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.
Friday 26 March 2010
Modulating Just Scale
I've designed a scale. It's like this:
In this scale you can modulate between I, II, IV and V chords with both minor and major thirds with absolute just accuracy.
Frankly I don't know what's going on with the minor seventh, so I've included all the options. Obviously to facilitate modulation we need the the 18/10 Fifth Minor Third. There is open (and probably unanswerable debate) as to whether the 12-TET minor seventh is approximating a septimal (i.e. derived from the 7th partial) interval or is the complement to the major second. It's probably both, so I've included both. But I don't think anyone needs three B flats, so there's work to be done here. Perhaps with more experience I would be able to make a firmer choice.
This scale will play modal forms such as C major and D dorian adequately well, although better scales could be designed for that purpose.
So that's basically it. I believe this scale is capable of playing most 20th century popular music and most folk more beautifully than 12-TET (it won't play III, VI or VII chords or 7th chords, except the root seventh). Now I need to build an instrument to test it.
Apologia
Some of these intervals are as small as 21 cents. Some are over two semitones. If you're playing a melody it would make no sense to run up and down the notes, just as it makes little sense to play a 12-TET chromatic melody. It might take a while to feel which notes are right, but it may also be possible to suggest harmonic change just by playing a melody. How effective this is remains to be tested.
Some of these intervals appear to be more dissonant than perhaps would be expected. This is because they are intended to suggest a shift in the root note ("key change"). Accepting this more firmly than is reasonable, the largest dissonance that remains is 9/8.
1/1 C Tonic 9/8 D Supertonic / Fifth Fifth 6/5 Eb Minor Third 5/4 E Major Third 4/3 F Fourth 54/40 = 27/20 F Super Minor Third 45/32 F# Super Major Third 3/2 G Fifth 24/15 G# Fourth Minor Third 20/12 = 10/6 A Fourth Major Third 27/16 A Super Fifth 7/4 Bb Septimal 16/9 Bb Subtonic 18/10 = 9/5 Bb Fifth Minor Third 15/8 B Fifth Major Third 2/1 C Octave
In this scale you can modulate between I, II, IV and V chords with both minor and major thirds with absolute just accuracy.
Frankly I don't know what's going on with the minor seventh, so I've included all the options. Obviously to facilitate modulation we need the the 18/10 Fifth Minor Third. There is open (and probably unanswerable debate) as to whether the 12-TET minor seventh is approximating a septimal (i.e. derived from the 7th partial) interval or is the complement to the major second. It's probably both, so I've included both. But I don't think anyone needs three B flats, so there's work to be done here. Perhaps with more experience I would be able to make a firmer choice.
This scale will play modal forms such as C major and D dorian adequately well, although better scales could be designed for that purpose.
So that's basically it. I believe this scale is capable of playing most 20th century popular music and most folk more beautifully than 12-TET (it won't play III, VI or VII chords or 7th chords, except the root seventh). Now I need to build an instrument to test it.
Apologia
Some of these intervals are as small as 21 cents. Some are over two semitones. If you're playing a melody it would make no sense to run up and down the notes, just as it makes little sense to play a 12-TET chromatic melody. It might take a while to feel which notes are right, but it may also be possible to suggest harmonic change just by playing a melody. How effective this is remains to be tested.
Some of these intervals appear to be more dissonant than perhaps would be expected. This is because they are intended to suggest a shift in the root note ("key change"). Accepting this more firmly than is reasonable, the largest dissonance that remains is 9/8.
Tuesday 23 March 2010
Mathematically Modelling Progression and Resolution
Partch makes an offhand statement that a move from one of his 43-tone notes to a certain other is a move from dissonance towards resolution. This was the first moment in which I saw the link between consonance and resolution.
Resolution is when a melody or harmony comes cleanly to an end. Typically it means returning to the root note or chord at the end of a phrase. The second most resolved place to finish a melody is on the fifth. Progression is when you move away from the root note, developing a melody or harmony into something interesting. Guitarists often say that progression is when you move forward around the circle of fifths, and resolution is when you move back. In both cases, what you are doing is controlling the amount of dissonance relative to the tonic. More dissonance is progression, less dissonance is resolution. In the case of the circle of fifths, you are both playing with powers of 3/2 and increasing the dissonant error in your tuning when you move forwards.
We can do maths with this. Let's assume a simple single note melody (if you like you can imagine a single note drone underneath).
At any time the dissonance is equal to the product of the ratio of the interval between the current note and the tonic.
Let D = dissonance, o = odentity (the numerator of the ratio), u = udentity (the denominator of the ratio)
D = ou
The progression or resolution of a melody is the rate of change of D by time.
Let m = progression/resolution, t = time
m = ΔD/Δt
As an example, let's assume a melody playing four equal-length notes on C, G, D and C. As the melody plays the second note, G, the dissonance is equal to 2x3 = 6 (D = ou). When it was playing the first note, D = ou = 1x1 = 1. So the change in D is 5, and the change in t is 1 unit. m = 5/1 = 5. Coming to the third note, D = ou = 9*8 = 72. m = (72-5)/1 = 67. Finally, at the last note, m = (1-72)/1 = -71.
The negative value of m at t = 4 indicates that it is resolution, not progression, that is occuring. It will be noted the m is not zero-sum, just as in musical practice.
I believe that it's not possible to do proper calculus with this because the series involved are discrete and not continuous.
Obviously this is an abstraction, in real music you do not get isolated one note sine-wave melodies. We need to take into account:
Polyphony
Moving root notes ("key changes")
Timbre
But none of these problems are insurmountable.
It should be obvious that by reversing this method, it would be possible to design a melody or harmonic sequence in which exactly the desired degree of progression and resolution are manufactured. Perhaps you are trying to illustrate a story or lyrics and you want your song to have a certain shape. No longer do you have to have a memory for trial-and-error and tradition generated chord sequences. You could trivially design a program to create them for you, perfectly. I imagine such a program would require a defined set of notes (a key) and some rhythmic ideas to get going, but you could really come up with some cunning harmonies far more complex than a human can efficiently design.
Resolution is when a melody or harmony comes cleanly to an end. Typically it means returning to the root note or chord at the end of a phrase. The second most resolved place to finish a melody is on the fifth. Progression is when you move away from the root note, developing a melody or harmony into something interesting. Guitarists often say that progression is when you move forward around the circle of fifths, and resolution is when you move back. In both cases, what you are doing is controlling the amount of dissonance relative to the tonic. More dissonance is progression, less dissonance is resolution. In the case of the circle of fifths, you are both playing with powers of 3/2 and increasing the dissonant error in your tuning when you move forwards.
We can do maths with this. Let's assume a simple single note melody (if you like you can imagine a single note drone underneath).
At any time the dissonance is equal to the product of the ratio of the interval between the current note and the tonic.
Let D = dissonance, o = odentity (the numerator of the ratio), u = udentity (the denominator of the ratio)
D = ou
The progression or resolution of a melody is the rate of change of D by time.
Let m = progression/resolution, t = time
m = ΔD/Δt
As an example, let's assume a melody playing four equal-length notes on C, G, D and C. As the melody plays the second note, G, the dissonance is equal to 2x3 = 6 (D = ou). When it was playing the first note, D = ou = 1x1 = 1. So the change in D is 5, and the change in t is 1 unit. m = 5/1 = 5. Coming to the third note, D = ou = 9*8 = 72. m = (72-5)/1 = 67. Finally, at the last note, m = (1-72)/1 = -71.
time, t | dissonance, D | progression/resolution, m 0 | 0 | 0 1 | 1 | 1 2 | 6 | 5 3 | 72 | 67 4 | 1 | -71
The negative value of m at t = 4 indicates that it is resolution, not progression, that is occuring. It will be noted the m is not zero-sum, just as in musical practice.
I believe that it's not possible to do proper calculus with this because the series involved are discrete and not continuous.
Obviously this is an abstraction, in real music you do not get isolated one note sine-wave melodies. We need to take into account:
Polyphony
Moving root notes ("key changes")
Timbre
But none of these problems are insurmountable.
It should be obvious that by reversing this method, it would be possible to design a melody or harmonic sequence in which exactly the desired degree of progression and resolution are manufactured. Perhaps you are trying to illustrate a story or lyrics and you want your song to have a certain shape. No longer do you have to have a memory for trial-and-error and tradition generated chord sequences. You could trivially design a program to create them for you, perfectly. I imagine such a program would require a defined set of notes (a key) and some rhythmic ideas to get going, but you could really come up with some cunning harmonies far more complex than a human can efficiently design.
