The Bass in Fifths

[Jeff Moote]

Originally Posted by David Powell

I've been really curious about how this harmonics / resonance phenomenon works with 5th tuning, so I am doing some ****ysis on the frequencies in the circle of fifths vs. those in the reciprocal circle of fourths. So far it appears that there may indeed be a difference. I'll let you know when I have something more definite. It's all math the way I see it, but how it it heard is more important.

I agree that the perceived effect is more important, but moreso I too am into the math of it all. The tricky part is when you start mixing temperament systems which is inevitable since you choose the intonation based on what you hear. I think that's why some people look at the claims made by fifths players and think "no, that's all wrong because a fifth is just a fourth upside down" - well that's only so if you temper your intervals that way. This is also why I think practicing with a tuner to work on intonation is the worst idea ever - you end up with a 12ET sense of intonation which is useless unless you're playing with other 12ET instruments. Even though we often play with pianos, usually the strings grossly outweigh the piano in both numbers and sound, so you end up with a sort of mixed temperament where the pitch center is defined by the piano in whatever range it's playing, but all the interval relationships are locally tempered according to something more like just temperament. This is the reason I don't get too much into the actual math of trying to prove this or that about a given tuning for the bass - it all comes down to subjective factors which are simply the result of our subjective sense of intonation, even though each temperament can be explained easily with math when taken on its own.

[David Powell]

I don't regard ET as harmony, and therefore, very strictly speaking, not pure music. I know people write good music with / for it, but it seems like one of the first perversions to come along. It's a system designed by bureaucrats, not mathematicians. In It can be mathematically described by ascending or descending by the 12th root of 2, but is not a harmonic relation, so it is mostly wrong harmonically except when you get to the octave. All of matter would be destabilized if one tried to impose ET on atoms and molecules. One way of looking at radioactivity is that the atom is trying to lose internal dissonance. It keeps degrading until finally all the parts are "in tune". The whole atomic structure stays together because of harmony.

Just harmony is about the only thing I can hear. It's the whole reason I started getting into fret-less instruments. A guitar is impossible to tune because b cannot simultaneously be the 3rd of g and the 5th of e. I wanted to hear some thirds that sounded like perfect constructive wave interference instead of noise. Perfectly constructive wave interference is how I define harmony. But can you really have that with a perfect third between a fundamental and a fifth? The 3rd has to be related perfectly to both the tonic and the fifth. How many notes can you put into a chord before something interferes destructively? Poly-chords are going to be very problematic. So this is not really an easy project, you see. As notes move through a melody, others must adjust. It's easy with just two notes, but 3 or more get complicated.

I changed teachers once because I just could not dig training myself to hear and play as if there were frets. The teacher was trying to get me to hear and play ET. My practice is to tune the notes to whatever the other instruments are at with slight adjustment of position. I can't hear ET. I can listen for perfect constructive interference. It is more subconscious than conscious and I know in practice most notes go by faster than I can hear them just right. But it gets better with practice. As far as the 5ths, they are very close whether one is in equal temperament or in just harmony. 4ths are also not too far out. The rest is pretty much a mess.

Here's a cool chart that ****yzes the differences between ET and pure harmonic music in a couple of different ways. The ratio column is the one I will be using mostly to do the ****ysis. There are a couple of things that can be gleaned right away if you look at G and A on that chart. Before we get into that it is important to recognize that the scale (to me, there is really only one scale per fundamental pitch and then different modes of the one scale) is comprised of simple note relations that follow the series 1:2, 2:3, 3:4, 4:5, 5:6, 6:7, 7:8, 8:9. That is the basis of the scale. It seems 8:9 is the whole step if I remember right. 15:16 is the half-step. Where it tends to be problematic I think, is when one plays chords with stopped notes as the fundamental when another instrument plays the fundamental as an open string. That's the direction to work in to get to why it might sound more like constructive interference with all the instruments tuned in 5ths. But beyond that we need to know what pitches the strings of all the instruments in the ensemble are tuned to. If I get that and get my head right, I can derive some relations that might show something or not. We should divide the work load. Here's what we do. Take the violin, viola, cello, and DB. Make a chart with all the strings pitches tuned in 5ths, and then derive the partials of the different strings in terms of Hz. Then we can use the Hz. to compare the harmonic relations across the strings of the orchestra as different chords are played. What should emerge is a set of preferred positions that play the notes of chords right on a just harmonic position as opposed to one that is off. If we make one chart with the DB in 5ths and another with it in 4ths, that should give us some ideas about what goes on in terms of the constructive interferences we hear, whether these are pure or off the ratios a bit.

[Jeff Moote]

David,

While I respect what you're doing, I really think you're over thinking it - or at least over thinking certain aspects and missing others.

