Harmonic Quarter-Tone Project

Subject: Harmonic Quarter-Tone Project
From: M. Ted (inactive)
Date: 13 Mar 00 - 02:22 PM

In another thread, there was a discussion of the harmonic overtone series, as it presented it self in natural brass instruments--

This discussion though not nearly as important to folk music as, say gun-control, drew some interest--

I mentioned that, in my musical studies (which some claim are just an excuse that allows me to call myself a composer without actually creating much of anything) I recently spent a lot of time studying Hindemith's calculations that allow for the derivation of the chromatic scale from the harmonic overtone series--

I also mentioned my observation that, with each succeeding octave, the number of tones that present themselves increases so that by the time you reach the fifth octave, there are enough notes to make a diatonic scale--

The octave that follows would include enough notes to make a chromatic scale (though not the one that Hindemith derived), and the one that follows that would have enough notes to create a quarter-tone type scale--

My interest is in creating a musical instrument that could be tuned to these pitches, and played--

Bert suggested that I use a Santur--which is a Catspaw like instrument-and a Santur-like instrument is a definite possibility-Harry Partch actually used an instrument based on the Kanu'un(which is a similar instument, but plucked instead of hammered) for his 39 pitch scale--

The only thing is that the Santur is intended for playing melodic music, the bridge really functions as a way of altering the pitch of any given note--since my intent is to work with harmonies, I need to have all the pitches available all the time--

It would be very possible to just tune a Santur or Kanu'un to each of the fractional pitches that I derive--and it would work for demonstrating intervals, but I am not sure how easy to play this instrument would be--

On an extremely wonkish note--Neither Persian, Arabic, nor Turkish music use actual "Quarter-tones"-- The intervals between pitches are allocated in a completely different way than in what we call "Western" music--and there are additional steps that are possible--some of which are considerably smaller than the smallest half step that we use--

I have been told that in Turkish classical music, there are about 180 different scales that are used--The scales typically have seven steps, and each scale has a name, which describes the pattern of intervals in the scale, the lengths of the intervals themselves are tradtionally set by the players themselves--

At any rate, the pitches are not intended for use in any sort of harmony--and, at least to my knowledge, the quarter tones scales used by twentieth century composers, such as Barth, Haba, and Carillo, were derived by using pitches that were in between the diatonic pitches, rather than pitches that came from an over arching harmonic scheme--

The resulting harmonies have been described as contrived--and the potentials of quarter tone music has been dismissed based on the idea that the harmonies are artificial--

Well, as far as I am concerned, since there is a place in the harmonic series where a "real" quarter tone like dvision of the octave appears, and since melodic music has been played (though in different musical traditions than our Western one) that uses smaller intervals than the half step--it is not so far off to suppose that it there might be some good harmonies out there, too--

So, if you are with me through this, I have an ongoing interest in creating some sort of instrument that can be used to play these notes, and also interested in creating a set of rules for using harmonies and melodies within this peculiar set of scales--

Any comments, questions, or other contributions would be welcomed--

Subject: RE: Harmonic Quarter-Tone Project
From: Amos
Date: 13 Mar 00 - 02:48 PM

Thanks, MTed, for a most interesting and educational post. I imagine you could generate the scales you are interested in digitally, for th epurposes of exploring harmonies and governing rules.

A

Subject: RE: Harmonic Quarter-Tone Project
From: M. Ted (inactive)
Date: 13 Mar 00 - 03:05 PM

I will probably do that at a fairly early stage--but the sounds aren't really natural sounds--so an instrument of some sort is necessary--

Subject: RE: Harmonic Quarter-Tone Project
From: Barbara
Date: 13 Mar 00 - 04:00 PM

The contemporary composer Alan Hohvannes (sp?) created a vocal piece to be sung by soprano, where her overtones supply the missing note. I can't recall the name of the piece right now (it might be "Missing Note" or a play on words on that subject). The tibetan monks routinely use overtones when they chant. I suppose you don't want the human voice, but rather something more reliable/predictable, right M. Ted?
Blessings
Barbara

Subject: RE: Harmonic Quarter-Tone Project
From: Homeless
Date: 13 Mar 00 - 04:11 PM

MTed - to be quite honest most of your post went right over my head (I've never heard of any of the instruments you named, but am somewhat familiar with the physics of music) but tripped a couple of thought that expressed themselves to me as questions. Maybe my questions will help you focus your thinking.
Since you've used the term "Western" a couple different times, I assume you are referring to the 12 tone normalized scale instead of the natural, simple-fractions scale?
If you are actually refering to the latter, could you not just add a few more complex fractions and create your rules based on the math, and test them aurally?
Do you intend for your scale to be based on the overtones of only 1 base note?
Are you looking at only a stringed instrument of some kind, or are you leaving other options open (woodwind, percussive, etc.)?
Have you considered retuning an autoharp, and cutting new chord bars for it?

