-C7 octave will sound pure but the triple octave C4-C7 will be quite narrow and beat in a manner that would be found objectionable to virtually any musician.

For the Triple Octave to sound pleasing therefore, each of the octaves between C4 and C7 must be tuned slightly wider than the point where they sound perfectly beatless. There has been much controversy about this lately, not only on this forum but within PTG to the point where angry letters have been written to the Journal editor and members have threatened to quit the organization over the issue.

Some technicians claim that they can tune single, double and triple octaves so that they all sound "pure". The key to understanding what those people are saying is the word, "sound". They all sound pure. That means that the beat that is put between the octaves is so slight that it is virtually unnoticed. So, the argument is really over a choice of words, not what technicians actually do.

If one technician says, "I can get 'em all pure", another may say and rightly so in his own mind, based on the words alone, "That is impossible" and thus the fist fight ensues.

Bearing in mind how I described the tuning of the notes from C4 to C7, the temperament, regardless of what temperament that is, ET or any other, the temperament will naturally be "distorted" as Byron aptly and accurately imagines. The whole arrangement will be stretched out.

The intervals, 4ths, 5ths, M3s and M6s which are heard in the midrange can no longer be perceived in the treble and high treble. M3s and M6s cannot be perceived much beyond A4, 4ths cannot be perceived much beyond F5 and 5ths cannot be perceived much beyond C6.

So, to check for accuracy, technicians use multiples of these intervals: M10, M17 and M12. That is, octave-third, double octave-third and octave-5th.

Take the F3-A3 M3 interval for example. If the F3-A3 M3, the F3-A4 M10 and the F3-A5 M17 were to all beat the same, other intervals such as 5ths, octaves, octave-5ths and double octaves would be narrow and objectionable. Therefore, the M10 and M17 must beat progressively faster as a compromise.

Starting with the F3-F4 octave being tuned, (usually with a slight beat between F3 and F4) one goes to tune F#4. It is known that the F#3-F#4 must also have a slight beat but the question is, "How much?" To answer Cashley's question about what is meant by comparing the 4th and the 5th, I will put it this way: The octave is tuned very slightly wide of a pure sound but that amount of width must be very carefully controlled.

Tuning F#4 from F#3 as a slightly wide octave will cause the B3-F#4 5th to be less tempered. If one makes the beat in the F#3-F#4 octave be the same as it is in the 5th down from the top note of the octave, (B3-F#4), it is considered an optimum compromise. It will also cause the C#-F# 4th to beat faster.

One only needs to make sure that the faster 4th is not altogether too fast. It should not exceed 2 beats per second. 4ths are Slowly Beating Intervals (SBI) as opposed to Rapidly Beating Intervals (RBI) such as M3s and M6s. I would say that any amount of beating that is 2 beats per second or less, is a slow beat. Any amount that is 3 beats per second or more is a rapid beat. That is just my impression. One could go either way with it.

There has to come a point at which a beat is very slow and I would consider that to be synonymous with a "slight" beat. I would consider a "slight" beat to be any amount of one beat per second or less. These are important because they lie within the realm of what are generally considered to be imperceptible by musicians. A very slow or slight beat is taken in a musical context as "pure".

Again, Cashley, as you learn more about tuning, you will come to understand that creating these very slow beats is a way to "hide" them from perception. They can be used to find a way to solve the inevitable dilemma in piano tuning that is caused by inharmonicity.

Please take note of my choice of word, "dilemma". It implies that there really is no solution. That much is true. For every action taken, there is a consequence, an equal and opposite reaction. Any interval that is deemed to be "improved" will always mean that another is made worse. The art of tuning involves finding the exact compromise between what is made better and what is made worse.

That is what I mean by comparing the 4th and the 5th. When ascending from the temperament octave, the 5th below the top not of the octave being tuned is always compared to the 4th below the top note of the octave being tuned. When descending from the temperament octave, the 5th above the bottom not of the octave being tuned is always compared to the 4th above the bottom not of the octave being tuned.