One thing to mention is that you seem to be equating constructive interference of waves and consonant sounds, if I'm not reading you incorrectly. Even if one ignores that a note sounds with significant amplitude on more than just the fundamental, a more constructive relationship between fundamentals doesn't necessarily mean a more consonant sound. Psychoacoustics is the major factor at play, and unfortunately that brings this way outside the realm of physics.

The point is that to use your chord example, you can tune a chord (say, a major triad) and that is very different than tuning a major third and a minor third on their own - so the intervals you choose in the chord to create the most consonant sound will not be the same you'd choose for either on their own.

To take this discussion further off topic, I'll mention the notion that composers and musicologists have held that different keys have different "colours". They absolutely DO NOT - with the emphasis on "absolutely" On a piano tuned to 12ET (I'll use it for reference at first) all keys sound identical. On an instrument of flexible temperament such as strings, as long as the player continues to play just temperament all keys will sound the same, in terms of dissonance. Wind instruments are kind of in between since they generally use just temperament but are somewhat limited by the key of their instrument (unless for example you have a set of natural trumpets - one for every key!). All this combined still points to keys being identical - so where does the idea come from? My guess is that it's more related to instrumentation and variation in timbre: certain keys lend themselves to various instruments playing in different parts of their range. This means that depending on the key, certain instruments are limited to some subset of the timbres they are capable of. When this is summed in an ensemble context some keys may begin to take on a "colour" which leads to the ideas mentioned above. This also explains why there is not universal agreement on this - one composer hears a given key as one thing, and another differently. This could be as simple as one composer preferring a different orchestration leading to that key taking on a certain colour. Even if we limit both composers to the use of their pianos, both pianos are not identical and are certainly tuned differently - for this reason the dissonance may be distributed differently giving them an impression of something about keys, as they hear them.


To bring this back on topic, I think one reason why fifths work better on the bass in the context of other string instruments is that you have a broader range of notes in each position on the neck allowing for different timbres which blend differently. This doesn't explain anything about the supposed increased resonance of the instrument itself though. For that I would come back to the physics of the harmonic relationships, so if you want to do any ****ysis this would be where I'd do it, and not between the bass and other instruments. In doing that we cannot leave out the natural resonances of our instrument either. It could be just that the instrument resonates differently when tuned in fifths as the wood is under different types of stresses, and this is another barrier to ****ysis because it is not as objective. Once you add the varying harmonic response of a single vibrating string to the simple fundamental ****ysis, and then consider the resonating wood (which is not unchanging over time) the task becomes increasingly complicated. There's easily enough material for 10 theses in mechanical engineering, but is this really worth the trouble?? Maybe if you need a topic for your doctoral thesis! As a musician it seems sensible to try the tuning myself, or at bare minimum listen to others using it (which is how it started for me) and if you like the results then it's good. If not, then keep using fourths.

[David Powell]

Originally Posted by Jeff Moote

David,

While I respect what you're doing, I really think you're over thinking it - or at least over thinking certain aspects and missing others.

One thing to mention is that you seem to be equating constructive interference of waves and consonant sounds, if I'm not reading you incorrectly. Even if one ignores that a note sounds with significant amplitude on more than just the fundamental, a more constructive relationship between fundamentals doesn't necessarily mean a more consonant sound. Psychoacoustics is the major factor at play, and unfortunately that brings this way outside the realm of physics.

The point is that to use your chord example, you can tune a chord (say, a major triad) and that is very different than tuning a major third and a minor third on their own - so the intervals you choose in the chord to create the most consonant sound will not be the same you'd choose for either on their own.

To take this discussion further off topic, I'll mention the notion that composers and musicologists have held that different keys have different "colours". They absolutely DO NOT - with the emphasis on "absolutely" On a piano tuned to 12ET (I'll use it for reference at first) all keys sound identical. On an instrument of flexible temperament such as strings, as long as the player continues to play just temperament all keys will sound the same, in terms of dissonance. Wind instruments are kind of in between since they generally use just temperament but are somewhat limited by the key of their instrument (unless for example you have a set of natural trumpets - one for every key!). All this combined still points to keys being identical - so where does the idea come from? My guess is that it's more related to instrumentation and variation in timbre: certain keys lend themselves to various instruments playing in different parts of their range. This means that depending on the key, certain instruments are limited to some subset of the timbres they are capable of. When this is summed in an ensemble context some keys may begin to take on a "colour" which leads to the ideas mentioned above. This also explains why there is not universal agreement on this - one composer hears a given key as one thing, and another differently. This could be as simple as one composer preferring a different orchestration leading to that key taking on a certain colour. Even if we limit both composers to the use of their pianos, both pianos are not identical and are certainly tuned differently - for this reason the dissonance may be distributed differently giving them an impression of something about keys, as they hear them.


