Sax on the Web Forum banner
1 - 20 of 21 Posts

·
Registered
Soprano: 1983 Keilwerth Toneking Schenklaars stencil
Joined
·
926 Posts
Discussion Starter · #1 · (Edited)
A few months ago, I was intrigued by a forum debate about whether the frequency of a pitch was always the same. One group argued correctly that the exact frequency of a note varied depending where it was in a given scale (Just temperament). The other group argued that when A4=440 Hz, C4, for example, is always 261.63 Hz. As it turns out this is also correct from a certain point of view (Equal temperament). To make my illustration even more perverse, If we start a C4 scale at 261.63 Hz with just temperament A4 is in tune at 336.05 Hz.

For a sax player, just temperament, tuning by ear, is the rule. For a piano or fretted guitar, it is all about equal temperament. Most tuners also tune to equal temperament. there is a lot on the internet about all of this and why. Here is a good explanation.

I wondered if I was playing in tune by ear (just temperament) and I looked at a tuner (equal temperament) at that moment, what would it say? I didn't find this chart on the internet, so I posted it here for reference.

So, if I play a major scale and my assistant notes on my tuner that I am 4 cents sharp on the Maj 2nd and 13.5 cents flat on the Maj 3rd, I am spot on.

Interval __________ Cent Difference
Fundamental _______ 0.00
Minor Second ______ 11.8
Major Second ______ 3.9
Minor Third ________ 15.6
Major Third ________ -13.7
Fourth ____________ -1.9
Augmented Fourth ___ -9.7
Diminished Fifth ______ 9.8
Fifth ______________ 1.9
Minor Sixth ________ 13.7
Major Sixth ________ -15.6
Minor Seventh ______ 17.6
Major Seventh _____ -11.7
Octave ____________ 0.0

This is one reason experienced saxophonists recommend putting away the tuners and learning to listen.

It also makes me wonder how many intonation problems are due to tuner operator error rather than the saxophone.

[Edit] There are a number of methods used to calculate the "correct" ratios for just temperament. The ratio for the minor second was originally calculated as 25/24 but appears now as 16/15. Both are correct, but the latter is more common. I also added the augmented fourth.

The cent system is exponential, so the difference between 0 and 1 cent is much smaller than the difference between 99 and 100 cents. That said, most people cannot distinguish two notes played sequentially less than about 5-6 cents, and simultaneously with about 2-3 cents.
 

·
Registered
Joined
·
334 Posts
Hmmm...if you are playing sax (or another instrument with just temperament as suggested above) with a guitar or piano that is equal temperament, then the just will have to be equal as well...so it would seem...agree that listening is the key regardless.
 

·
Registered
Soprano: 1983 Keilwerth Toneking Schenklaars stencil
Joined
·
926 Posts
Discussion Starter · #3 ·
You are right. Listening is the key, or so I hear.
 

·
Super Moderator
Joined
·
26,299 Posts
Right you are. Listening is the key.

If you are playing using a form of just intonation, you must make the adjustments according to the key that you're playing in. It becomes second nature after awhile.
 

·
Registered
Joined
·
4,725 Posts
I tune pianos. After I set my starting A which I get from my metronome the rest is by ear. I do use equal temperment. I set The octave from F to F around middle C using 4ths, 5ths, major
3rds and 6ths until the major 3rds starting with F-A increase beating frequency gradually. The octace F's then will line up. To tune the rest I use octaves, 4ths, 5ths, Maj. 3rds, 10ths, double and triple 10ths. This stretches the tuning using the harmonics. There is always a compromise up and down. Some pianos like saxs have better or worse inherent tuning problems, but it's not one note but certain areas that don't line up well with the rest of the piano.

There are no perfect intervals in a piano.
Even the unisons vary.
 

·
Super Moderator
Joined
·
26,299 Posts
I tune pianos. After I set my starting A which I get from my metronome the rest is by ear. I do use equal temperment. I set The octave from F to F around middle C using 4ths, 5ths, major
3rds and 6ths until the major 3rds starting with F-A increase beating frequency gradually. The octace F's then will line up. To tune the rest I use octaves, 4ths, 5ths, Maj. 3rds, 10ths, double and triple 10ths. This stretches the tuning using the harmonics. There is always a compromise up and down. Some pianos like saxs have better or worse inherent tuning problems, but it's not one note but certain areas that don't line up well with the rest of the piano.

There are no perfect intervals in a piano.
Even the unisons vary.
Not to mention that the hammer on the piano strikes multiple strings. The piano techs I know purposely slightly detune them to make the piano sound 'bigger'. It's the same effect that you get from many string players playing a unison, but with slightly different vibratos. It sounds 'big'.
 

