Many scales have been defined by propagating a certain interval up or down until an multiple-octave interval (or an approximation thereof) from the root is arrived at. Each of the intermediate notes is then transposed through all of the octaves to fill in all of the notes of the scale. Equal Temperament is defined like this, using an interval of a "fifth" where seven octaves equal twelve "fifths". Meantone Temperaments tried to even out the error between in tune fifths and in tune thirds by propagating with flattened fifths. There are many interesting just (and non-just) intervals that can be used to create scales in this manner.
I'm particularly fascinated by the minor seventh at the minute. I came to a greater understanding of the thirds by creating scales propagated on those intervals, so perhaps I will get somewhere propagating a scale using the minor seventh? The problem is that the equally tempered minor seventh is powerful and fascinating because the brain reads it as both 7/4 and the complement of 9/8. You cannot have a just version of the equally tempered minor seventh, because the minor seventh is a product of equal temperament. (An approximation would be possible, but what would be the point in abstracting it one further step?)
So let's define a scale by propagating the equally tempered minor seventh. What happens? You get 12-TET normal everyday standard equal temperament. This is true for any equally tempered interval. In fact, you could define notes intervals in any given equal temperament as those notes which when propagated produce each other. This is because the temperament is equal. That, at least, is interesting in itself.
Sunday 18 April 2010
Friday 9 April 2010
Notes on a Modulating Just Scale
I have now had the chance to play a keyboard tuned to the scale I proposed here.
The difference between the Fourth of 4/3 and the Super Minor Third of 27/20 was perceivable but too small to be of any value. I would suggest that a piece composed with many II chords and few IV chords would do well to adopt 27/20 rather than 4/3, but that otherwise 4/3 is quite suitable. For example, much Godspeed You! Black Emperor would benefit from this adjustment. I think the powerful identity of 4/3 (it is after all the complement of a very low numbered partial) is such that one tends to hear 27/20 as an out of tune 4/3. This relative identity is an interesting concept that should be explored further.
Two Bbs were used, 7/4 and 9/5. The 7/4 was indeed useful and appropriate, although it did not ring as clearly in the I7 as I would have expected. In some ways it had more of the quality of a 7th chord, and in another way it did not. I think by denying the minor seventh an identity as the complement of the wholetone, part of the leading feeling of the 7th chord is lost, but that by isolating and perfecting the identity of the minor seventh as the seventh partial, another aspect to the character of the 7th chord is distilled and amplified. This is a tool to use with discretion. The two tones were readily identifiable and it was entirely appropriate to have both on the same instrument. I would consider placing the 7/4 on the B and the 9/5 on the Bb and doing away with the major seventh altogether.
I would like to make more observations of the two As in the scale.
Overall, the whole thing held together and rang beautifully. The modulations worked just as I had intended, and it was very pleasant as a melodic scale. This proves that the ideas I have about abstract numbers on paper translate properly into real world musical situations. I know what I am doing.
The difference between the Fourth of 4/3 and the Super Minor Third of 27/20 was perceivable but too small to be of any value. I would suggest that a piece composed with many II chords and few IV chords would do well to adopt 27/20 rather than 4/3, but that otherwise 4/3 is quite suitable. For example, much Godspeed You! Black Emperor would benefit from this adjustment. I think the powerful identity of 4/3 (it is after all the complement of a very low numbered partial) is such that one tends to hear 27/20 as an out of tune 4/3. This relative identity is an interesting concept that should be explored further.
Two Bbs were used, 7/4 and 9/5. The 7/4 was indeed useful and appropriate, although it did not ring as clearly in the I7 as I would have expected. In some ways it had more of the quality of a 7th chord, and in another way it did not. I think by denying the minor seventh an identity as the complement of the wholetone, part of the leading feeling of the 7th chord is lost, but that by isolating and perfecting the identity of the minor seventh as the seventh partial, another aspect to the character of the 7th chord is distilled and amplified. This is a tool to use with discretion. The two tones were readily identifiable and it was entirely appropriate to have both on the same instrument. I would consider placing the 7/4 on the B and the 9/5 on the Bb and doing away with the major seventh altogether.
I would like to make more observations of the two As in the scale.
Overall, the whole thing held together and rang beautifully. The modulations worked just as I had intended, and it was very pleasant as a melodic scale. This proves that the ideas I have about abstract numbers on paper translate properly into real world musical situations. I know what I am doing.
Wednesday 7 April 2010
Phenomenology of Scales
I want to start work on a phenomenology of scales.
Some attributes that I think might be useful are as follows:
1. "Limit": this is the highest partial involved in the scales. For example, Partch's 43-tone scale is one flavour of an 11-limit scale. Western music is largely an approximation of 5-limit. Organum is 3-limit.
2. Symmetry: what happens beneath the tonic? Is a scale mirrored (complementary) or rotationally symmetrical or non-symmetrical? The Dorian and Ionian modes are unique in being rotationally symmetrical about the tonic (rather than the logarithmic mid-point). Partch's 43-tone scale is mirrored due to an error in understanding the nature of utonality. Organum is both mirrored and rotationally symmetrical. Most western modes and western "minor" are non-symmetrical (I think this is a result of being 5-limit; 3-limit and 7-limit reflect more easily. In general, if the limit is one less than a power of 2 the scale may be easily designed to exhibit good symmetry. Is this related to methods for searching for prime numbers?).
