"too hot to hoot" is palindromic.
"too[b]h ot[/b] to[b]h oot[/b]"
"[b]too h[/b]ot [b]to h[/b]oot"
"too[b]h ot[/b] to[b]h oot[/b]"
"[b]too h[/b]ot [b]to h[/b]oot"
Gordon's Progress
Posted: 7/19/2006 10:40:03 AM
And, bringing it back to a musical theme, from the totally brilliant book Gödel, Escher, Bach: An Eternal Golden Braid by Douglas R Hofstadter (which I read when I was fifteen):
[b]Crab Canon[/b]. http://www.barryland.com/canon.html
First link on the page is to the score for Bach's musically palindromic Crab Canon.
[b]Crab Canon[/b]. http://www.barryland.com/canon.html
First link on the page is to the score for Bach's musically palindromic Crab Canon.
Posted: 7/25/2006 5:52:06 PM
I'm still trying to get my head around harmony as it applies to the theremin. The basic principles are very simple:
1. We hear frequencies exponentially - doubling any frequency increases the pitch perceptually by a constant amount.
2. The simpler the ratio between two frequencies, the more harmonious they sound one after the other.
Two notes that are the same frequency - in the ratio 1:1 (unison) - sound perfect together. Two notes in the ratio 2:1 (octave) sound very good. Two notes in the ratio 3:2 (perfect fifth) sound good. And so on...
And then there's the catch:
Doing arithmetic with ratios tends to make them less simple. For example, say the first note is 440Hz. Going up a perfect fifth (half as much again) gives the next note as 660Hz. Going up a perfect fifth from that note leads to 990Hz. Another similar step leads to 1485Hz. Stepping down two octaves (half and half again) leads to 371.25Hz. To reach the first note from here in a single step would require a ratio of 27:32 - not very simple, and consequently not very harmonious, even if every step on the way there was harmonious.
And this is a problem because of the physical constraints of most musical instruments:
1. You can't have an unlimited number of frequencies available - there are only so many keys you can squeeze into an octave on a piano keyboard, for instance, so you have to pick a set of keys that work together and stick with them because:
2. You can't go retuning your instrument in the middle of a song.
And this leads to the work-around:
People cannot hear the difference between very similar frequencies, so one can move the notes a tiny bit from simple ratios and get away with it. There are lots of different ways of jiggling the frequencies, the most successful of which is called 12 tone equal temperament - the familiar piano keyboard notes. This allows a whole variety of step sizes, all very close to simple ratios, and for the greater part it sounds just fine. It's good enough, but not actually perfect.
Did you spot the get-out clause? The theremin is a continuous tone instrument - the first constraint does not apply.
So how do the basic principles apply to the theremin? The desirable quality of linearity in the pitch response of the theremin means there is a one to one correspondence between the distance of a hand movement and the resultant change in pitch.
Before continuing, please open this image (http://www.charlton.demon.co.uk/octave.gif) on a new page or tab so it can be referred to whilst you read.
The diagram at the top represents, horizontally, a span of one octave in the pitch field of an idealised (i.e. perfectly linear) theremin. For the sake of argument the left hand side could be 440Hz and the right 880Hz, but it could equally well be from 673Hz to 1346Hz. (In other words, like the markings on a binary slide rule.) The white numbering along the top - x1 x1.5 x2 indicates how much you need to multiply the lowest frequency by to reach that position.
The short green bars indicate the positions of the semitones of a piano keyboard (12-TET), from unison to octave (for instance from A0 to A1). You will see that there is constant increase in spacing from left to right. That is the secret of 12-TET's success.
The thin white bars indicate points where there is a simple ratio between the lowest frequency and the point marked. The blue fractions show what proportion of the lowest frequency would have to be added to the lowest frequency to reach that point. (Normally transitions between notes are described by ratio of their frequencies so the point marked 1/2 would be called 3/2. If you prefer it that way, just add 1 to each fraction. I prefer this way as it highlights the fact that this is a Farey Sequence.)