·
Super Moderator
Joined
·
26,299 Posts
Can we get a recount on this??
Yeah, thanks Carl. Twocircles, it's Cents (1/100 of a semitone), not Hertz (vibrations/second). I think you used Hz for the calculations. Remember, a semitone is the 12th root of an octave, and a cent is .01 of that.

If you look at the mean tone temperment comparison, you find that the greater differences in Hz occur at the higher frequencies
 

·
Registered
Joined
·
495 Posts
A few months ago, I was intrigued by a forum debate about whether the frequency of a pitch was always the same. One group argued correctly that the exact frequency of a note varied depending where it was in a given scale (Just temperament). The other group argued that when A4=440 Hz, C3, for example, is always 261.63 Hz. As it turns out this is also correct from a certain point of view (Equal temperament). To make my illustration even more perverse, If we start a C3 scale at 261.63 Hz with just temperament A4 is in tune at 336.05 Hz.
Sorry, your calculations are way off. Starting with C3=261.63Hz and going up by fifths (C - G - D - A) in just temperaments we end up with A5 at 883HZ. One octave below means A4 = 441.5Hz.
 

·
Registered
Soprano: 1983 Keilwerth Toneking Schenklaars stencil
Joined
·
926 Posts
Discussion Starter · #10 ·
Thanks for the comments.

Can we get a recount on this??
Sorry, typo. It should be C4 not C3. I'll see if I can correct it. For details see this page

@adamk I probably should not have been so black and white about pianos and guitars. They don't sound their best when strictly tuned with equal temperament. And, that was basically one point. If we are trying to tune every note of our saxes and other instruments with an equal temperament tuner, we are making efforts in the wrong direction.

What got me thinking about all of this was that I happened upon a site that charted the blind intonation of saxes with the ET pitch being 0. This seemed biased to me. I thought a better way would be to see how much a pitch could be bent sharp and flat for a given fingering with a given player, sax and setup. However, I did not know how many cents of out of tune a JT pitch might be from its ET counterpart.
 

·
Registered
Joined
·
4,725 Posts
Good pun: black and white.
I just had to repair a set of ivory keys and re-black the blacks.

What about that Japanese mechanical sax player?
Could it be a way to measure intonation on various saxs since the support and "lip" pressure are a constant?
 

·
Registered
Soprano: 1983 Keilwerth Toneking Schenklaars stencil
Joined
·
926 Posts
Discussion Starter · #12 ·
What about that Japanese mechanical sax player?
Could it be a way to measure intonation on various saxs since the support and "lip" pressure are a constant?
I haven't seen this yet.
 

·
Out of Office
Grafton + TH & C alto || Naked Lady 10M || TT soprano || Martin Comm III
Joined
·
30,105 Posts
What about that Japanese mechanical sax player?
Could it be a way to measure intonation on various saxs since the support and "lip" pressure are a constant?
I think something with a variable lip pressure such as the Pillinger artificial embouchure would be better as you could measure at varying dynamics, which is what people would want I think.

 

·
Distinguished Technician & SOTW Columnist. RIP, Yo
Joined
·
17,204 Posts
Good pun: black and white.
I just had to repair a set of ivory keys and re-black the blacks.

What about that Japanese mechanical sax player?
Could it be a way to measure intonation on various saxs since the support and "lip" pressure are a constant?
Are they constant??? I don't believe they are for an accomplished player. If I were making a robot I would not make them constant. (IMO pressing down an octave key is not all that is needed to rise an octave while maintaining good tone. It's just an assistant, and this becomes more obvious with the larger instruments - and to any flute player)
 

·
Distinguished Technician & SOTW Columnist. RIP, Yo
Joined
·
17,204 Posts
Interesting OP. And we get customers and ubertechnicians making a fuss over a couple of cents here and there.
 

·
Registered
Joined
·
28 Posts
Equal and just temperaments are based on wrong math models. Equal one consider octaves as turns of spiral but calculated as turns of circle. The paper below introduces Pi-octave that is a bit wider than regular octave.
The paper is russian but I hope the google-translator will help you.
The experiments with piano tuning show that this model is at least more adequate and ready to use for real tuning.
http://chernov-trezin.narod.ru/ZS_1_borbat.htm
http://translate.google.com/transla...rezin.narod.ru/ZS_1_borbat.htm#serebrchastoti.
 

·
Registered
Soprano: 1983 Keilwerth Toneking Schenklaars stencil
Joined
·
926 Posts
Discussion Starter · #19 · (Edited)
Equal and just temperaments are based on wrong math models. Equal one consider octaves as turns of spiral but calculated as turns of circle. The paper below introduces Pi-octave that is a bit wider than regular octave.
The paper is russian but I hope the google-translator will help you.
The experiments with piano tuning show that this model is at least more adequate and ready to use for real tuning.
http://chernov-trezin.narod.ru/ZS_1_borbat.htm
http://translate.google.com/transla...rezin.narod.ru/ZS_1_borbat.htm#serebrchastoti.
I had not seen this post on this thread until today.

