In defense of equal temperament

Greetings to the list. I have been an interested researcher in Just Intonation theory for a few years, after having been introduced to Partch and "Genesis of a Music" some years back by a friend of mine. I am still doing active research, and would like to post some of my results at a later date.
However, seeing the recent posts about low limit, I think it is important to suggest that 1. the tendency to emphasize low limit can be limiting, as is 2. the tendency to dismiss equal temperaments as being arbitrary and unnatural - not that this is a problem in this group. Much of what I have to say most of you will know already, but perhaps some of it may be interesting.

1. Just Intonation ain't so just

Although low limit intervals guarantee a certain amount of consonance, they guarantee a certain amount of prejudice if you are trying to play certain suspensions of chords that are based on related harmonic series.
There are, to my limited knowledge, at least four basic types of modulation or chord suspension of this type in music:
1. Those based on overtones of undertones (IV for example, since it is a major chord based on 4/3, an undertone).
2. Those based on undertones of overtones (i, for example, because the minor triad 1/1, 6/5, 3/2 is the undertone series based on 3/2. Oddly, iv is not a modulation at all, but the undertone series based on 1/1. This explains why it sounds so good.)

These are the usual contents of a tonality diamond, because the numerators and denominators cancel nicely when you multiply, for example, 8/5 by 5/4, or 3/2 by 4/3.

However, two species of modulation are discriminated against:
3. Overtones of overtones, for example II, since it is the overtone series based on 9/8, an overtone. The Lydian #4, or 45/32, a mainstay of jazz, is excluded from consideration sometimes, even though it is very consonant, because it is the 5/4 of 9/8. Unfortunately, 5/4 times 9/8 does not reduce. This problem affects all overtones of overtones, and as a result some very consonant overtones of overtones have a high limit.
4. Undertones of undertones, for example vii. This for the same reason as above - the fractions don't reduce.

None of this is to claim that these other two species of chord suspension are as consonant as the first two - some are, some aren't. they just happen to be a part of music, and should not be unfairly discriminated against.
Within the next two months I will be posting several "tone maps" based on the relations between all of the permutations of a 37 limit just scale, in the "four directions" of modulation. This should show the real complexity of just intonation, should it be taken to an (absurd?) extreme.

2. In defense of equal temperament

One of the results of my research into just intonation was the discovery that in my tone maps, the various symmetry lines converged on an unique interval, the square root of 2 over 1, which equates to the equal tempered augmented fourth. The reason why this unique equal tempered interval shows up in the middle of just scales is that all just scales are based on fractions, and sqrt 2 is the one interval which is its own fractional inversion. Given any over and undertone series based on a single fundamental, and a high enough limit there will always be a clusterf*** of related tones around the augmented fourth, because the various inversions (11/8 - 16/11, 3/2 - 4/3, 45/32 - 64/45) will tend to approximate each other the closer they get to sqrt 2/1. This is not to say that the inversions are consonant with each other while played simultaneously - 5/4 and 8/5 are not particularly happy together. However, vi to I is one of the most common sequential modulations in music, and it employs 5/4 and 8/5 in starring, complimentary and warring roles (5/4 wins, of course). Broadly speaking, if you are talking about musical interrelationships, the one between major and minor harmonic series is absolutely fundamental, and underlies many suspension and modulation functions. The tension and shift between the two is what music is all about.

A cursory study of fugue concepts will bear this out, because to invert a melody intervalically is to mathematically invert it as well - to turn major to minor and light to dark, or dark to light. This has an immediate musical function that is plain to hear - the inverted melody is emotionally inverted from the original, in many cases. A great fugue subject is one that has 4 faces: normal, inverted, retrograde (backwards) and retrograde inverted.

Coming back again to the mysterious interval sqrt 2/1, it lies squarely in the middle of all the interrelationships between the tones in a combination overtone and undertone series, and thus between very many modulations in music. It is mathematically a hybrid and mediator, being neither an undertone nor an overtone by definition, since it is irrational, and also both, since it is its own inversion.

Musically, the augmented fourth is the interval of tension par-excellence, especially in classical music, which for years has, I gather, employed an interval equal to or close to sqrt2/1.
It seems almost sure to me that equal temperaments as a whole share the mediating, musical tension-filled nature of sqrt2/1 in music. I would wager that the emergence of equal temperament ushered in the era of musical tension and drama, due to the influence of irrational numbers.
Finally, I would suggest one cannot rule out the possibility that other irrational number systems could contain a musicality unfamiliar to the human ear, but perhaps familiar to the ears of God.

The article at :
http://en.wikipedia.org/wiki/Square_root_of_2
makes clear some of the magic of this number, and may suggest some of the esoteric significance ascribed to it by the Pythagoreans. It seems to be a number of cosmological significance.

Sincerely,
Mike Nolley

Greetings to the list. I have been an interested researcher in Just Intonation theory for a few years, after having been introduced to Partch and "Genesis of a Music" some years back by a friend of mine. I am still doing active research, and would like to post some of my results at a later date.
However, seeing the recent posts about low limit, I think it is important to suggest that 1. the tendency to emphasize low limit can be limiting, as is 2. the tendency to dismiss equal temperaments as being arbitrary and unnatural - not that this is a problem in this group. Much of what I have to say most of you will know already, but perhaps some of it may be interesting.