The height of each line is related to the simplicity of the fraction, and corresponds to the harmoniousness of the note in r
1. We hear frequencies exponentially - doubling any frequency increases the pitch perceptually by a constant amount.
2. The simpler the ratio between two frequencies, the more harmonious they sound one after the other.
Two notes that are the same frequency - in the ratio 1:1 (unison) - sound perfect together. Two notes in the ratio 2:1 (octave) sound very good. Two notes in the ratio 3:2 (perfect fifth) sound good. And so on...
And then there's the catch:
Doing arithmetic with ratios tends to make them less simple. For example, say the first note is 440Hz. Going up a perfect fifth (half as much again) gives the next note as 660Hz. Going up a perfect fifth from that note leads to 990Hz. Another similar step leads to 1485Hz. Stepping down two octaves (half and half again) leads to 371.25Hz. To reach the first note from here in a single step would require a ratio of 27:32 - not very simple, and consequently not very harmonious, even if every step on the way there was harmonious.
And this is a problem because of the physical constraints of most musical instruments:
1. You can't have an unlimited number of frequencies available - there are only so many keys you can squeeze into an octave on a piano keyboard, for instance, so you have to pick a set of keys that work together and stick with them because:
2. You can't go retuning your instrument in the middle of a song.
And this leads to the work-around:
People cannot hear the difference between very similar frequencies, so one can move the notes a tiny bit from simple ratios and get away with it. There are lots of different ways of jiggling the frequencies, the most successful of which is called 12 tone equal temperament - the familiar piano keyboard notes. This allows a whole variety of step sizes, all very close to simple ratios, and for the greater part it sounds just fine. It's good enough, but not actually perfect.
Did you spot the get-out clause? The theremin is a continuous tone instrument - the first constraint does not apply.
So how do the basic principles apply to the theremin? The desirable quality of linearity in the pitch response of the theremin means there is a one to one correspondence between the distance of a hand movement and the resultant change in pitch.
Before continuing, please open this image (http://www.charlton.demon.co.uk/octave.gif) on a new page or tab so it can be referred to whilst you read.
The diagram at the top represents, horizontally, a span of one octave in the pitch field of an idealised (i.e. perfectly linear) theremin. For the sake of argument the left hand side could be 440Hz and the right 880Hz, but it could equally well be from 673Hz to 1346Hz. (In other words, like the markings on a binary slide rule.) The white numbering along the top - x1 x1.5 x2 indicates how much you need to multiply the lowest frequency by to reach that position.
The short green bars indicate the positions of the semitones of a piano keyboard (12-TET), from unison to octave (for instance from A0 to A1). You will see that there is constant increase in spacing from left to right. That is the secret of 12-TET's success.
The thin white bars indicate points where there is a simple ratio between the lowest frequency and the point marked. The blue fractions show what proportion of the lowest frequency would have to be added to the lowest frequency to reach that point. (Normally transitions between notes are described by ratio of their frequencies so the point marked 1/2 would be called 3/2. If you prefer it that way, just add 1 to each fraction. I prefer this way as it highlights the fact that this is a Farey Sequence.)
The height of each line is related to the simplicity of the fraction, and corresponds to the harmoniousness of the note in r
Posted: 7/26/2006 4:47:22 AM
I lost you about half way through your spiel, but it all sounds incredibly interesting and absorbing. I have read a few tomes on the topic of the physics of music, and lots of it (especially the matter of tuning) I found very intriguing.
I listened to the mp3, and could readily detect that the later sequences were not equal temperament. The seventh, thirds and sixths sound completely off in the last octave, as do loads of the other notes. I must confess to being an equal-temperament sort of person.
It may be of interest to you to learn that the Venetian Music theorists in the early 16th century (around the time when our tone systems were being developed) had fifteen-note scale harpsichords with multiple levers which would facilitate quick transitions between various tunings.