3. Utonal limit: which partials are considered suitable for transposing tones into the home octave? In Pythagorean tuning, only the Octave and Fifth are suitable (it has a Utonal 3-limit). In Partch's 43-tone scale it is an 11-limit. Organum is also a 3-limit. (In these latter two examples the Utonal limit is the same as the Otonal limit.) When I started this, it was because I wanted to create scales with a Utonality of exactly 19. I now see that this is nonsense, but scales could be constructed using this method. Scales with a high Otonal limit and a low Utonal limit will be close to the Harmonic series (the Harmonic series is a scale with an infinite Otonal limit and a Utonal limit of 1).
4. Dissonance-limited: rather than limiting scales by partials, they could be limited less crudely by measuring the dissonance of their intervals.
5. Respect for the Octave: does a scale repeat every 2/1? A pentave, repeating on the 3/2 could be used instead, or even other intervals.
I am sure there are more things that would be useful that will present themselves as time goes on.
Some attributes that I think might be useful are as follows:
1. "Limit": this is the highest partial involved in the scales. For example, Partch's 43-tone scale is one flavour of an 11-limit scale. Western music is largely an approximation of 5-limit. Organum is 3-limit.
2. Symmetry: what happens beneath the tonic? Is a scale mirrored (complementary) or rotationally symmetrical or non-symmetrical? The Dorian and Ionian modes are unique in being rotationally symmetrical about the tonic (rather than the logarithmic mid-point). Partch's 43-tone scale is mirrored due to an error in understanding the nature of utonality. Organum is both mirrored and rotationally symmetrical. Most western modes and western "minor" are non-symmetrical (I think this is a result of being 5-limit; 3-limit and 7-limit reflect more easily. In general, if the limit is one less than a power of 2 the scale may be easily designed to exhibit good symmetry. Is this related to methods for searching for prime numbers?).
3. Utonal limit: which partials are considered suitable for transposing tones into the home octave? In Pythagorean tuning, only the Octave and Fifth are suitable (it has a Utonal 3-limit). In Partch's 43-tone scale it is an 11-limit. Organum is also a 3-limit. (In these latter two examples the Utonal limit is the same as the Otonal limit.) When I started this, it was because I wanted to create scales with a Utonality of exactly 19. I now see that this is nonsense, but scales could be constructed using this method. Scales with a high Otonal limit and a low Utonal limit will be close to the Harmonic series (the Harmonic series is a scale with an infinite Otonal limit and a Utonal limit of 1).
4. Dissonance-limited: rather than limiting scales by partials, they could be limited less crudely by measuring the dissonance of their intervals.
5. Respect for the Octave: does a scale repeat every 2/1? A pentave, repeating on the 3/2 could be used instead, or even other intervals.
I am sure there are more things that would be useful that will present themselves as time goes on.
Tuesday 6 April 2010
Beautiful Numbers
So I've been converting a lot of values between cents and hertz, between musical tone and physical frequency. The way the numbers work out is absolutely beautiful.
I've been working with 440Hz as my baseline because modern practice is to tune A=440Hz. It seems that this was a very sensible choice given the biases of modern music. I find the following pattern:
440Hz Tonic
550Hz Major third
660Hz Fifth
(770Hz Septimal)
880Hz Octave
That's really quite beautiful. Is this a product of choosing 440 as the root?
If we pick 330Hz for the tonic, we get:
330Hz Tonic
440Hz Fourth
550Hz Fourth + Major Third
660Hz Octave
So there's different patterns for different starting values. What we are doing here is logarithmically modifying our selected utonal rules.
Or, with 100Hz as the tonic:
100Hz Tonic
110Hz ?
120Hz Minor Third
130Hz ?
140Hz (0.63Hz out from the Super Major Third)
150Hz Fifth
160Hz Fourth + Minor Third
170Hz ?
180Hz Fifth + Minor Third
190Hz ?
200Hz Octave
I could go on picking out different patterns forever. I realise that I am using the same ratio method I have been using but viewed from the opposite angle. Nonetheless, this is a quite different way to think about things, an alternate cognition of tone. Perhaps it will be fruitful.
I've been working with 440Hz as my baseline because modern practice is to tune A=440Hz. It seems that this was a very sensible choice given the biases of modern music. I find the following pattern:
440Hz Tonic
550Hz Major third
660Hz Fifth
(770Hz Septimal)
880Hz Octave
That's really quite beautiful. Is this a product of choosing 440 as the root?
If we pick 330Hz for the tonic, we get:
330Hz Tonic
440Hz Fourth
550Hz Fourth + Major Third
660Hz Octave
So there's different patterns for different starting values. What we are doing here is logarithmically modifying our selected utonal rules.
Or, with 100Hz as the tonic:
100Hz Tonic
110Hz ?
120Hz Minor Third
130Hz ?
140Hz (0.63Hz out from the Super Major Third)
150Hz Fifth
160Hz Fourth + Minor Third
170Hz ?
180Hz Fifth + Minor Third
190Hz ?
200Hz Octave
I could go on picking out different patterns forever. I realise that I am using the same ratio method I have been using but viewed from the opposite angle. Nonetheless, this is a quite different way to think about things, an alternate cognition of tone. Perhaps it will be fruitful.
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