At some point, when ascending, the 4ths can no longer be heard because their partials are too high and faint. At that point, they no longer matter. That point is generally at D5-F5. There is a similar point in the Bass. Off hand, I would say it is at or about F2-A#2.

I hope this helps but I am sure it will raise yet more questions. I will answer any more questions as they may come up when I can.

Thank you, Mark, for attempting to explain. You have explained inharmonicity in general, but I'm afraid we're looking at different issues. Of course, my question concerns inharmonicty, so any mention of inharmonicity would seem related at first glance.



I could sense a 'big' jump from your earlier explanation about inharmonicity. It appears to be a conclusion, but what is the justification ? This is the problem I have always faced in this forum.

When you say 'even though octaves are pure', do you mean pure as in beatless or pure mathematically ? If the octaves are beatless, doesn't that mean the inharmonicity has already been accounted for ?

I concur, if you want to be grammatically correct. But I believe you knew what I meant.



I think we need a co-understanding of the meaning of the word 'stretch' for a start. I just don't understand how the stretching is related to the comma.

I think we're slowly moving away from the question. Many seem to think that my question is 'what is inharmonicity ?' or 'how does inharmonicity affect ET ?'

It's the wording of the article that throws me into confusion. The 'gap' in ET has already been accounted for - with or without the article mentioned.

Cashley:

“ I think we need a co-understanding of the meaning of the word 'stretch' for a start. I just don't understand how the stretching is related to the comma.”

OK, let’s look at what a comma actually is. There are a number of different ones. The Pythagorean comma is how much you “overshoot” seven octaves when tuning 12 contiguous fifths.

Let’s say that the chosen octave is not a 2:1 ratio. And say we “stretch” the octave to a ratio larger than 2:1. Since the octave is no longer a just interval, it could be considered “tempered”. Now when we tune 12 contiguous fifths the comma is smaller and the tempered fifths are tempered less than for a 2:1 octave ratio ET. BUT the fourths are tempered more.

I will stop here to see how far this takes you.

Thanks, Mr Bremmer. I appreciate your time and patience.



This is my question. Why ? Why usually with a slight beat ? I would have thought usually meant beatless.



Less tempered as in less narrowed so that it inclined towards pure ?



Which 5th are you referring to ? And why is it considered an optimum compromise ?



Which C＃ and which F# ?

Frankly, I don't where we're heading

Does stretching means instead of a 2:1, it's a 4:2 or 6:3 ? That was my question when I asked for a co-understanding of the term 'stretch'.

I do understand that even if each octave is tuned pure (so that for that particular octave concerned inharmonicity has been accounted for), when you compare double octaves the higher partials from the lowest note in the lower octave would have overshot the fundamental partial of the highest note in the higher octave; and the situation worsens if you compare 3 octaves.

Please don't forget this phrase as this is the root of my question:

'It uses the piano's natural inharmonicity to fill that gap'

The word 'it' refers to the suggested method of tempering octaves; and the word 'gap' refers to the 24-cent comma.

And if my understanding [of piano tuning, and not North American English] is correct, the 24-cent comma has already been accounted for and distributed evenly across the keyboard. Each note in the ET now occupies a 'fixed' position. Any tempering that deviates from the fixed position would make it un-ET. Am I right ? And any tempering of the octaves would move the tempered note away from the 'fixed' position.

Cashley:

I think we are getting closer.

”Does stretching means instead of a 2:1, it's a 4:2 or 6:3 ? That was my question when I asked for a co-understanding of the term 'stretch'.?”

Yes, the higher the matching partials that are used, the greater the stretch.

”I do understand that even if each octave is tuned pure (so that for that particular octave concerned inharmonicity has been accounted for), when you compare double octaves the higher partials from the lowest note in the lower octave would have overshot the fundamental partial of the highest note in the higher octave; and the situation worsens if you compare 3 octaves.”