Intonation 8: What is Minor?
I have argued at length that I think that the western use of a bastardised version of the aeolian mode as "minor" is the result of an authoritarian practice approaching the dorian by trial and error.
Partch thinks that "minor" is relationships within utonality, by which he means the denominators of ratios. What does this mean? It means that all the partials which are brought into the home octave by bringing them down a number of octaves are related. It means that all the partials which are brought into the home or second pentave by bringing them down a number of fifths are related. It means that all the partials which are brought into the first three 3-aves by bringing them down thirds are related. It means very little.
At first I thought he had found a justification for why A minor is relative to C major; I thought he had defined this relationship. But it doesn't line up.
Then I thought he had struck upon some kind of inversion/fourths-based/lydian harmony which he had confused with minor. But I can't make sense of that, it's just wishful thinking on my part wanting things to be simple.
This can't just be nonsense, can it? Perhaps I have misunderstood. I suspect that Partch has tried to adopt a fashionable theoretical apology for the minor mess without questioning it, but really I am in ignorance.
Utonality refers to nothing more than the unit by which the scale repetition is defined. Partch is unique in giving equal gravitas to all available options simultaneously. I think he jumped in too early with the numbers and just plain didn't know what he was doing.
I would like some more advanced guidance in this. I tried drafting emails to music professors, but what could they possibly tell me?
Partch thinks that "minor" is relationships within utonality, by which he means the denominators of ratios. What does this mean? It means that all the partials which are brought into the home octave by bringing them down a number of octaves are related. It means that all the partials which are brought into the home or second pentave by bringing them down a number of fifths are related. It means that all the partials which are brought into the first three 3-aves by bringing them down thirds are related. It means very little.
At first I thought he had found a justification for why A minor is relative to C major; I thought he had defined this relationship. But it doesn't line up.
Then I thought he had struck upon some kind of inversion/fourths-based/lydian harmony which he had confused with minor. But I can't make sense of that, it's just wishful thinking on my part wanting things to be simple.
This can't just be nonsense, can it? Perhaps I have misunderstood. I suspect that Partch has tried to adopt a fashionable theoretical apology for the minor mess without questioning it, but really I am in ignorance.
Utonality refers to nothing more than the unit by which the scale repetition is defined. Partch is unique in giving equal gravitas to all available options simultaneously. I think he jumped in too early with the numbers and just plain didn't know what he was doing.
I would like some more advanced guidance in this. I tried drafting emails to music professors, but what could they possibly tell me?
Intonation 7: Consonance and Dissonance
It's self evident that some combinations of notes sing pleasantly and other clash nastily. In between, there are endless shades of grey. At what point does consonance become dissonance?
For medieval church music, the answer was that only harmonies of octaves, fifths and fourths were permissible. For Harry Partch, ratios built out of prime numbers less than 11 are acceptable (he also includes 9, "because it is an odd number". He mentions 15 in this same context and it is clear to me that he is saying that numbers with two prime factors have a value greater than other numbers but less than primes). He halted at 11 and did not proceed to 13 simply because he thought that gave him enough to be getting on with.
A simple mathematical explanation of consonance is that it is a measure of how often frequency intervals line up. An octave is extremely consonant because it has the maximum possible agreement between two different frequencies (the wavelength is half or double that of the other). A fifth is very consonant because three wavelengths of the fifth frequency line up with two wavelengths of its tonic.
So, we can measure consonance by multiplying together a ratio's numerator and denominator (Partch calls these the odentity and udentity, I don't know if this is common practice). It's important to have the simplest equivalent fraction, for example, 2/1 is correct but 4/2 is not. This is because what we're actually after is the product of the identities (prime factors) of the note. Here are some examples:
2/1 (octave) - dissonance 2
3/2 (fifth) - dissonance 6
It makes no sense to try and use this method to calculate the consonance of equally tempered intervals because the ratios involved are irrational. The values that result only reflect your level of accuracy in calculating them.
I'm no closer to defining the point at which consonance becomes dissonance, but I now have a method for objectively comparing intervals and stating whether one is more or less consonant than another.
I wonder if modern music theory understands this as plainly as I have set it out? Partch understands the general principle but survives only by rules of thumb. I need a more advanced textbook.
A simple mathematical explanation of consonance is that it is a measure of how often frequency intervals line up. An octave is extremely consonant because it has the maximum possible agreement between two different frequencies (the wavelength is half or double that of the other). A fifth is very consonant because three wavelengths of the fifth frequency line up with two wavelengths of its tonic.
So, we can measure consonance by multiplying together a ratio's numerator and denominator (Partch calls these the odentity and udentity, I don't know if this is common practice). It's important to have the simplest equivalent fraction, for example, 2/1 is correct but 4/2 is not. This is because what we're actually after is the product of the identities (prime factors) of the note. Here are some examples:
2/1 (octave) - dissonance 2
3/2 (fifth) - dissonance 6
5/4 (just major third) - dissonance 20
45/32 (a small just tritone) - dissonance 1440 (this is universally agreed to be dissonant)It makes no sense to try and use this method to calculate the consonance of equally tempered intervals because the ratios involved are irrational. The values that result only reflect your level of accuracy in calculating them.
I'm no closer to defining the point at which consonance becomes dissonance, but I now have a method for objectively comparing intervals and stating whether one is more or less consonant than another.
I wonder if modern music theory understands this as plainly as I have set it out? Partch understands the general principle but survives only by rules of thumb. I need a more advanced textbook.
Intonation 6: Constructing a Tonic/Supertonic scale
Western music gave up just intonation because it wanted to modulate between keys. One reason you might want to do this is laziness. You can make a whole album of very similar songs if you make small alterations to the key and rhythm: fast blues in G, slow blues in D, shuffle in E etc. I am chiefly interested in exploring dronal music (using only one chord), but I also recognise that there's all sorts of awesome thing you can do with it modulation. So let's see if we can construct a justly-tuned scale that allows for some limited modulation.
As an example I'm going to pick one very simple modulation: that between the tonic and the supertonic. The supertonic is simply the chord one tone above the tonic. So we might be moving from C to D and back again. This is a useful technique often used in post rock. It sounds very epic.
I believe that the wholetone (major second) is the difference between two fifths stacked on top of each other and an octave. This is 3/2 * 3/2 * 1/2/1 = 9/8. Harry Partch gives both 9/8 and 10/9 for his major second(s), but this is a product of his methodology: he first found all ratio combinations of small numbers, and then looked for identities for them. It is possible that there is a good argument for using 10/9 or both, but I don't know it. The ratio used in traditional Pythagorean tuning is also 9/8 so I think it's pretty safe ground making that assertion, but I'd still like to be very explicit that this is totally subjective.
Let's say I want to make ordinary major chords using the tonic, the fifth and the third. The fifth and the major third correspond to the third and fifth partials respectively, and are universally agreed to be 3/2 (the fifth) and 5/4 (the major third).
If we make a note one tone above each of these, we get a six tone scale as follows:
1/1 Tonic
9/8 Supertonic
3/2 Major Third
45/32 Super Major Third (this is somewhere between the perfect fourth and the lower just tritone)
3/2 Fifth
27/16 Super Fifth (this note is less than the limit of human hearing out from the Major Sixth)
So it should be easy to construct more complex scales that allow for a wider range of modulation, such as a justly intoned blues.
As an example I'm going to pick one very simple modulation: that between the tonic and the supertonic. The supertonic is simply the chord one tone above the tonic. So we might be moving from C to D and back again. This is a useful technique often used in post rock. It sounds very epic.
I believe that the wholetone (major second) is the difference between two fifths stacked on top of each other and an octave. This is 3/2 * 3/2 * 1/2/1 = 9/8. Harry Partch gives both 9/8 and 10/9 for his major second(s), but this is a product of his methodology: he first found all ratio combinations of small numbers, and then looked for identities for them. It is possible that there is a good argument for using 10/9 or both, but I don't know it. The ratio used in traditional Pythagorean tuning is also 9/8 so I think it's pretty safe ground making that assertion, but I'd still like to be very explicit that this is totally subjective.
