Perfect 5ths' aren't Really Perfect

[MrTuner]

no interval is truly "in tune" on a piano, or fretted string instrument.

Modern piano tuning theory has developed pure 12th stretch tuning. I have a method that can produce pure P11/12/19/22. (That's octave plus P4, octave plus P5, two octaves plus P5, and triple octave. But because of limits in human hearing AND the variability of frequency over the sustain, true pure intervals are never really achieved. It's an illusion, close enough to get 'that' sound. )

I assume when you say "in tune", you are referring to "pure", meaning no beating.

 

[MrTuner]

Yes, stretch tuning is done to balance the overtones that are heard behind the fundamental tone.
When a piano string is hit, it is hit on a harmonic node. It rings out the fundamental tone plus many other tones at lower volumes but higher pitches.
Those harmonics are not in tune because the thickness and length of the string has to be compromised from its theoretical ideal. (A grand piano would need to be like 45’ long)
The more compromised the more out of tune the overtones become. That is why a small spinet piano needs to be stretched more than a grand. The overtones are sharp on the low notes and flat on the high notes. So for the higher notes a tuner might push the tuning of the fundamental well sharp to get the overtones closer to tune. If you think of every piano note actually being a chord made up of the fundamental and all overtones, stretching gets the whole chord closer to being in tune, even thought the fundamental tone is not.

Good stuff. I'll just clarify:
- Harmonics are ideal. The piano is not ideal, so they are called "partials". Same note names, but harmonics have specific frequencies - whole number multiples of the fundamental. Partials do not have that strict a definition.
- Upper partials are always sharper in all strings, bass and treble. The only string that has ideal partials that are harmonic, has zero thickness, infinite tension, and infinite length. Even guitars and basses have inharmonic strings.

 

[Papageno]

Now, the unison and octave are called perfect as well, (PU and P8) and they too are consonant. It is also interesting to note that in the equal-tempered system, (the most common tuning system for instruments and pianos) the PU and P8 are beatless, the P4 beats around 1 bps, the P5 around 0.6 bps, but the M2/M3/M6/M7 all beat much, much faster.

It does not make sense to state that an equally tempered interval beats at a particular frequency (e.g. "P4 beats 1 bps"), because the beat frequency is given by a frequency difference whereas the interval is given by a frequency ratio. Because of this, the beating frequency is proportional to the frequency of the notes considered, and the higher you go, the greater the beating frequency is (it doubles every octave).

 

[MrTuner]

But if it's bass notes are tunes flat and treble notes sharp, when it plays those notes, at the same time as bass a treble instruments, they are out of tune.

Depends on your definition of out of tune.

When we play a P22 (triple octave) on the piano and the top note is 20 cents sharp of what it should be mathematically, you could say it is out of tune if your definition was a theoretical one.

However, if the top note is exactly 8x the bottom frequency (i.e. harmonic) it would sound flat and the triple octave would sound ugly. If your definition has to do with how it sounds, then that interval would be out of tune.

That's from the piano's point of view. When the piano plays with an orchestra, all bets are off. And you are right. The piano is the most out of tune instrument

 

[MrTuner]

It does not make sense to state that an equally tempered interval beats at a particular frequency (e.g. "P4 beats 1 bps"), because the beat frequency is given by a frequency difference whereas the interval is given by a frequency ratio. Because of this, the beating frequency is proportional to the frequency of the notes considered, and the higher you go, the greater the beating frequency is (it doubles every octave).

You are absolutely right! I only give it as a reference. Piano tuners use this reference to try and get ‘close’ when tuning the temperament octave F3-F4.

 

[La Faro]

I pulled away from this thread after digging in a little, as I was meant to be working on something else, and realised I get into a deeper topic that required outside reading, when I just wanted a succinct satisfactory answer.

