Just intonation intervals really sweeter than 12et?

[Jerry Freeman]

I've looked up "partial" as it applies to harmonics, and I think I understand what it is, and by extension, what a common partial would be as it applies to an interval.

It appears to mean precisely this:

ALL just intervals are pure in the sense that a component of the waveform will be in perfect synchrony with the root frequency, as well as with every other pitch in the scale.

The component of the waveform in each just interval that is in perfect synchrony is the common partial.

Best wishes,
Jerry

[Lorenzo]

What's a "common partial"?

As I understand it, in the string theory world anyway , it's an common overtone that is produced by both strings. Harmonics. Sometimes this will cause a string, tuned to the same overtone--and higher up--to sympathize and to start vibrating inadvertently, unintentionally. Common overtones would be, "producing the same tone (overtone)." In the interval C-E, both strings would sound the same overtone (I don't remember what that note would be, but probably G, a 5th above the C. Can an E produce an overtone of G? Yes, I think so. And it even might be in the next octave.

Yesterday, while tuning a piano, it dawned on me what all the various tests are, that are used by tuners, to double check accuracy (other than the machine). I ended up checking each note for purity by playing, for example, C4 against E5, and used this wide wide interval all the way up and down throughout the keyboard. It's even more accurate than simply playing C4 against C5.

EDITED IN: from Answers.com (wiki)

• In pitched instruments, these shorter, faster waves are reflected between the two ends of the string or air column. As the reflected waves interact, frequencies whose wavelengths do not divide evenly into the length of the string or air column are suppressed, and the vibrations that persist are called harmonics. Their wavelengths are 1, 1/2, 1/3, 1/4, 1/5, 1/6, etc. of the length of the string or air column. http://www.answers.com/topic/harmonic-series

[BoneQuint]

Is there any chance you could create clean samples of that same chord with the technique you've been using?

Here's an A major chord in just and equal temperament (each note added one at a time), then a comparison of just the chords side-by-side:

A Major Just
A Major Equal
Just vs. Equal

Jerry referenced a page on Pat Missin's Harmonica Tuning site, he does a direct comparison of equal vs. just major chords using a harmonica, which is a good real-world example with a fairly pure-toned instrument.

Holes 2, 3 and 4 draw in Just Intonation
Holes 2, 3 and 4 draw in Equal Temperament

This is a G major triad - you should be able to hear that the JI version is a lot smoother than the ET version (there should really be no beating at all, but few things in this world are totally perfect...!). In fact, many (most?) people would say that the tempered version sounds "out of tune".

And I thought it was interesting what the difference looked like, here's a screen capture from audacity of the two chords (again, just followed by equal, although I'm sure I don't have to tell you that in this case.)

[Jerry Freeman]

<a href="http://www.skytopia.com/project/scale/t ... mp3">Pitch 1</a>
<a href="http://www.skytopia.com/project/scale/t ... mp3">Pitch 2</a>

There are 3 tests each for the above two mp3s. Let me know if you would like the samples stuck end to end, so there's no pause between each arpeggio.

Hi, Don.

I can't really tell the difference between the tunings.

I listen several times and begin to think I can tell, then I come back and it sounds different to me. If get a chance, I'll try slowing them down by half and see if that helps. These are definitely easier to listen to, though. With chords, I can usually hear a difference pretty easily, but not with arpeggios, it seems.

Best wishes,
Jerry

[dwhite]

Hi Jerry,

Hmm.. perhaps the old ones with lots of partials was easier to tell

Anyway, try speeding these up as that seemed to help me distinguish between them. If you use winamp, then there's a brilliant utility plugin called 'Pacemaker' which allows you to change the speed, pitch and tempo at will.

[Jerry Freeman]

Hi Jerry,

Hmm.. perhaps the old ones with lots of partials was easier to tell

Try speeding them up too as that seemed to help me distinguish between them. If you use winamp, then there's a brilliant utility plugin called 'Pacemaker' which allows you to change the speed, pitch and tempo at will.

I have The Amazing Slowdowner, which does the same things.

Best wishes,
Jerry

[StevieJ]

This has been a fascinating discussion - despite the absence of contributions from the learned talasiga.

[Jerry Freeman]

I went back to the beginning of this post and rediscovered Peter's link to some harpsichord recordings in the original tunings Bach used.

Some are available for free download, so I've posted one so you can just click and listen:

Modulation Canon from the Musical Offering

You can access more samples here:
http://www.last.fm/music/Bradley+Lehman ... %27s+fancy

Here's some background:

Tuning

In J. S. Bach's obituary it was reported: "In the tuning of harpsichords, he achieved so correct and pure a temperament that all the tonalities sounded pure and agreeable. He knew of no tonalities that, because of impure intonation, one must avoid."