Let's say I want to make ordinary major chords using the tonic, the fifth and the third. The fifth and the major third correspond to the third and fifth partials respectively, and are universally agreed to be 3/2 (the fifth) and 5/4 (the major third).
If we make a note one tone above each of these, we get a six tone scale as follows:
1/1 Tonic
9/8 Supertonic
3/2 Major Third
45/32 Super Major Third (this is somewhere between the perfect fourth and the lower just tritone)
3/2 Fifth
27/16 Super Fifth (this note is less than the limit of human hearing out from the Major Sixth)
So it should be easy to construct more complex scales that allow for a wider range of modulation, such as a justly intoned blues.
Intonation 5: Limited Scales
It would appear that a common approach used by "microtonal" musicians (the last thing I want to do is get involved with microtonal music; I just want to tune better) is to take a set of prime numbers and find all the combinations of these as ratios. It's so ingrained that you can say something like "11-limit scale" and it is understood that all the combinations of prime numbers less than eleven are used to make up the ratios which define the scale.
That's all well and good but you end up with the sound of maths rather than something more sophisticated. All that abstract 20th century music seems pretty pointless from where I'm sitting, although I can see that it had merit in context.
Why use prime numbers only? Sure, there's something special about the prime-numbered partials, but that doesn't make the decision for you. Why not use only secondary partials and suggest the primes through harmony? The contrapuntal possibilities of this approach are exciting. You could use two lines harmonious to the fundamental to make a third, purer line ring out. With the right instrument that would sound eerily beautiful like icicles on a crisp winter day.
I think there's a fair case for assessing 12-TET (modern tuning) as an approximation of a selective 19-limit scale (with no 7, 11 or 13) filled in with secondary and ternary tones. That can never be more than opinion, but it's where I'm working from at present. Another valid approach is that 12-TET is an approximation of a 3-limited scale, but with each number used up to 12 times.
So I want to design scales. Here's some approaches that can be used:
1. Traditional microtonal prime-limited sets
2. Secondary-limited sets, ternary sets etc.
3. Sets with more limited denominators than numerators (e.g. Pythagorean tuning is a subset of a scale which is 3-limited in the denominator but not really limited at all in the numerator. My initial interest in this was because I wanted to create a 5-denominator-limited scale, although I did not know that was what it was at the time)
4. Do not use a denominator at all. One way this could work would be to take the partials that exist in the fifth octave above the fundamental.
5. Use a function like the product of the primes as the limit. This would mean that that if you used a low prime in the ratio, you could use a larger number for the other part, but you couldn't have two large numbers.
6. Introduce a lower limit, for example 3-19 limited or somesuch.
7. Sets limited by the number of repetitions of each prime in some way.
8. Pick a set of harmonies that just sound good together. I suspect this will yield the best music. How am I going to find these? By experimenting with the other options I've laid out in 1-7.
A more distant goal would be to aim at rotationally symmetrical just-derived scales. An approximation would be to introduce an extra 3/2 (fifth) to create the mirrored notes, but I think there will be a more perfect way.
That's all well and good but you end up with the sound of maths rather than something more sophisticated. All that abstract 20th century music seems pretty pointless from where I'm sitting, although I can see that it had merit in context.
Why use prime numbers only? Sure, there's something special about the prime-numbered partials, but that doesn't make the decision for you. Why not use only secondary partials and suggest the primes through harmony? The contrapuntal possibilities of this approach are exciting. You could use two lines harmonious to the fundamental to make a third, purer line ring out. With the right instrument that would sound eerily beautiful like icicles on a crisp winter day.
I think there's a fair case for assessing 12-TET (modern tuning) as an approximation of a selective 19-limit scale (with no 7, 11 or 13) filled in with secondary and ternary tones. That can never be more than opinion, but it's where I'm working from at present. Another valid approach is that 12-TET is an approximation of a 3-limited scale, but with each number used up to 12 times.
So I want to design scales. Here's some approaches that can be used:
1. Traditional microtonal prime-limited sets
2. Secondary-limited sets, ternary sets etc.
3. Sets with more limited denominators than numerators (e.g. Pythagorean tuning is a subset of a scale which is 3-limited in the denominator but not really limited at all in the numerator. My initial interest in this was because I wanted to create a 5-denominator-limited scale, although I did not know that was what it was at the time)
4. Do not use a denominator at all. One way this could work would be to take the partials that exist in the fifth octave above the fundamental.
5. Use a function like the product of the primes as the limit. This would mean that that if you used a low prime in the ratio, you could use a larger number for the other part, but you couldn't have two large numbers.
6. Introduce a lower limit, for example 3-19 limited or somesuch.
7. Sets limited by the number of repetitions of each prime in some way.
8. Pick a set of harmonies that just sound good together. I suspect this will yield the best music. How am I going to find these? By experimenting with the other options I've laid out in 1-7.
A more distant goal would be to aim at rotationally symmetrical just-derived scales. An approximation would be to introduce an extra 3/2 (fifth) to create the mirrored notes, but I think there will be a more perfect way.
Intonation 4
(cont. from earlier post)
We've made a lot of progress on the just intonation front. Tim has lent me a sound module with a "Just" preset that plays in a form of just intonation. I don't know which flavour of just intonation they used, but it disrespects the octave and is centred on middle C. It rings and sings and sounds a bit mediaeval and it's exactly how I expected it to be even though I'd never heard it and all I had was maths and faith.
I had noticed there was a pattern in the arrangement of partials in the harmonic series. To recap, I had seen that:
1.1 All overtones have overtones of themselves.
1.2 There is some kind of complex exponential relationship surrounding the position of overtones of each particular order in the series.
1.3 Some partials, like the 15th partial (commonly identified as the major seventh), can be seen to be made up of two earlier partials, in this case the fifth and major third identified as partials 3 and 5 respectively.
Mark was talking about how the odd numbered partials were the interesting ones. I had seen this written on the internet but it sounded like guff: the early even numbered partials are very useful. Mark also mentioned that he had got "carried away" with prime numbers. Then it all fell into place and I saw how it worked.
Each overtone in a prime numbered position in the harmonic series (we have been calling them "prime partials") has a strong identity. We get the following identities:
2 - Octave
3 - Fifth
5 - Major Third
7 - unassigned (this is the forbidden note on a natural trumpet)
11 - unassigned
13 - unassigned
17 - Minor Second (semitone)
19 - Minor Third
etc.
If you get too far into the harmonic series it gets diluted to the point where it may as well be dissonance, so we'll arbitrarily stop there.
If you multiply these partials together you get an interval which is musically the sum of the elements you put into it. So 3rd partial x 5th partial = 15th partial, and fifth + major third = seventh. This is true for all intervals. You can find the elements that make up an interval by finding the prime factors. How awesome is that?
Next we tried generating ratios to plug into software synths Mark was programming as we went. We weren't happy with the ratios given on wikipedia so we decided to find better ones ourselves. We realised that there wasn't a good prime seventh interval that corresponded with modern tuning, so we used the 15th partial instead. This shows that modern tuning approximates a mixture of prime and second-degree partials. There is thus nothing particularly archetypal about prime partials, but they are elemental in a way.
We were interested in the 7th partial (which is the 7th note playable on a natural trumpet using only embouchure, that is the shape of your lips and how hard you blow). We quickly found that it occupied a place in the scale somewhere around the minor seventh in modern tuning. (It's a coincidence that the 7th partial makes a "seventh").
At this point Mark made a machine with three keys. It produced a constant tone until you pressed the keys, at which point the tone changed. The lowest key brought it up a major third. The middle key did a fifth and the third key the interval of the 7th partial. If you held down combinations of the keys, it altered the pitch by both intervals. If you wanted a major seventh you held down the major third and fifth keys. In this way we produced an eight note scale spread over several octaves. If you had one in both hands you could begin to make open-handed harmonies. Very exciting. Afterwards, when we played a normal keyboard to check our tuning and identifications it sounded disgusting. The normal intervals used in modern tuning sounded dissonant and ugly. The entire history of recorded music needs to be thrown away and it all recorded again. You would hear it too once you had heard how it is supposed to be.