The details of the arguments are difficult to understand in the Google translate, but I think I got the gist. The 12-TET is based on the silver ratio 1:1+(12 root of 2). Basically, the author conceives each octave as a spiral rather than a linear spectrum. He references a physicist, Chernov, who says the ratio of the circumference to the diameter of a spiral should equal to pi, which he calls the true silver ratio of the spiral. This is basic geometry, so far.

He derives the Chernov equation (pi - 1)*x^9 - x^6 - x^3 - 1 = 0 to map the spiral, which, in sound, suggests using a minor third (x^3) as the diameter of the spiral, this stretches each octave from a multiple of 2 to 2.00201022913774 based on the pi ratio.

Since the frequency of the wave is based on pi, he suggested that Pythagorean doubling of the frequency is only an approximation of an octave, which creates commas and Wolf intervals to compensate for the approximation. His method suggests a more precise calculation. The alternative method to compute the semitone (x^1) results in 1.05955179365421 versus 1.05946309 of the 12 TET system and then each progressive semitone in an equal-tempered stretched octave is based on pi rather than the twelfth root of 2.

The author claims this will eliminate some inherent audible beats and many tuning issues between octaves as well as the otherwise lifelessness of the 12 TET system. In one octave this does not make an audible difference, but over the range of a piano or ensemble of instruments it does.

Here is the difference in Hz over the audio spectrum. The first note is the currently accepted standard octave; the second in pi-based. The third number is the difference in cents. It takes a difference of 3-5 cents to be audible to the most discerning ears.
A0 = 27.5, 27.39, -6.94
A1 = 55, 54.83, -5.36
A2 = 110, 109.78, -3.47
A3 = 220, 219.78, -1.73
A4 = 440, 440 *, 0
A5 = 880, 880.88, 1.73
A6 = 1760, 1763.54, 3.48
A7 = 3520, 3530.62, 5.22
A8 = 7040, 7068.35, 6.96

It is an interesting premise, but he will have a lot of physicists, acousticians and musicians standing on their heads. He does a similar exercise for light and color. I'd like to see harmonics, which tend to go sharp, calculated with this method.

He claims that piano tuning technicians already do something similar to this, but I thought this had something to do with string size or something. The author says that it is more than the design of the piano, but that the theoretical octave needs to be recalculated. It would be nice to hear from the piano techs.

Wow!
 

·
Registered
Soprano: 1983 Keilwerth Toneking Schenklaars stencil
Joined
·
926 Posts
Discussion Starter · #20 ·
Sorry, I find the notion of harmonics, consonance and disconsonace based on scales really interesting, right now. And the notion of stretching octaves to achieve better consonance is fascinating. My daughter is a piano tuner, among other things. Discussing this she was like, of course, you stretch octaves and make other compromises when tuning piano, duh Dad.

Here are the frequencies of stretched octaves of A=440 suggested by a piano tuner who bases the stretching on a log and powers of 2 scale “to compensate for string stiffness.” I have compared to standard octaves and calculated cents.

A0 = 27.5, 26.95, - 35.98
A1 = 55, 54.43, - 18.04
A2 = 110, 109.50, - 7.89
A3 = 220, 219.65, - 2.76
A4 = 440, 440 *, 0
A5 = 880, 881.42, 2.71
A6 = 1760, 1768.09, 7.94
A7 = 3520, 3556.27, 17.75

I have done this primarily to compare the magnitude of stretching between the two methods of stretching octaves. The pi-based stretching is much more mild that the piano tuner’s. Playing with the numbers of this for a while, there is some appeal there.

After digesting this and contemplating for a few days what this might mean to a sax player, I have decided that the impact is trivial. A bass sax player might have to tune his low end as much as 35 cents to play with a piano with stretched octaves. A soprano might have to sharpen as much a 17 cents, but that is not much in a range that tends to go sharp anyway. Once again, it comes back to learning to listen and tuning by ear. As said before, it makes those who obsess over a couple of cents in the intonation of a sax appear ridiculous. In fact, there is some argument in favor of a sax with easy flexibility in intonation versus one that is “too locked-in.”

There was also some discussion here some time ago about tuning octaves, adjusting the volumes of this and that on a sax to get it in tune with itself. Some sax tech will make some major alterations to the sax to achieve this. The notion of a pi-based octave and stretched octaves, in general, suggests that these octaves might best tune if upper octaves are a few cents sharp relative to lower octaves.

It is time to test this in the real world to see how it really affects the consonance.
 
1 - 20 of 21 Posts
This is an older thread, you may not receive a response, and could be reviving an old thread. Please consider creating a new thread.
Top