To bring this back on topic, I think one reason why fifths work better on the bass in the context of other string instruments is that you have a broader range of notes in each position on the neck allowing for different timbres which blend differently. This doesn't explain anything about the supposed increased resonance of the instrument itself though. For that I would come back to the physics of the harmonic relationships, so if you want to do any ****ysis this would be where I'd do it, and not between the bass and other instruments. In doing that we cannot leave out the natural resonances of our instrument either. It could be just that the instrument resonates differently when tuned in fifths as the wood is under different types of stresses, and this is another barrier to ****ysis because it is not as objective. Once you add the varying harmonic response of a single vibrating string to the simple fundamental ****ysis, and then consider the resonating wood (which is not unchanging over time) the task becomes increasingly complicated. There's easily enough material for 10 theses in mechanical engineering, but is this really worth the trouble?? Maybe if you need a topic for your doctoral thesis! As a musician it seems sensible to try the tuning myself, or at bare minimum listen to others using it (which is how it started for me) and if you like the results then it's good. If not, then keep using fourths.

Well, I'm not sure what you are talking about when you say consonant sounds, so I'm definitely missing that. I do know what I mean about constructive sound wave interference of two or more frequencies. When the sine waves are laid over each other in a perfect simple ratio, one can hear that as a new wave form. Did you look at the chart on the bazantar site?

If you bow two strings you can find some of those off the half step (sometimes called quarter tones). Anyway, it is easy to find some of the more unusual ratios just by ear while gradually changing the pitch of one string while droning the other string. I know how those harmonizing ratios sound and it has everything to do with constructive interference characteristics of the sound waves. I really don't know if psycho-acoustics has anything to do with what I'm talking about. It may. The phenomenon I'm talking about is the same thing that allows me to tune one note on an instrument and then tune the whole thing. It is the same constructive interference I'm hearing when two notes are in tune. I don't think the "color" thing about keys is easy to talk about in any certain terms. Nor do I think that it is of any consequence in the way that I'd like to look at the resonance / constructive interference differences between the tuning.

I would like you to explain the "consonant sound" definition. I couldn't find anything that referenced that terminology or explained it. Depending on the definition, it may or may not have anything to do with what I'm talking about.

I really think if the increased resonance is something that is experienced in an ensemble with other strings, we will miss what's happening without including all the strings in the ensemble in the physical ****ysis. Sure, there should be a pattern just looking at one instrument tuned in fifths, but we need to consider the implications of all of them playing chords at once. Not chords on one instrument, but chords where one plays the tonic, another the third, another the fifth. What changes on the bass between the two tunings is not the pitch played, but the position and string it winds up being played on. Tuning in fifths should not change the stresses on the instrument if we choose strings that give the same overall tension result. That must be kept constant. If there is anything that seems "instrument dependent", in other words it works differently on this bass than that one, the the idea is pretty much invalid. If the tuning itself generally improves something, the effect should be universal across instruments as long as we hold everything else constant. If tension is part of the effect, then the effect is not due to tuning in 5ths and could be achieved by 4th tunings and just using a different set of strings. If an instrument resonates differently, it has to be because of the different series of partials that result on the open strings and on the relation to a specific stopped note before it has much to do with 5th tunings. FYI, I wasn't going to use any instruments in the ****ysis, I was going to chart the partial series of the notes and look at the frequency relationships / ratios. If this turns into something that is instrument dependent in any way, the concept is lost completely and it is just a novelty tuning. I don't think it will take ten mechanical engineers to show what I'm talking about. It will take some time and a calculator, that's about it. In my mind it is not complicated if we stick to a theoretical ****ysis. When we leave the theoretical ****ysis and start considering different instruments, etc., then any meaningful ****ysis is impossible. If a theoretical ****ysis shows up nothing, then I don't think there is anything there dependable. You can try it on this or that instrument and see what you get in the real world, but constructive interference is something that I have found (and how, given physics, could it be different? Sound has it's laws and it abides them.) is the same no matter what one uses to generate the pitches, as long as the pitches generated also generate a partial series of resonances based on the pure harmonic series. To me what wouldn't be worth the trouble is stringing up my instrument differently without having some theoretical foundation for why.

What I am looking for is a general improvement in resonance that can be specifically related to nothing but the tuning difference. That is a problem worked out on paper. My hunch is that there could be something there, that it will probably be somewhat key dependent;- that it also could require playing the notes on each instrument in a specific position and string. It won't prove or disprove anything, but it might point in a direction.