One other thing we did was to understand more thoroughly how to make ratios. I used the following method to generate some simple ratios, scrawling out a table on a whiteboard. I took the numbers of prime partials and used them as the numerators. For the denominators, I used 2 raised to the power of the number of octaves above the fundamental the partial sounds in (this is always the power of 2 below the value of the partial). It is necessary to divide by this amount to bring all of the notes into the same octave together. This gives, in ascending order:
tonic 1/2^0 = 1/1
minor second 17/2^4 = 17/16
minor third 19/2^4 = 19/16
major third 5/2^2 = 5/4
? 11/2^3 = 11/8
fifth 3/2^1 = 3/2
? 13/2^3 = 13/8
? 7/2^2 = 7/4
octave 2/2^0 = 2/1
etc.
Mark and I noticed that all of the denominators were powers of 2 before we understood what it meant. We turned the relationship around to try and find the meaning given above, and found the relationship (sounding octave) = log2 (denominator).
We were struggling with some intervals, particularly the fourth. The fourth is one of the most important intervals, especially in the 20th century when the guitar dominated music. I understand the fourth as the anti-fifth (a fourth and a fifth make an octave exactly, even in just temperament), but what does that mean in terms of ratios? We couldn't work it out.
I noticed that in traditional Pythagorean tuning many of the denominators were powers of 3. I saw that this meant they were related to the third partial (the fifth) rather than the second partial (the octave). I modified my spreadsheet to respect the pentave as the basic unit of harmony rather than the octave and it fell into place. By pentave I mean that every seven semitones is considered to be the same note as the fundamental. There's a much stronger argument for using twelve semitones as we do (the octave), but why not find overtones of the third partial as well as overtones of the fundamental and the second partial? Here is another set of intervals which are just as valid, and when we trialled it, they sounded sweetly compatible.
I made an important error. I assumed we would have to use powers of 3 for the relationship because it was the third partial instead of the second that we were operating from. It is actually powers of 3/2 that you must use. This is because we have defined the fifth in the home octave and not in the next octave where it's partial belongs. If you use 3, you are actually using an interval of an octave plus a fifth as your basic unit of repetition. I could see by trial and error that I was out by a factor of 2 but it took a long while to understand why.
Using the pentave rather than the octave, all of the partials get new enharmonic identities.
2 - Fourth
3 - Fourth
5 - Pentave
7 - unassigned
11 - unassigned
13 - unassigned
17 - Major Second (wholetone)
19 - Minor Third
etc.
I shouted, "I've found the fourth!" In the first pentave (so already a fifth up) the interval left to the first partial is a fourth. If you use octaves the first partial is a whole octave away. Fifth + Fourth = Octave.
fourth 2/(3/2)^1 = 4/3
Which is exactly where it is in Pythagorean tradition. We made another key on our machine for this interval and it sounded great. It was 2am and we stopped there.
One final thing, when we were making the machine we needed a process to drop the pitch back down when you release the key. We made a mistake here at first, using the ratio of the note in the octave below in the belief that this would negate the effect of the ratio increase. We made this mistake because in common musical practice, if you go up an interval you can go back down the same interval by moving down the keyboard. What we actually needed was the inverse of the ratio we had gone up by.
This may seem trivial, but a week or two ago I posted about my big idea that the Dorian mode is the genuine minor mode and that western practice got it wrong when it picked the Aeolian. Well, the way we mistakenly tried to come back down the scale was rotationally symmetrical in exactly the way I had meant when I tried to justify why Dorian was the bee's knees. Here is the objective proof that the Dorian mode is the real minor. Before, all I had on that was aesthetics. Now I have maths.
(cont. in part 5)
Original comments from posting at woodideas.blogspot.com:
3 comments:
Anonymous said...
Have you look into the work of Harry Partch? He was really into microtonal scales and stuff for similar reasons to yours.
18 Feb 2010 16:51:00
This I Did Not Do said...
I've never heard of him before. Reading the wikipedia article, he seems very similar in character to me, and his ideas seem a very good match to what we're exploring at the minute. I'm sufficiently excited to have ordered his book Genesis of a Music off amazon. Thanks for that.
I'm listening to some of his music and I'm disappointed. He's really not using this to its potential. I have a vision of something quite different.
Also, I am turned off by the idea of microtonality. I don't want to piddle around in tiny insignificant details. I want to fix our broken system and draw something purer from it.
18 Feb 2010 18:11:00
mark said...
I now understand the revelations that you had while I was hacking away trying to get the synth working... nice write-up.
19 Feb 2010 09:12:00
We've made a lot of progress on the just intonation front. Tim has lent me a sound module with a "Just" preset that plays in a form of just intonation. I don't know which flavour of just intonation they used, but it disrespects the octave and is centred on middle C. It rings and sings and sounds a bit mediaeval and it's exactly how I expected it to be even though I'd never heard it and all I had was maths and faith.
I had noticed there was a pattern in the arrangement of partials in the harmonic series. To recap, I had seen that:
1.1 All overtones have overtones of themselves.
1.2 There is some kind of complex exponential relationship surrounding the position of overtones of each particular order in the series.
1.3 Some partials, like the 15th partial (commonly identified as the major seventh), can be seen to be made up of two earlier partials, in this case the fifth and major third identified as partials 3 and 5 respectively.
Mark was talking about how the odd numbered partials were the interesting ones. I had seen this written on the internet but it sounded like guff: the early even numbered partials are very useful. Mark also mentioned that he had got "carried away" with prime numbers. Then it all fell into place and I saw how it worked.
Each overtone in a prime numbered position in the harmonic series (we have been calling them "prime partials") has a strong identity. We get the following identities:
2 - Octave
3 - Fifth
5 - Major Third
7 - unassigned (this is the forbidden note on a natural trumpet)
11 - unassigned
13 - unassigned
17 - Minor Second (semitone)
19 - Minor Third
etc.
If you get too far into the harmonic series it gets diluted to the point where it may as well be dissonance, so we'll arbitrarily stop there.
If you multiply these partials together you get an interval which is musically the sum of the elements you put into it. So 3rd partial x 5th partial = 15th partial, and fifth + major third = seventh. This is true for all intervals. You can find the elements that make up an interval by finding the prime factors. How awesome is that?
Next we tried generating ratios to plug into software synths Mark was programming as we went. We weren't happy with the ratios given on wikipedia so we decided to find better ones ourselves. We realised that there wasn't a good prime seventh interval that corresponded with modern tuning, so we used the 15th partial instead. This shows that modern tuning approximates a mixture of prime and second-degree partials. There is thus nothing particularly archetypal about prime partials, but they are elemental in a way.
We were interested in the 7th partial (which is the 7th note playable on a natural trumpet using only embouchure, that is the shape of your lips and how hard you blow). We quickly found that it occupied a place in the scale somewhere around the minor seventh in modern tuning. (It's a coincidence that the 7th partial makes a "seventh").
At this point Mark made a machine with three keys. It produced a constant tone until you pressed the keys, at which point the tone changed. The lowest key brought it up a major third. The middle key did a fifth and the third key the interval of the 7th partial. If you held down combinations of the keys, it altered the pitch by both intervals. If you wanted a major seventh you held down the major third and fifth keys. In this way we produced an eight note scale spread over several octaves. If you had one in both hands you could begin to make open-handed harmonies. Very exciting. Afterwards, when we played a normal keyboard to check our tuning and identifications it sounded disgusting. The normal intervals used in modern tuning sounded dissonant and ugly. The entire history of recorded music needs to be thrown away and it all recorded again. You would hear it too once you had heard how it is supposed to be.
One other thing we did was to understand more thoroughly how to make ratios. I used the following method to generate some simple ratios, scrawling out a table on a whiteboard. I took the numbers of prime partials and used them as the numerators. For the denominators, I used 2 raised to the power of the number of octaves above the fundamental the partial sounds in (this is always the power of 2 below the value of the partial). It is necessary to divide by this amount to bring all of the notes into the same octave together. This gives, in ascending order:
tonic 1/2^0 = 1/1
minor second 17/2^4 = 17/16
minor third 19/2^4 = 19/16
major third 5/2^2 = 5/4
? 11/2^3 = 11/8
fifth 3/2^1 = 3/2
? 13/2^3 = 13/8
? 7/2^2 = 7/4
octave 2/2^0 = 2/1
etc.
Mark and I noticed that all of the denominators were powers of 2 before we understood what it meant. We turned the relationship around to try and find the meaning given above, and found the relationship (sounding octave) = log2 (denominator).