The present album and its companion set for organ (LaripS 1002, "A Joy Forever") explore the specific temperament I have proposed as reconstruction of his practice, the discovery of his documented expectations. This tuning method is deduced from historical context and Bach's written evidence: his drawing at the top of the Well-Tempered Clavier's title page (1722) plus an analysis of his music--in that book and elsewhere. The article about this discovery is published in the February and May 2005 issues of Early Music (Oxford University Press), and further details are at <www.larips.com>. For the reasons I have stated there, I believe that Bach probably preferred this specific system for much or all of his career, instead of equal temperament or the various unequal methods around him. This also solves classic problems in the music of his sons, as to harmonic flexibility and surprises within their styles.

The layout is:

F-C-G-D-A-E 1/6 comma narrow 5ths;
E-B-F#-C# pure 5ths;
C#-G#-D#-A# 1/12 comma narrow 5ths;
A#-F a residual diminished 6th, 1/12 comma wide.

In this tuning, every major scale and minor scale sounds different from every other, due to the subtle differences of size among the tones and semitones. This allows music to project a different mood or character in each melodic and harmonic context, with a pleasing range of expressive variety as it goes along. It builds drama into musical modulations.

The result sounds almost like equal temperament in its smoothness, and it similarly allows all keys to be used without problem, but it has much more personality and color. In scales and harmony it sounds plain and gentle around C major (most like regular 1/6 comma temperament), mellower and warmer in the flat keys such as A-flat major (most like equal temperament), and especially bright and exciting in the sharp keys around E major (like Pythagorean tuning, with pure 5ths). Everything is smoothly blended from these three competing systems, emerging with an emphasis on melodic suavity.

(for more, http://www-personal.umich.edu/%7Ebpl/larips/cd1003.html)

Looking at the tuning scheme, we see that Bach preserved some of the just intonation effect while at the same time, tempering to allow modulation and chord changes.

Best wishes,
Jerry

[Brian Boru]

I actually heard equal temperament as hopelessly out of tune. It's built in. The reasons have a hell of a lot to do with mathematics, and require about 5 volumes of very thick books ...

For the mathematically inclined who don't want to wade through the 5 volumes of thick books here is a quick explaination. Every time you go up an octave you double the frequency of the note. If you go up by a perfect fifth you increase the frequency by 3/2. So once you go through the circle of fifths you will get a note whose frequency is (3/2) raised to some power. But getting to the same note using octaves would give an note whose frequency is 2 raised to some different power. There is no way to make these notes the same unless you sacrifice either the octave (easily heard) or the fifth (less easily heard).

[Jerry Freeman]

I actually heard equal temperament as hopelessly out of tune. It's built in. The reasons have a hell of a lot to do with mathematics, and require about 5 volumes of very thick books ...

For the mathematically inclined who don't want to wade through the 5 volumes of thick books here is a quick explaination. Every time you go up an octave you double the frequency of the note. If you go up by a perfect fifth you increase the frequency by 3/2. So once you go through the circle of fifths you will get a note whose frequency is (3/2) raised to some power. But getting to the same note using octaves would give an note whose frequency is 2 raised to some different power. There is no way to make these notes the same unless you sacrifice either the octave (easily heard) or the fifth (less easily heard).

Please note:

It's not necessary to use the circle of fifths to construct the scales of the various keys. (For example, it's not the way singers, string ensembles, and even whole orchestras arrive at their pitches in the various chords and harmonies they perform.) This is where there seems to be a lot of confusion in this thread.

Using the circle of fifths is only one of the possible ways to construct scales. The circle of fifths comes into play when you're trying to create all-purpose, fixed notes that can't be adjusted during a performance, which will work in all keys/chords. This is necessary with fixed pitch instruments, but it is not necessary or desirable when every pitch can be adjusted to create pure/just intervals in every chord and key.

Best wishes,
Jerry

[Lorenzo]

Jerry, I think you missed your calling. You should seriously consider becoming a piano tuner. Several of my cousins do it full time (I use to) and make about $120 on each piano. They give schools and churches a break, but still, many of these pianos can be tuned in 1-1½ hrs! By tuning sometimes five or six piano in a day (schools)--well, do the math!

[Jerry Freeman]

Jerry, I think you missed your calling. You should seriously consider becoming a piano tuner. Several of my cousins do it full time (I use to) and make about $120 on each piano. They give schools and churches a break, but still, many of these pianos can be tuned in 1-1½ hrs! By tuning sometimes five or six piano in a day (schools)--well, do the math!

Did you have a chance to listen to that Bach piece, and look at the tuning he used?

Very interesting!