Subject: RE: Harmonic Quarter-Tone Project
From: GUEST,Bruce O.
Date: 13 Mar 00 - 04:36 PM

180 is a lot of scales. With just 12 pitches per octave there are 2047 possible scales or modes. 148 of such for English, Scots, and Irish tunes are noted in file CODEMATR.TXT on my website. I show in file CODETEXT.TXT how to unambibuously number them so one doesn't need names for any of them in order to identify them. www.erols.com/olsonw

Subject: RE: Harmonic Quarter-Tone Project
From: Charlie Baum
Date: 13 Mar 00 - 05:52 PM

If you wanted to create a music instrument, you might try to find a zither, which is functionally a hammer dulcimer without any bridges. You could then tune it as you wished.

If you're looking for another instrument that's capable of hitting those unusual notes, would a fretless banjo work? The problem is that you would have to know exactly what pitch you wanted to make sure you hit it preciselyt, but you could (at least theoretically) play your composition on the fretless banjo (or any other fretless stringed instrument, including the strings of the violin family.)

For precision, though, I guess you'd want a computer program capable of generating mathematically-derived pitches and feeding those pitches to some sort of MIDI-output device. Once the formula for generating the scale had been determined, perhaps an electronic MIDI-compatible keyboard could be used to play those compositions. using those scales.

--Charlie Baum

Subject: RE: Harmonic Quarter-Tone Project
From: M. Ted (inactive)
Date: 13 Mar 00 - 07:35 PM

The trouble with having a discussion on a musical topic is that people tend to ask questions that you must think about before you answer them--

The Kanu'un is a zither, which is basically an autoharp without the auto----The santur is the parent instrument of the hammered dulcimer, with the addition of a bar that can be used to alter the pitch by a fractional of a tone-Charlie, as for the banjo, ther are a number of middle eastern instruments which are basically banjoes, and the fretless part isn't even necessary, because the frets are moveable wrapped wire--

When I talk about western scales, I mean the even tempered one--which is based on harmonic intervals--as opposed to the arabic or middle eastern sort of scale, which are derived by a whole different principle--

As to the natural fractions scale, I am not sure what you mean--if you are talking about creating some sort of even division of the string through a mathematical formula, this would miss my point, which is to have a scale made up of pitches that had a harmonic relationship to a fundamental tone--and yes, the calculation of a scale based on harmonics does use one fundamental pitch as the basis for calcuations--

And Bruce, I am told that the each of the 180 or so scales(Makams) has it's own name and adherents--there are also attendent rhythm patterns that go with each makam-- Needless to say, most players only work with a few makams, all though I remember an Egyptian friend who complained about a Druse Oud player that he worked with, "One hundred and sixty three makams, and he uses the same one for every solo--"" (apparently there are only 163 used in Arabic music)

Subject: RE: Harmonic Quarter-Tone Project
From: Homeless
Date: 13 Mar 00 - 08:13 PM

Go back a couple hundred years and you'll find that the western scale was a bit different. Rather than being based on evenly spaced semitones, it was based on a given tone (C=256 {as opposed to A=440 nowadays}) and the other pitches were based on simple fractions (1/3, 1/4, 2/3, 3/5) of the pitch of the base. If you changed keys, all the other tones would change - then being based on the new root, for example D=287. And an F# was not the same as a Gb - a few hertz difference. That is what I meant by a simple fraction scale.

I want to make sure I've got everything straight. You want to take a note, for discussion's sake A = 110. You then want to take the first harmonic - 220, A'. Then take the second harmonic - 330, E'. The third harmonic is 440 - A''. So at the second octave above your root, you have 3 tones within the scale (A', E', A'')? I'm assuming when you say scale you mean octave? Then continue adding harmonics until you can get 48(?) distinct tones within a one octave span. Does this accurately describe what you're wanting?