We were struggling with some intervals, particularly the fourth. The fourth is one of the most important intervals, especially in the 20th century when the guitar dominated music. I understand the fourth as the anti-fifth (a fourth and a fifth make an octave exactly, even in just temperament), but what does that mean in terms of ratios? We couldn't work it out.
I noticed that in traditional Pythagorean tuning many of the denominators were powers of 3. I saw that this meant they were related to the third partial (the fifth) rather than the second partial (the octave). I modified my spreadsheet to respect the pentave as the basic unit of harmony rather than the octave and it fell into place. By pentave I mean that every seven semitones is considered to be the same note as the fundamental. There's a much stronger argument for using twelve semitones as we do (the octave), but why not find overtones of the third partial as well as overtones of the fundamental and the second partial? Here is another set of intervals which are just as valid, and when we trialled it, they sounded sweetly compatible.
I made an important error. I assumed we would have to use powers of 3 for the relationship because it was the third partial instead of the second that we were operating from. It is actually powers of 3/2 that you must use. This is because we have defined the fifth in the home octave and not in the next octave where it's partial belongs. If you use 3, you are actually using an interval of an octave plus a fifth as your basic unit of repetition. I could see by trial and error that I was out by a factor of 2 but it took a long while to understand why.
Using the pentave rather than the octave, all of the partials get new enharmonic identities.
2 - Fourth
3 - Fourth
5 - Pentave
7 - unassigned
11 - unassigned
13 - unassigned
17 - Major Second (wholetone)
19 - Minor Third
etc.
I shouted, "I've found the fourth!" In the first pentave (so already a fifth up) the interval left to the first partial is a fourth. If you use octaves the first partial is a whole octave away. Fifth + Fourth = Octave.
fourth 2/(3/2)^1 = 4/3
Which is exactly where it is in Pythagorean tradition. We made another key on our machine for this interval and it sounded great. It was 2am and we stopped there.
One final thing, when we were making the machine we needed a process to drop the pitch back down when you release the key. We made a mistake here at first, using the ratio of the note in the octave below in the belief that this would negate the effect of the ratio increase. We made this mistake because in common musical practice, if you go up an interval you can go back down the same interval by moving down the keyboard. What we actually needed was the inverse of the ratio we had gone up by.
This may seem trivial, but a week or two ago I posted about my big idea that the Dorian mode is the genuine minor mode and that western practice got it wrong when it picked the Aeolian. Well, the way we mistakenly tried to come back down the scale was rotationally symmetrical in exactly the way I had meant when I tried to justify why Dorian was the bee's knees. Here is the objective proof that the Dorian mode is the real minor. Before, all I had on that was aesthetics. Now I have maths.
(cont. in part 5)
Original comments from posting at woodideas.blogspot.com:
3 comments:
Anonymous said...
Have you look into the work of Harry Partch? He was really into microtonal scales and stuff for similar reasons to yours.
18 Feb 2010 16:51:00
This I Did Not Do said...
I've never heard of him before. Reading the wikipedia article, he seems very similar in character to me, and his ideas seem a very good match to what we're exploring at the minute. I'm sufficiently excited to have ordered his book Genesis of a Music off amazon. Thanks for that.
I'm listening to some of his music and I'm disappointed. He's really not using this to its potential. I have a vision of something quite different.
Also, I am turned off by the idea of microtonality. I don't want to piddle around in tiny insignificant details. I want to fix our broken system and draw something purer from it.
18 Feb 2010 18:11:00
mark said...
I now understand the revelations that you had while I was hacking away trying to get the synth working... nice write-up.
19 Feb 2010 09:12:00
Temperament 3
(cont. from previous post)
There is no mystery to the harmonic series. Let's be plain about it, it's just 1/1, 1/2, 1/3, 1/4... that's all it is. Let's say you start off with an A note at 440Hz (that means the sound wave runs at 440 cycles per second). If you go down the harmonic series from there the next note is 1/2 that frequency, that is 220Hz, another A, followed by 1/3 of 440Hz, which is 146.66Hz, a D, and so on. If you go up it's the same thing, but you have to flip the ratio over so for the second term you multiply 440Hz by 2, then by 3 for the third note and so on. Simple. This is how a natural trumpet produces different notes without pistons.
One problem that takes a bit more thought is that your brain perceives tone logarithmically. The size of a tone in hertz is different depending on where the tone is in the musical spectrum. If you want to look up the maths you can google it. I'm not great with logarithms so I won't attempt an explanation, but I've got the equations working which is sufficient for me at present.
I found myself swimming in a sea of information, so I made a spreadsheet to try and gain control of it. You can download it to your computer in the file menu in google docs and have a play with it if you like.
A few notes about the spreadsheet:
You can adjust the absolute frequencies and the note identification names by changing the note in the purple box labelled "Tonic:" Everything will follow through automatically from there.
"Partials" is the name of the harmonic elements that go into making a tone. It includes both overtones and the fundamental.
"12-TET" means 12-Tone Equal Temperament, our modern system of tuning.
The "Diff. (Cents)" column tells you how much a harmonic is out of tune in our modern system.
The purple box labelled "Identification Resolution:" specifies the maximum value allowed in the "Diff. (Cents)" column to lead to the following columns being filled in. I've set it at 14 (14 cents out of tune!) so that major thirds are identified, but 14 is a completely unacceptable level of compromise. If you set it to 50, all harmonics are identified as their closest diatonic equivalent, even if they're a quarter tone out. If you set it to 2 you get medieval organum.
So what can we do with this information? We can use the values produced to tune to Pythagorean tuning or other flavours of just temperament. We can use it to examine the relationship between harmony and equal temperament. But the two things that excite me are:
1. All of those unused harmonies which have no equivalents in modern tuning. The enharmonic possibilities and unfelt emotional affects that may be hidden in these gaps between consensus reality really inspire me. There are legitimate notes here that no one plays and few people have ever consciously heard.
2. I have noticed a pattern in the harmonic series. I'm sure other people have spotted it, but I need to know exactly how this works. The fifteenth partial, identified as a seventh, is actually a major third of a fifth. You can justify this by looking at their tuning. A fifth is 1.96 cents out of tune, a major third is -13.69 cents out. The fifteenth partial is 1.96-13.69=-11.73 cents out. This is because each of the overtones of the fundamental have their own overtones. The relationship of their position in the series follows the pattern 1, 2, 4, 8... in complex ways. This is what I will explore today.
(part 4 to follow)
There is no mystery to the harmonic series. Let's be plain about it, it's just 1/1, 1/2, 1/3, 1/4... that's all it is. Let's say you start off with an A note at 440Hz (that means the sound wave runs at 440 cycles per second). If you go down the harmonic series from there the next note is 1/2 that frequency, that is 220Hz, another A, followed by 1/3 of 440Hz, which is 146.66Hz, a D, and so on. If you go up it's the same thing, but you have to flip the ratio over so for the second term you multiply 440Hz by 2, then by 3 for the third note and so on. Simple. This is how a natural trumpet produces different notes without pistons.
One problem that takes a bit more thought is that your brain perceives tone logarithmically. The size of a tone in hertz is different depending on where the tone is in the musical spectrum. If you want to look up the maths you can google it. I'm not great with logarithms so I won't attempt an explanation, but I've got the equations working which is sufficient for me at present.
I found myself swimming in a sea of information, so I made a spreadsheet to try and gain control of it. You can download it to your computer in the file menu in google docs and have a play with it if you like.
A few notes about the spreadsheet:
You can adjust the absolute frequencies and the note identification names by changing the note in the purple box labelled "Tonic:" Everything will follow through automatically from there.
"Partials" is the name of the harmonic elements that go into making a tone. It includes both overtones and the fundamental.
"12-TET" means 12-Tone Equal Temperament, our modern system of tuning.
The "Diff. (Cents)" column tells you how much a harmonic is out of tune in our modern system.
The purple box labelled "Identification Resolution:" specifies the maximum value allowed in the "Diff. (Cents)" column to lead to the following columns being filled in. I've set it at 14 (14 cents out of tune!) so that major thirds are identified, but 14 is a completely unacceptable level of compromise. If you set it to 50, all harmonics are identified as their closest diatonic equivalent, even if they're a quarter tone out. If you set it to 2 you get medieval organum.