Best wishes,
Jerry

[Brian Boru]

Please note:

It's not necessary to use the circle of fifths to construct the scales of the various keys. (For example, it's not the way singers, string ensembles, and even whole orchestras arrive at their pitches in the various chords and harmonies they perform.) This is where there seems to be a lot of confusion in this thread.

Jerry

Granted. I just used it in my example as a simple mathematical explanation.

[Lorenzo]

Did you have a chance to listen to that Bach piece, and look at the tuning he used?

Very interesting!

Yes. I listened to the piece and enjoyed it. Did you find anything particular of interest?

If you looked at his tuning layout....

• F-C-G-D-A-E 1/6 comma narrow 5ths;
  E-B-F#-C# pure 5ths;
  C#-G#-D#-A# 1/12 comma narrow 5ths;
  A#-F a residual diminished 6th, 1/12 comma wide.

...you see this is not pure by any means, only the E-b and F#-C fiths are pure. I'm not sure why (or if) he's calling B-F# a 5th. Bradley Lehman calls this scheme a "modified meantone" tuning. Notice how he describes its limitations...

• Modified meantone, or "circulating" or "irregular" or "ordinary" (17th-19th centuries): The series of 5ths is regular in the midsection (on the natural notes of the C major scale, ...C-G-D-A-E...), but increasingly wide toward the outsides. This sharpens the sharps and/or flattens the flats gradually, so they can serve passably as one another, and it reduces or eliminates the "wolf" intervals. Some of these schemes also have a flattened F or a raised B, as transition into the flats or sharps. -link to--Bach's Art of Temperament http://www.larips.com/

Lehman goes on to say...

• "Tempering" refers to a bit of impurity that is introduced deliberately into an otherwise perfectly harmonious interval, such as a 5th or a 4th on the keyboard. The pitches are put slightly off their expected positions, giving a vibrato-like wobble when both notes are played together. There is impurity, asymmetry, and subtle variety: if used carefully, all of these features can strengthen the musical effects and enliven the sound.
  
  Why are clever tuning schemes necessary at all, on keyboards? The short answer is: nature has not provided any definitively obvious way in which all the available material (within a musical octave) fits most neatly into twelve small packages, to serve all possible musical needs. If we try to tune everything directly as well as possible, we quickly run into dead-ends where other note combinations do not sound as good, since they were not given such direct attention. We must add some careful refinements and deliberate impurities, along the way, for the final result to work out well.

Two things: one is the classical style the piece is written in--not many simultaneous chords. And the other is the harpsichord. Back in Bach's day, not much tension could be put on the poor quality strings, nor on the weakly built instrument itself. As a result, not as much inharmonicity could be weeded out, meaning it isn't likely Bach could hear or appreciate truely pure intervals we hear on pianos (or on today's harpsichords--and I've tuned a few). Plus, with that style of music, and the lack of dampers, the sustaining notes played earlier in the piece are still ringing through the next measure--making it hard, if not impossible to hear the true dissonance in the intervals--on the mp3 sample.

You've got to try tuning keyboards, Jerry. The math is one thing. The ability to apply it correctly and precisely to an instrument (non-electrical) is quite another. To illustrate this point, the Piano Technician's Guild won't even let students take their test on a Steinway Grand, because the instrument is so well built it eliminates nearly all inharmonicity. They make them tune on the cheaper ones, which are far more difficult. The cheaper ones, even though far far better than any harpsichord (for eliminating inharmonicity), are much harder to judge in-tuneness.

If you don't have the tools (they're not that expensive) use a wrench and some cloth for mutes and try it on your piano. But here's what a starter set looks like. The red felt coil is inserted between each of the 13 string in the central octave so that only the single center string plays. On eBay, search "piano tuning tools." They're only $47-85 per set. This one pictured is the $85 set.

[Jerry Freeman]

You've got to try tuning keyboards, Jerry. The math is one thing. The ability to apply it correctly and precisely to an instrument (non-electrical) is quite another. To illustrate this point, the Piano Technician's Guild won't even let students take their test on a Steinway Grand, because the instrument is so well built it eliminates nearly all inharmonicity. They make them tune on the cheaper ones, which are far more difficult. The cheaper ones, even though far far better than any harpsichord (for eliminating inharmonicity), are much harder to judge in-tuneness.

If you don't have the tools (they're not that expensive) use a wrench and some cloth for mutes and try it on your piano. But here's what a starter set looks like. The red felt coil is inserted between each of the 13 string in the central octave so that only the single center string plays. On eBay, search "piano tuning tools." They're only $47-85 per set. This one pictured is the $85 set.

That's interesting.

It's not as if I don't have enough to do.

But we do have a piano here that hasn't been tuned in ages. Maybe someday ...

Best wishes,
Jerry