The question of stetched tuning came for using VSTi in my DAW. One critism of say rhaodes is they will never sound as good as a real rhodes, because sampled rhodes are too tuned. BUt that doesn't have to be, (And I'm not sure is a good argument anyway). I have XLN Audio's Addictive Keys, and that gives me the option to choose stretched tuning and to manually tune. BUt made me think, if I had stretched tuning on my sampled piano/rhodes, won't my synthesizers be out of tune, and how you I tune my bass. My intonation isn't great on sax and flute, but I have melodyne, so I can change (fix or just improve a little) that byt do I match the stretched tunes keys or not. I just stayed in standard equal temperament in the end.

Interestingly melodyne allows you to change the amplitude of the first range of the harmonic series. the first I think.

Here's Addictive keys set to Pythagerian tuning and a little stretch

 

[MrTuner]

As we get close to “perfection” in tuning - pure stretch, matched frequencies, etc - we achieve a “sound” that allows us to express emotion through music, without the imperfection that is always in music getting in the way.

But perfection can never be achieved.

People have commented, “How can my bass be in tune with a piano if the stupid thing isn’t even in tune with itself?!“ (I’m paraphrasing of course)

Well, it can’t. It’s like the first time you learned about equal temperament. “What!? Octaves and fifths can’t be in tune at the same time?”

But still, orchestras are able to create piano concertos that make people cry. Jazz groups with piano and bass out of tune with each other play songs that make people happy to be alive. It’s like walking a tight rope. A slight move one way or the other and poor intonation breaks the spell but the better intonation, that creates the magic, is not perfect.

You see? We don’t need to be perfect. Just good.

 

[Papageno]

As we get close to “perfection” in tuning - pure stretch, matched frequencies, etc - we achieve a “sound” that allows us to express emotion through music, without the imperfection that is always in music getting in the way.

But perfection can never be achieved.

People have commented, “How can my bass be in tune with a piano if the stupid thing isn’t even in tune with itself?!“ (I’m paraphrasing of course)

Well, it can’t. It’s like the first time you learned about equal temperament. “What!? Octaves and fifths can’t be in tune at the same time?”

But still, orchestras are able to create piano concertos that make people cry. Jazz groups with piano and bass out of tune with each other play songs that make people happy to be alive. It’s like walking a tight rope. A slight move one way or the other and poor intonation breaks the spell but the better intonation, that creates the magic, is not perfect.

You see? We don’t need to be perfect. Just good.

This ^^^ !

Furthermore, for the music to be good and generate emotions, the requirement on intonation is not the same for all notes.

For the notes which are the harmonic foundation of the tune, one should clearly strive to be as much in tune as possible, within the limitations and compromises discussed in the post above.

On the other hands, for passing notes (non chord tones, or chromatic notes) in melodic lines, there is a much broader range of possibilities for the benefit of expressivity, provided the harmonic notes you resolve to are well in tune.

An extreme example is the following: consider a jazz blues in Bb, i.e.:
Bb7 / Eb7 / Bb7 / Fm7 Bb7 /
Eb7 / Eb7 / Bb7 / G7 /
Cm7 / F7 / Bb7 / F7 //
In bar 7, to walk from Bb7 to G7, you can simply walk down from Bb to open G, by dividing the minor third in 4 equal steps! Although all the passing notes are horribly out of tune (and would be impossible to notate on a staff), this constitutes a very convincing and smooth motion towards the G, which the ear accepts as musically logical. If you outline, by contrast, a strong, in-tune, G7 arpeggio upon landing, nobody will ever complain about your intonation.

 

[MrTuner]

We also have instances of fifths being played purposely out of tune (very slightly) to create tension, like for example Wagner in the trombone section. (No direct experience with this. Just postulating)

And then there's the many examples of singers and jazz musicians playing/singing purposely out of tune for effect. Billy Holiday, John Coltrane, etc.

 

[bfields]

You go into this looking for mathematical perfection, but what you find is a bunch of rough compromises that are just close enough to be useful.

It's a strange and imperfect and interesting world....