So what can we do with this information? We can use the values produced to tune to Pythagorean tuning or other flavours of just temperament. We can use it to examine the relationship between harmony and equal temperament. But the two things that excite me are:
1. All of those unused harmonies which have no equivalents in modern tuning. The enharmonic possibilities and unfelt emotional affects that may be hidden in these gaps between consensus reality really inspire me. There are legitimate notes here that no one plays and few people have ever consciously heard.
2. I have noticed a pattern in the harmonic series. I'm sure other people have spotted it, but I need to know exactly how this works. The fifteenth partial, identified as a seventh, is actually a major third of a fifth. You can justify this by looking at their tuning. A fifth is 1.96 cents out of tune, a major third is -13.69 cents out. The fifteenth partial is 1.96-13.69=-11.73 cents out. This is because each of the overtones of the fundamental have their own overtones. The relationship of their position in the series follows the pattern 1, 2, 4, 8... in complex ways. This is what I will explore today.
(part 4 to follow)
Temperament 2
(cont. from previous post)
When I was first experimenting with pitch as a teenager (I was completely off the rails with my wild experimentation back then) I analysed a number of songs by Deftones and other metal bands and I found that often they would use a scale which went tone, semitone, tone, semitone, tone, semitone... At the time I thought they just didn't know any better, but this is actually a very valuable idea.
One of the features of this scale is that it takes two octaves to come back to the root note. Every note in the second octave is out of key with every note in the home octave. You would think this would cause problems, especially when the only way this pattern can manifest is when you go up into the second octave, and considering the fact that a bass guitar is an octave below a normal guitar. But it doesn't mess everything up, it creates interesting effects. There is very much you can do by disrespecting the octave. It's a short cut to incredibly complex modulation.
What this proves is that you can abandon one of the most basic assumptions of western music. Notes which are an octave apart do not have to modelled as the same note. It's just a harmony like any other. A very clean and pure harmony, but just a harmony.
Let's say we're tuning in just temperament starting on D, and we're moving up from the home octave to the second octave. How should we tune A in the second octave? It's a fourth above a fifth above a fifth. This gives us a note 2 cents out from equal temperament. But what if it's five minor thirds and a major third up? This time, the note is 22 cents out. It is a different note.
There's four solutions to this:
0. Use the most fundamental intervals you can to construct each note. Octave > fifth > fourth > major third etc. This is the historical solution.
1. Build a keyboard laid out in such a way as to offer all possible combinations. You would need 12 times the number of combinations of 12 intervals for every octave. That's a huge number of buttons.
2. Build an instrument that retunes itself based on the root of whatever you're currently playing. Programming this would be an exciting challenge. Googling reveals several attempts have been made.
3. Optimise your scale for a certain root note and a certain favoured interval.
I want to explore 2 and 3.
For 3, the root note will be D = 294 Hz. If we optimise for the fifth or the fourth instead of the octave, we end up with a scale that is very close to equal temperament. I hope that by optimising for either the thirds, the tritone or the seconds, we may be able to build new musics with different harmonies that are emotionally affecting in unexpectedly nuanced ways. There's a whole realm out there that has been very ill explored.
The conclusion of DIftW is a violin duel with the devil. The devil plays all this fancy filigree stuff, but the simple woodsman just plays a single harmony of unexpected nuance and purity. The secret chord that David played that pleased the lord. I want to find new chords, and I have found a gap in which they may exist.
(part 3 to follow)
Original comments from posting at woodideas.blogspot.com:
2 comments:
madKharman said...
I'm on board - I estimate option 3 should take a days' programming using virtual control voltages and some mildly complex logarithmic equations. Option 2 is, of course, grail territory, but not out of the question.
12 Feb 2010 14:23:00
This I Did Not Do said...
For option 2 we would have to do two things:
2.1 Design an algorithm to find the root note based on which keys were being pressed and the active motion of the performance (not impossible, but tricky... some parameters in the algorithm could be altered with knobs during performance.)
2.2 Retune the keys as we go, which is just algebra.
I'm sitting down with pen and paper now playing with stuff. I can't find enough information on the harmonic series from the music end of things (the first 30 terms are not enough) so I'm going to calculate it all from scratch. There's wonderful shapes in here.
12 Feb 2010 16:27:00
When I was first experimenting with pitch as a teenager (I was completely off the rails with my wild experimentation back then) I analysed a number of songs by Deftones and other metal bands and I found that often they would use a scale which went tone, semitone, tone, semitone, tone, semitone... At the time I thought they just didn't know any better, but this is actually a very valuable idea.
One of the features of this scale is that it takes two octaves to come back to the root note. Every note in the second octave is out of key with every note in the home octave. You would think this would cause problems, especially when the only way this pattern can manifest is when you go up into the second octave, and considering the fact that a bass guitar is an octave below a normal guitar. But it doesn't mess everything up, it creates interesting effects. There is very much you can do by disrespecting the octave. It's a short cut to incredibly complex modulation.
What this proves is that you can abandon one of the most basic assumptions of western music. Notes which are an octave apart do not have to modelled as the same note. It's just a harmony like any other. A very clean and pure harmony, but just a harmony.
Let's say we're tuning in just temperament starting on D, and we're moving up from the home octave to the second octave. How should we tune A in the second octave? It's a fourth above a fifth above a fifth. This gives us a note 2 cents out from equal temperament. But what if it's five minor thirds and a major third up? This time, the note is 22 cents out. It is a different note.
There's four solutions to this:
0. Use the most fundamental intervals you can to construct each note. Octave > fifth > fourth > major third etc. This is the historical solution.
1. Build a keyboard laid out in such a way as to offer all possible combinations. You would need 12 times the number of combinations of 12 intervals for every octave. That's a huge number of buttons.
2. Build an instrument that retunes itself based on the root of whatever you're currently playing. Programming this would be an exciting challenge. Googling reveals several attempts have been made.
3. Optimise your scale for a certain root note and a certain favoured interval.
I want to explore 2 and 3.
For 3, the root note will be D = 294 Hz. If we optimise for the fifth or the fourth instead of the octave, we end up with a scale that is very close to equal temperament. I hope that by optimising for either the thirds, the tritone or the seconds, we may be able to build new musics with different harmonies that are emotionally affecting in unexpectedly nuanced ways. There's a whole realm out there that has been very ill explored.
The conclusion of DIftW is a violin duel with the devil. The devil plays all this fancy filigree stuff, but the simple woodsman just plays a single harmony of unexpected nuance and purity. The secret chord that David played that pleased the lord. I want to find new chords, and I have found a gap in which they may exist.
(part 3 to follow)
Original comments from posting at woodideas.blogspot.com:
2 comments:
madKharman said...
I'm on board - I estimate option 3 should take a days' programming using virtual control voltages and some mildly complex logarithmic equations. Option 2 is, of course, grail territory, but not out of the question.
12 Feb 2010 14:23:00
This I Did Not Do said...
For option 2 we would have to do two things:
2.1 Design an algorithm to find the root note based on which keys were being pressed and the active motion of the performance (not impossible, but tricky... some parameters in the algorithm could be altered with knobs during performance.)
2.2 Retune the keys as we go, which is just algebra.
I'm sitting down with pen and paper now playing with stuff. I can't find enough information on the harmonic series from the music end of things (the first 30 terms are not enough) so I'm going to calculate it all from scratch. There's wonderful shapes in here.
12 Feb 2010 16:27:00
Temperament
ou may or may not be aware that the system of tuning used in virtually all the music you've ever heard is a fudge. It's a pretty good fudge, made for good reasons, but it's not right. The trouble is that none of the smaller intervals fit exactly into an octave. All of the notes we use are out by a few hundredths of a semitone. The worst note is out by 11.73 cents compared to its harmonic resonance. I can easily hear when something is out of tune by about 5 cents, so it is a big deal. Almost all the music that has ever been recorded is out of tune.
These different tuning systems are called temperaments. We commonly use equal temperament, in which octaves are divided into 12 equally sized semitones. Other temperaments that have been used historically include meantone temperament and well temperament. What I'm interested in exploring are different varieties of just temperament, or "Pythagorean" tuning.
If you play a pure tone, it naturally resonates at an a number of different frequencies, with a number of overtones. (Actually the number of overtones is infinite, but as you get further away from the main tone they get quieter until they're insignificant.) There's no such thing as a tone without overtones; it's impossible, that's just the nature of waves.