If what I understand is correct, then it would be possible to take something like a 10 foot long steel beam and string a wire down the length. Somehow add frets every 3 inches to give your 40 different harmonics of the original string's tone. A tuned resonator (or pickup) to amplify the sound. You could pluck it, a la harmonics on a guitar, or bow it (fretted) for a pitch at the tone of that harmonic. Add a few strings, possibly with bridges set at various places in order to keep all your harmonics within reach when playing. Possible, but feasible? Definitely not very transportable.

I just proofread this before sending it and realized that you could take a guitar, banjo, etc., pull off the existing neck (or fingerboard) and just make a new one putting frets only where the harmonics would be, e.g. 1/2 the distance (d) from nut to bridge, 1/3d, 1/4d, 1/5d. With 2 strings tuned in octaves you could get the full range, but the higher string would be missing every other tone.

Is this clear as mud? Does it give you any food for thought? Do you want me to go away and pretend I never read the question?

Subject: RE: Harmonic Quarter-Tone Project
From: catspaw49
Date: 13 Mar 00 - 08:38 PM

Back to the HD/Santir idea........If you had movable bridges and a two course set up you could tune each string in the harmony of the Q-tone and also adjust the bridges in any manner to produce the tone desired and the other side at cetain ptches would be the harmonic.

Let me think about that again...Out in the shop I have a trial horse dulcimer I used for box design and also for rail heights,,,,,,,,Somewhere I have a can of pagoda bridges too. I might try something here and then get some additional input.

Geeziz Ted...Where'd you come up with this thought??? Damn.....

Also.....Any of this relate to NA Indian flutes which were made to "sing their own song?" Many had no relationship familiar to western tonal patterns.

Spaw

Subject: RE: Harmonic Quarter-Tone Project
From: Homeless
Date: 13 Mar 00 - 08:48 PM

Spaw - from my readings on said flutes, basically they bored a hollow tube, and put the holes where ever their fingers fell naturally along the length. So no two flutes had the same tones. An unavoidable effect of that was that each person had to "play the music that was in his heart" - no playing someone else's song, since no two flutes could be counted on to play the same tones.

Subject: RE: Harmonic Quarter-Tone Project
From: GUEST,Bruce O.
Date: 13 Mar 00 - 08:50 PM

I take a lot of notes to make a true just intonation scale (using the 4:5:6 ratio for major chords and 10:12:15 for minor ones). See the 'Extended Just Intonation' file on my website. I now have English, Scots and Irish tunes in 145 modes tunes coded in file COMBCOD2.TXT on my website. You can display them all with their scoring bases using option 9 of the program there. Some tunes are scored in as many as 3 different scoring modes. Scoring modes and tune modes are only the same for tunes in the 7 'Greek' modes, and even then one finds some strange scorings, e.g, lydian as ionian with sharp 4th. "John Hays' Bonnie Lassie" in 'Orpheus Caledonius', 1733, is a dorian scored in a mixolydian basis mode (except the flute score in the appendix is straight dorian).

Subject: RE: Harmonic Quarter-Tone Project
From: M. Ted (inactive)
Date: 14 Mar 00 - 01:27 AM

If it is any help, keep in mind that it isn't actually necessary to produce the notes as harmonics--

Homeless, after reading your post a couple times, I think see where you're going--you don't need frets where the harmonics sound, they will sound when you touch the string--it is possible to bow a string so the the harmonics sound in this way--(it takes a while to learn to produce clean sounds, though)

'Spaw, I came up with this idea after about the twentieth time though Hindemith's explanation of how he derived all of the 12 pitches from either harmonic overtones of C 64 or from harmonics of the first Six harmonics--

When he began to explain the problems with using the harmonics above the sixth, I began to think about how peculiar it was that this "scale" just sort of appeared in the fifth octave of the harmonic series, even though the pitches were off from what they needed to be, and that with each succeeding octave, there were enough notes for a chromatic scale and a quarter-tone like scale--

It then occurred to me that given that the pitches have a harmonic relationship., they ought to sound like something--the question is, what?