It depends on the partials of the octave and the double octave. Whenever two contiguous 4:2 octaves are tuned, the result will be a wide, not narrow, 4:1 double octave. But when two contiguous 2:1 octaves are tuned, it will usually result in a narrow 4:1 octave.

The actual effects of inharmonicity are different than many think they are.

” 'It uses the piano's natural inharmonicity to fill that gap' ”

I will leave the explanation of this phrase to the person who wrote it.

”And if my understanding [of piano tuning, and not North American English] is correct, the 24-cent comma has already been accounted for and distributed evenly across the keyboard. Each note in the ET now occupies a 'fixed' position. Any tempering that deviates from the fixed position would make it un-ET. Am I right ? And any tempering of the octaves would move the tempered note away from the 'fixed' position.”

I can think of a number of approaches to addressing this.

Equal Temperament means that all intervals of each type are tempered from just intonation equally. If desired, fifths could be tuned beatless and the result will still be ET because all intervals of each type are tempered equally.

With harmonic tones, this will produce octaves that are all stretched equally. On an actual piano, because of the many different effects of iH, the octaves will not be stretched equally. The strictest definition of ET is impossible to satisfy with a real piano. This is because if a particular just fifth has a ratio of, say, 3.1:2 another may have a ratio of, say, 3.2:2.

So tuning ET on a piano is not about tempering all intervals types the same amount from just intonation. It is about making it seem that way by tuning so that beatrates are progressive. And by progressive, this can include getting steadily faster and then steadily slower. And it is possible to do so with many different schemes for stretching.

I'll chime in for a bit


Since ET disguises the pythagorean comma by making the 5ths a bit narrow and the 4ths a bit wide, these intervals act like a kind of outer boundaries to your temperament stretch. If you stretch too much, the 4th will sound 'busy', 'twangy' (because it is too wide). If you stretch too little, the 5th will sound 'sour' (because it is too narrow).



Because you will have to deal with inharmonicity anyways in extending the temperament upwards and downwards, most tuners consider it's a good idea to make the initial temperament octave a little wider than a theoretical 2:1 octave, in order to make the transition from temperament to temperament extension as smooth as possible.

Most tuners tune at least a 4:2 octave, many tune wider than that (between 4:2 / 6:3, or simply a 6:3). Going wider, you'll start to hear a slight beat (a slow "wave" or roll) that - again, to most tuners - isn't really offensive… at least not until you get beyond 6:3.


Yes, exactly!



Bill is referring to the 5th a) below the upper note, in tuning upwards, b) above the lower note, in tuning downwards. That is, the 5th from the note you tune, towards the direction you came from

It is, by most tuners, considered the best compromise, because the two cornerstones of harmony - 1) the unison 'echoed' 12 half steps above/below, ie the octave (2nd partial), and 2) the fifth (the 3rd partial) - sound as good as they possibly can do in the same tuning.

With instruments of non-fixed pitch (string instruments, the human voice), you can adjust the intervals on-the-fly, and most musicians naturally intonate towards just intonation (this I think Bernhard Stopper would disagree on, but my musical experience tells me otherwise, so I just rely on that )

Dealing with the piano, however, we have to set the pitches so they work for all intervals. Hence the compromise Bill is talking about.



Bill is talking about tuning F#4 from F#3. The fourth (C#) follows the same logic as I described above. In this case it is C#4 - a 4th from the note you tune, towards the direction you came from.



Jeff is right on the money, and this should definitely straighten out many of your question marks!



Stretching is not only justified - it is desperately needed Just as Mark says, due to inharmonicity, if the octaves are matched on their 2:1 partials (= what Mark describes as pure), the bass is painstakingly sharp, and the treble is unbearably low. To the listener, in any musical context, they piano sounds horribly out of tune.


Mark uses the description 'pure octaves' for a 2:1 partial match.