The main frequency you hear is the note itself. The most prominent overtone is the octave of the main note. Equal temperament has been defined so as to preserve this ratio so the octave is the only interval which is not out of tune. The next overtone is the perfect fifth. It turns out that this is 2 cents different to 7/12 of an octave (the fifth is seven semitones above the root). 2 cents isn't a problem to my ears, but already we've compromised. Then we get a perfect fourth, which, again, is 2 cents out but flat instead of sharp. A fourth and a fifth together make an octave so we're now two octaves above the root note.
This is where medieval organum music stops. For the medieval church, anything beyond this point was dissonance. If you want to make something sound hauntingly ancient, stick to these three intervals. It's not a bad plan in terms of tuning either because you don't have to make any significant compromises.
Then comes the major third, 8 cents out of tune. 8 cents! That's easily perceptible to the human ear. Each of the intervals comes in turn, defined by the mathematics of waves. (Two curiosities are worth commenting on: the minor third is actually more in tune than the major third, which is one reason minor keys sound lusher than major keys. Secondly, the 11th harmonic, often fudged as an augmented fifth, is 49 cents out of tune. That's a quarter tone out. This is a note which does not exist in western music yet has a closer harmonic relationship to the fundamental than many notes which western music does use. It's a strong argument for Arabic style 24-tone systems.)
The trouble comes when you try to make an instrument that can play in any key. An instrument which only plays in, say, D, can be tuned in just temperament without any problems. But if you then try to play it in Bb or many other keys, it won't work. Each of the notes is tuned slightly differently until you get to Bb. If you go up from D to Bb, you get the diminished fifth, 12 cents sharper than in equal temperament. But if you go down from D to Bb, you get the augmented fourth, 12 cents flatter than in equal temperment. This makes a difference of 23 cents between the two notes. In traditional Pythagorean tuning, they used the mean of the two values, the interval was called "diabolus in musica" and it was forbidden to use it. We can do better than that by employing an extra note, and playing one or the other according to context.
With computer equipment, it should be trivial to change between tunings depending on the key or the chord being played. With electronic music, there is no need to maintain the ugly compromise of equal temperament any longer. Down with ET! We want music that's in tune. Maybe I'll make a T-Shirt promoting Just Temperament.
(part 2 to follow)
These different tuning systems are called temperaments. We commonly use equal temperament, in which octaves are divided into 12 equally sized semitones. Other temperaments that have been used historically include meantone temperament and well temperament. What I'm interested in exploring are different varieties of just temperament, or "Pythagorean" tuning.
If you play a pure tone, it naturally resonates at an a number of different frequencies, with a number of overtones. (Actually the number of overtones is infinite, but as you get further away from the main tone they get quieter until they're insignificant.) There's no such thing as a tone without overtones; it's impossible, that's just the nature of waves.
The main frequency you hear is the note itself. The most prominent overtone is the octave of the main note. Equal temperament has been defined so as to preserve this ratio so the octave is the only interval which is not out of tune. The next overtone is the perfect fifth. It turns out that this is 2 cents different to 7/12 of an octave (the fifth is seven semitones above the root). 2 cents isn't a problem to my ears, but already we've compromised. Then we get a perfect fourth, which, again, is 2 cents out but flat instead of sharp. A fourth and a fifth together make an octave so we're now two octaves above the root note.
This is where medieval organum music stops. For the medieval church, anything beyond this point was dissonance. If you want to make something sound hauntingly ancient, stick to these three intervals. It's not a bad plan in terms of tuning either because you don't have to make any significant compromises.
Then comes the major third, 8 cents out of tune. 8 cents! That's easily perceptible to the human ear. Each of the intervals comes in turn, defined by the mathematics of waves. (Two curiosities are worth commenting on: the minor third is actually more in tune than the major third, which is one reason minor keys sound lusher than major keys. Secondly, the 11th harmonic, often fudged as an augmented fifth, is 49 cents out of tune. That's a quarter tone out. This is a note which does not exist in western music yet has a closer harmonic relationship to the fundamental than many notes which western music does use. It's a strong argument for Arabic style 24-tone systems.)
The trouble comes when you try to make an instrument that can play in any key. An instrument which only plays in, say, D, can be tuned in just temperament without any problems. But if you then try to play it in Bb or many other keys, it won't work. Each of the notes is tuned slightly differently until you get to Bb. If you go up from D to Bb, you get the diminished fifth, 12 cents sharper than in equal temperament. But if you go down from D to Bb, you get the augmented fourth, 12 cents flatter than in equal temperment. This makes a difference of 23 cents between the two notes. In traditional Pythagorean tuning, they used the mean of the two values, the interval was called "diabolus in musica" and it was forbidden to use it. We can do better than that by employing an extra note, and playing one or the other according to context.
With computer equipment, it should be trivial to change between tunings depending on the key or the chord being played. With electronic music, there is no need to maintain the ugly compromise of equal temperament any longer. Down with ET! We want music that's in tune. Maybe I'll make a T-Shirt promoting Just Temperament.
(part 2 to follow)
Monday 22 March 2010
Rotationally Symmetrical Scales
I think yesterday's post was very bad in terms of the artistry of a blog post. What was it about?
I've got a theory that there are three musical scales which are epitomes of emotional resonance, and that the majority of music we create and consume relies on bastard approximations of these three scales.
There's a number of assumptions you have to make to get to this position. I'll list them:
1. A note with a frequency twice or half that of another note is somehow equivalent (the octave).
2. The set of possible notes is the set of notes produced when you extrapolate the circle of fifths through the octaves (this gives twelve semitones in an octave).
3. Divisions of three semitones (giant steps) are ugly (we must therefore pick six or more notes from the 12 semitones to use).
4. Wholetone scales with six notes in carry little emotional value except as a contrast to another scale.
5. The fewer notes in a scale, the greater the emotional affect (7 is therefore optimum).
This much is common to all western music, but it's good to state your assumptions because then you know where you are. There are flutes from the palaeolithic (the time of cave paintings) whose design debatably agrees with all of these points. Human beings have always used these ideas, and you use them every time you listen to music. But they're not just a product of emotions and evolution. If they're not objectively true on the level of proof and logic then they're at least sensible mathematical choices.
It happens that there are three meaningful mathematically possible combinations of two semitone and five wholetone intervals, each with seven flavours depending upon which note you start on (these are not trivial for reasons of temperament and instrumentation), each in turn with seven modes depending on which interval you begin at. There are one hundred and forty seven possible diatonic scales without giant steps.
My proposition is this: scales which are rotationally symmetrical about the octave are the purest. A quick glance at a piano or a fretboard will show that to maintain rotational symmetry and the rule of no giant steps, the fifth and the fourth necessarily have to be in the scale. That's a good start as these are the most commonly used intervals in all music where their absence is not a feature. The diminished second is not possible because to preserve the symmetry you would need to have the fourth, the tritone and the fifth, which is more note selections than there are available. So we must have the ordinary second, and then you have to have the sixth to maintain the symmetry. This leaves only the question of the third and the seventh. There are thus two rotationally symmetrical scales available without giant steps.
The first of these scales is the normal everyday major scale (called the "ionian" scale when you're talking about modes). This makes sense, as western (and many non-western) musical theories assume this as the basis for music without looking any deeper into it. If it's that easy to accept it as true, it may well be so.
The second scale is the dorian. It is my belief that western theory dropped the ball when it adopted the aeolian as its minor mode. You can see that something's not right when you look at all that classical fudging over harmonic and melodic minors (classical theorists straight-facedly insist that you should play different notes going up compared to coming down a minor scale, even in equal temperament). When you need to stick that many plasters on a system, it's broken. The only thing that has sustained the supremacy of the aeolian mode over the years has been too much education. Wherever musicians have been left to figure it out on their own, they've usually reverted to the dorian as their minor mode, often without realising they've done it. As examples, I present Scottish, Irish and English folk musics, bluegrass, the 60s, the 70s... even 90s dance music. Wherever musicians lack a man in a wig to tell them not to, they have consistently flattened the seventh.