Before we get to far into this (although it may be too late, already) I have to say that as peculiarly appealing as this may seem, I have a number of other, much closer to mainstream, musical projects that are higher priority--

That means that, at least for now, this project is strictly theoretical--

The only thing is, especially at 1:30 in the morning, the idea that "There is something out there" is strangely appealing--

Bruce--I have fooled around a bit with the modes and intonations that you've got, but my main synthesizer isn't set up yet, and the MIDI voices in my computer aren't very satisfactory--

Subject: RE: Harmonic Quarter-Tone Project
From: toadfrog
Date: 26 May 02 - 08:57 PM

M. Ted:
Just a bit curious; though I'm not learned about musical theory; I would think that the idea of searching new scales for new harmonies would be the antithesis of an interest in folk music.

Subject: RE: Harmonic Quarter-Tone Project
From: JohnInKansas
Date: 27 May 02 - 06:59 AM

M Ted (et. al.)

The reference to Hindemuth makes your motive extremely suspect. As a former member of a collegiate Concert Band that played a number of his compositions with some frequency, I can attest that his works will warp your brain.

The calculations that Hindemeth made were not original to him. They were simply a "sorting and re-thinking" of attempts to make sense of scales and harmonies, using the same methods that have been "established practice" since at least classical Greek and Roman times - in western music, at least.

A string, or an air column, can be made to vibrate "in parts" to produce higher pitches. Vibration in 2 parts gives an pitch twice the frequency of the "full length," and we have come to call that an octave. Vibration in 3 parts gives 3 times the frequency, and we call that a "fourth," simply because - traditionally - it's the fourth note of the scale.

If we go to high enough "parts," we get more closely spaced "notes," and we can always "divide by 2, 4, 8, etc to move things back down by octaves.

Simple objects, like strings and air columns can only vibrate naturally in "integer parts." You can split them into 2, 3, ...100 parts and get a vibration, but they "misbehave" if you try to split them so that some parts are unlike other parts.

The whole of "classical harmonic" scale(s) can be generated by taking any sequence of integers and dividing each by whatever "power of two" gives a result between 1 and 2. Conceptually, you are using a very long string - but only using a small part of it.

Interpreting the fifth octave to which you refer as having a frequency of 2E5 (2 with exponent 5, or 2 to the fifth power) or 2x2x2x2x2 = 32, so that the "tonic" note has a frequency 32 times that of your "conceptual long string."

The sequence 32/32, 33/32, 34/32, ... 64/32 would make a perfectly "natural" harmonic (super-chromatic?) scale of one octave, but would sound rather strange to most of us (but maybe not to Hindemuth?).

The first few notes of the scale, with the division completed, would be at:
32/32 = 1.000000, 33/32 = 1.031250, 34/32 = 1.062500, 35/32 = 1.093750, 36/32 = 1.125000
Compared to the first few notes of the equal tempered chromatic scale at:
1.000000, 1,059463, 1.122462, 1.189207,...

For the equal tempered C scale, the third note, D, at 1.122462 (times 256 Hz) comes fairly close to the "Hindemuth" fifth note at 1.125000 - but not much of anything really matches up. Hindemuth wrote some pieces for conventional instruments that applied approximations to "harmonies" based on "high octave entervals," but it just sounds "out of tune" to most people.

The classical western instrument that comes closest to being able to play in "Hindemuth style" is probably the French horn. The tube is very long, so that the "fundamental" of the instrument is lower than for a tuba or sousaphone; and most of the commonly used range is in the fourth(?) through seventh(?) octave harmonics. That's why the French horn player has to keep his fist in the bell to "push" the notes into tune with the other instruments - by physically changing the effective length of the tube.

John

Subject: RE: Harmonic Quarter-Tone Project
From: Pied Piper
Date: 27 May 02 - 09:00 AM

Most sound modules can be given different intonation by sending System exclusive Midi messages. This normally only give you access to the notes of the 12-note chromatic scale, maybe you could use more than one module to generate the in-between notes. I know you want a real instrument eventually but I think electronics would give you the chance to experiment with the resources more easily. All the best PP.

Subject: RE: Harmonic Quarter-Tone Project
From: English Jon
Date: 27 May 02 - 09:20 AM

Ever heard of Charles Lucy? Worth looking up.

EJ