1. Just Intonation ain't so just

Although low limit intervals guarantee a certain amount of consonance, they guarantee a certain amount of prejudice if you are trying to play certain suspensions of chords that are based on related harmonic series.
There are, to my limited knowledge, at least four basic types of modulation or chord suspension of this type in music:
1. Those based on overtones of undertones (IV for example, since it is a major chord based on 4/3, an undertone).
2. Those based on undertones of overtones (i, for example, because the minor triad 1/1, 6/5, 3/2 is the undertone series based on 3/2. Oddly, iv is not a modulation at all, but the undertone series based on 1/1. This explains why it sounds so good.)

These are the usual contents of a tonality diamond, because the numerators and denominators cancel nicely when you multiply, for example, 8/5 by 5/4, or 3/2 by 4/3.

However, two species of modulation are discriminated against:
3. Overtones of overtones, for example II, since it is the overtone series based on 9/8, an overtone. The Lydian #4, or 45/32, a mainstay of jazz, is excluded from consideration sometimes, even though it is very consonant, because it is the 5/4 of 9/8. Unfortunately, 5/4 times 9/8 does not reduce. This problem affects all overtones of overtones, and as a result some very consonant overtones of overtones have a high limit.
4. Undertones of undertones, for example vii. This for the same reason as above - the fractions don't reduce.

None of this is to claim that these other two species of chord suspension are as consonant as the first two - some are, some aren't. they just happen to be a part of music, and should not be unfairly discriminated against.
Within the next two months I will be posting several "tone maps" based on the relations between all of the permutations of a 37 limit just scale, in the "four directions" of modulation. This should show the real complexity of just intonation, should it be taken to an (absurd?) extreme.

2. In defense of equal temperament

One of the results of my research into just intonation was the discovery that in my tone maps, the various symmetry lines converged on an unique interval, the square root of 2 over 1, which equates to the equal tempered augmented fourth. The reason why this unique equal tempered interval shows up in the middle of just scales is that all just scales are based on fractions, and sqrt 2 is the one interval which is its own fractional inversion. Given any over and undertone series based on a single fundamental, and a high enough limit there will always be a clusterf*** of related tones around the augmented fourth, because the various inversions (11/8 - 16/11, 3/2 - 4/3, 45/32 - 64/45) will tend to approximate each other the closer they get to sqrt 2/1. This is not to say that the inversions are consonant with each other while played simultaneously - 5/4 and 8/5 are not particularly happy together. However, vi to I is one of the most common sequential modulations in music, and it employs 5/4 and 8/5 in starring, complimentary and warring roles (5/4 wins, of course). Broadly speaking, if you are talking about musical interrelationships, the one between major and minor harmonic series is absolutely fundamental, and underlies many suspension and modulation functions. The tension and shift between the two is what music is all about.

A cursory study of fugue concepts will bear this out, because to invert a melody intervalically is to mathematically invert it as well - to turn major to minor and light to dark, or dark to light. This has an immediate musical function that is plain to hear - the inverted melody is emotionally inverted from the original, in many cases. A great fugue subject is one that has 4 faces: normal, inverted, retrograde (backwards) and retrograde inverted.

Coming back again to the mysterious interval sqrt 2/1, it lies squarely in the middle of all the interrelationships between the tones in a combination overtone and undertone series, and thus between very many modulations in music. It is mathematically a hybrid and mediator, being neither an undertone nor an overtone by definition, since it is irrational, and also both, since it is its own inversion.

Musically, the augmented fourth is the interval of tension par-excellence, especially in classical music, which for years has, I gather, employed an interval equal to or close to sqrt2/1.
It seems almost sure to me that equal temperaments as a whole share the mediating, musical tension-filled nature of sqrt2/1 in music. I would wager that the emergence of equal temperament ushered in the era of musical tension and drama, due to the influence of irrational numbers.
Finally, I would suggest one cannot rule out the possibility that other irrational number systems could contain a musicality unfamiliar to the human ear, but perhaps familiar to the ears of God.

The article at :
http://en.wikipedia.org/wiki/Square_root_of_2
makes clear some of the magic of this number, and may suggest some of the esoteric significance ascribed to it by the Pythagoreans. It seems to be a number of cosmological significance.

Sincerely,
Mike Nolley

I think Mike there are many reasons to like ETs just for their simplicity.

I think though it is very difficult to characterize how people who use JI use it as each person i have witness seems to do for a different reason and in different ways. For instance i don't use it of think of it in the way you describe.
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I agree that 45/32 is a fairly consonant sounding interval, and I'm not exactly sure why Harry Partch's guideline for consonance being related to the simplicity of the intervals is a bit 'loose' there. I have two ideas, but I don't know how good they are:

That our hearing apparati (apparatuses?) interpret intervals that are overtones of overtones (essentially still an overtone) in a slightly biased way, perhaps related to the auditory illusion of the mosquito-whine-like difference tone.

That interference patterns in cochlea may be involved, too. Intervals close to sqrt(2) should produce nodes that are farther apart in the cochlea than intervals that are farther away from sqrt(2), making them sound more consonant than the interval simplicity implies.

I might be able to help this out with my handy psychoacoustics book that I have somewhere - but right now I have no idea where I stashed it.

- Andrew

--- In tuning@yahoogroups.com, "bigAndrewM" wrote:

> That our hearing apparati (apparatuses?) interpret intervals that are overtones of overtones (essentially still an overtone) in a slightly biased way, perhaps related to the auditory illusion of the mosquito-whine-like difference tone.

It's not just an overtone of an overtone, it's also 225/224 sharp of 7/5.