Interestingly, if you accept the presence of giant steps in the scale there is only one additional scale that preserves the symmetry. I do not have a name for this scale but it belongs to the family of "locrian majors" (which are neither locrian nor major). The scale is giantstep-semitone-semitone-tone-giantstep-semitone-semitone. This scale has a certain bluesy feel about it, as though the quarter-tone african scale that was the father of the blues is somehow related to this. It certainly looks that way on a fretboard, but I don't know enough to comment further.
However, there is some theoretical dissonance with this third scale. We needed to assume that giant steps were bad in order to chose to form seven-note scales. How can we then turn around and say that giant steps are OK at a later stage? If giant steps are OK, why not form five-note scales? In practice, that's exactly what those old bluesmen did.
One last comment: insisting that the diminished second be rotationally reflected in the tritone is a little pedantic as this reflection falls outside the home octave. If you ignore this consideration you end up with the lydian mode. So you could make an argument that there is something pure and special about the lydian mode, too. This does seem like a viable idea; there is a unique quality which is neither major nor minor that is probably strongest in the lydian. I wish some 50s jazz leader had looked into it.
If you accept both compromises (giant steps and diminished second) there are at least another two scales with rotational symmetry, one of which is the "double harmonic" scale from the Pulp Fiction theme. The other has two semitone intervals at the beginning and sounds like it's derived from the chromatic (that is, is sounds pretty characterless).
Coming soon: experiments in thirds-propagated temperaments.
I've got a theory that there are three musical scales which are epitomes of emotional resonance, and that the majority of music we create and consume relies on bastard approximations of these three scales.
There's a number of assumptions you have to make to get to this position. I'll list them:
1. A note with a frequency twice or half that of another note is somehow equivalent (the octave).
2. The set of possible notes is the set of notes produced when you extrapolate the circle of fifths through the octaves (this gives twelve semitones in an octave).
3. Divisions of three semitones (giant steps) are ugly (we must therefore pick six or more notes from the 12 semitones to use).
4. Wholetone scales with six notes in carry little emotional value except as a contrast to another scale.
5. The fewer notes in a scale, the greater the emotional affect (7 is therefore optimum).
This much is common to all western music, but it's good to state your assumptions because then you know where you are. There are flutes from the palaeolithic (the time of cave paintings) whose design debatably agrees with all of these points. Human beings have always used these ideas, and you use them every time you listen to music. But they're not just a product of emotions and evolution. If they're not objectively true on the level of proof and logic then they're at least sensible mathematical choices.
It happens that there are three meaningful mathematically possible combinations of two semitone and five wholetone intervals, each with seven flavours depending upon which note you start on (these are not trivial for reasons of temperament and instrumentation), each in turn with seven modes depending on which interval you begin at. There are one hundred and forty seven possible diatonic scales without giant steps.
My proposition is this: scales which are rotationally symmetrical about the octave are the purest. A quick glance at a piano or a fretboard will show that to maintain rotational symmetry and the rule of no giant steps, the fifth and the fourth necessarily have to be in the scale. That's a good start as these are the most commonly used intervals in all music where their absence is not a feature. The diminished second is not possible because to preserve the symmetry you would need to have the fourth, the tritone and the fifth, which is more note selections than there are available. So we must have the ordinary second, and then you have to have the sixth to maintain the symmetry. This leaves only the question of the third and the seventh. There are thus two rotationally symmetrical scales available without giant steps.
The first of these scales is the normal everyday major scale (called the "ionian" scale when you're talking about modes). This makes sense, as western (and many non-western) musical theories assume this as the basis for music without looking any deeper into it. If it's that easy to accept it as true, it may well be so.
The second scale is the dorian. It is my belief that western theory dropped the ball when it adopted the aeolian as its minor mode. You can see that something's not right when you look at all that classical fudging over harmonic and melodic minors (classical theorists straight-facedly insist that you should play different notes going up compared to coming down a minor scale, even in equal temperament). When you need to stick that many plasters on a system, it's broken. The only thing that has sustained the supremacy of the aeolian mode over the years has been too much education. Wherever musicians have been left to figure it out on their own, they've usually reverted to the dorian as their minor mode, often without realising they've done it. As examples, I present Scottish, Irish and English folk musics, bluegrass, the 60s, the 70s... even 90s dance music. Wherever musicians lack a man in a wig to tell them not to, they have consistently flattened the seventh.
Interestingly, if you accept the presence of giant steps in the scale there is only one additional scale that preserves the symmetry. I do not have a name for this scale but it belongs to the family of "locrian majors" (which are neither locrian nor major). The scale is giantstep-semitone-semitone-tone-giantstep-semitone-semitone. This scale has a certain bluesy feel about it, as though the quarter-tone african scale that was the father of the blues is somehow related to this. It certainly looks that way on a fretboard, but I don't know enough to comment further.
However, there is some theoretical dissonance with this third scale. We needed to assume that giant steps were bad in order to chose to form seven-note scales. How can we then turn around and say that giant steps are OK at a later stage? If giant steps are OK, why not form five-note scales? In practice, that's exactly what those old bluesmen did.
One last comment: insisting that the diminished second be rotationally reflected in the tritone is a little pedantic as this reflection falls outside the home octave. If you ignore this consideration you end up with the lydian mode. So you could make an argument that there is something pure and special about the lydian mode, too. This does seem like a viable idea; there is a unique quality which is neither major nor minor that is probably strongest in the lydian. I wish some 50s jazz leader had looked into it.
If you accept both compromises (giant steps and diminished second) there are at least another two scales with rotational symmetry, one of which is the "double harmonic" scale from the Pulp Fiction theme. The other has two semitone intervals at the beginning and sounds like it's derived from the chromatic (that is, is sounds pretty characterless).
Coming soon: experiments in thirds-propagated temperaments.
Musical Ramblings
1. I found myself in Crookes on my bicycle after running some errands (I live in Sheffield, England). As I'd already gained so much altitude I thought I may as well go out into the Peak District and use the moors like a ring road and come home that way. There's a number of questions I need to answer about DiftW narrative events taking place on moors so I hoped that I'd get some answers out in the countryside.
Instead, I found myself meditating on two phrases:
"[Music is] the texturization of the deliquescence of time" (Jason Martineau, The Elements of Music) and
"Music is Rotted One Note" (the title of a Squarepusher album; I assume this is an allusion to the Miles Davis-style modal approach used on the album - for example, you could say that the dorian or locrian modes were rotted one note from the major/ionian).
I recently did a show with my Ibly Dy project. I have always described Ibly Dy as "apocalyptic" but I had begun to think about what that meant in physical rather than emotional and philosophical terms. After the end of time; what would music sound like? How can you have a music without time? I sat down and tried to make the sounds that I thought would be an appropriate approximation for this idea, and then I realised that I was trying to make the sound of static on a drum kit. Which is highly appropriate as when we listen to sounds from before the beginning of time (for example radio signals from the big bang) we hear static. I tried to communicate all of this on stage. The performance met with limited success.
It is clear to me that music and time are related but are different orders of the same system. How might we relate these two orders using calculus? I think this is a reasonable approach to take as a thought experiment, although I have a list of reservations it would probably never lead to anything mathematically useable. I've sat down with a pen and some equations and things feel promising but I can't define my terms successfully. This needs more work, but not today.
2. Last night I did some work with my classic rock band Firegarden. We rehearsed some things that needed tightening up performance-wise. Then we talked about the songs. I think it's reasonable to say that the three or four songs we wrote immediately after the last line-up change aren't up to scratch. We were too keen to move on quickly and no one really knew how the balance, the ebb and flow of things, would work. Recently Jake has bought some great tunes to the band, and we've worked them up very successfully. But I am worried that they could have been more successful had I felt more secure and hadn't had to defend my opinion at every turn. I think one of my greatest skills (whether musically or with the DIftW or in any other context) is as an arranger. I've got a very quick and accurate taste when it comes to the detail of constructing pieces, and 90% of the time when decisions go against my gut reaction there comes a consensus a few months down the line that I was actually right (the other 10% of the time I make mistakes).
I am also worried that it is only Jake who brings material to the table. Back in the day every member contributed raw ingredients (mine were largely and correctly ignored). When you have two or three writers you get the magical situation of competition which allows only the best to get through. Without that you're left scrabbling around using every scrap of idea the sole writer can come up with. I think that applies especially in a band situation where there's a few guys standing around in a room expectantly waiting for a riff, but it probably isn't a problem with writing a novel because you don't have that social and performance-emotional pressure.
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