The Italian Nicola Vicentino surpassed this further by inventing the archicembalo, a keyboard with 36 notes to the scale. Using this he was able to play musically 'correct' intervals in any key (because of course some scales work better than others with equal temperament).
Another thing that interests me is the Mediaeval pet-hate for both parallel fifths, thed closed fifth and of the third and sixth sounding with the tonic. It is obvious that without being brought up with equal temperament, such intervals sound 'wrong' unless in meantone.
I listened to the mp3, and could readily detect that the later sequences were not equal temperament. The seventh, thirds and sixths sound completely off in the last octave, as do loads of the other notes. I must confess to being an equal-temperament sort of person.
It may be of interest to you to learn that the Venetian Music theorists in the early 16th century (around the time when our tone systems were being developed) had fifteen-note scale harpsichords with multiple levers which would facilitate quick transitions between various tunings.
The Italian Nicola Vicentino surpassed this further by inventing the archicembalo, a keyboard with 36 notes to the scale. Using this he was able to play musically 'correct' intervals in any key (because of course some scales work better than others with equal temperament).
Another thing that interests me is the Mediaeval pet-hate for both parallel fifths, thed closed fifth and of the third and sixth sounding with the tonic. It is obvious that without being brought up with equal temperament, such intervals sound 'wrong' unless in meantone.
Posted: 7/26/2006 8:39:59 AM
Case in point:
The lovely curves on Mrs Diggy could be expressed mathematically. It would take only a few eauations to define them and graph them.
To do so would entail losing track of what is really important about her, though.....
She's a great cook!
The lovely curves on Mrs Diggy could be expressed mathematically. It would take only a few eauations to define them and graph them.
To do so would entail losing track of what is really important about her, though.....
She's a great cook!
Posted: 7/27/2006 1:00:40 PM
Charlie.
Ooh, so I've caught up with the 15th C Venetians. Neat.
Yeah, that note marked 3/4 on my diagram. The one furthest from any of the even temperament notes. Apparently it's a harmonic seventh.
Wikipedia had this to say about it: "The 7:4 interval (the harmonic seventh) has been a contentious issue throughout the history of music theory; it is 31 cents flatter than an equal-tempered minor seventh. Some assert the 7:4 is one of the blue notes used in jazz."
So as an even tempered guy (and rightly so if you are studying classical music!) do I guess correctly that you don't get on too comfortably with jazz?
Diggy, or should I say Lucky Dog, indeed I get that being a scientist won't make you a chef, but at the same time knowing a bit of kitchen science will certainly improve your cookery.
It's about having a mental image of what happens when I stick my hand in the pitch field.
Ooh, so I've caught up with the 15th C Venetians. Neat.
Yeah, that note marked 3/4 on my diagram. The one furthest from any of the even temperament notes. Apparently it's a harmonic seventh.
Wikipedia had this to say about it: "The 7:4 interval (the harmonic seventh) has been a contentious issue throughout the history of music theory; it is 31 cents flatter than an equal-tempered minor seventh. Some assert the 7:4 is one of the blue notes used in jazz."
So as an even tempered guy (and rightly so if you are studying classical music!) do I guess correctly that you don't get on too comfortably with jazz?
Diggy, or should I say Lucky Dog, indeed I get that being a scientist won't make you a chef, but at the same time knowing a bit of kitchen science will certainly improve your cookery.
It's about having a mental image of what happens when I stick my hand in the pitch field.
Posted: 7/27/2006 1:28:56 PM
I don't really like Jazz. I can bear cocktail piano and black 'hot' jazz, but I lose it when you you get random improvised swarms of notes that don't seem to make musical sense. The sort of jazz that has jazzaholics closing their eyes and bobbing back and forth as though they're having some sort of spiritual revalation usually leaves me feeling confused, or at best, uninspired.
It's a shame really. I'd love to love jazz.
It's a shame really. I'd love to love jazz.
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