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The musical connection
Triads, Harmonic Polyrhythms and the 3D Lissajous
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The Musical Connection

The original Lissajous patterns were based on musical tones

The original Lissajous patterns way back in the nineteenth century were generated by bouncing light off mirrors attached to tuning forks playing musically related pitches. You get simple Lissajous patterns when the tuning forks play pure harmonies - with the frequencies at small integer ratios to each other, e.g. 2 : 3 etc.

These musical intervals are used for music therapy.

The Lissajous 3D program arose out of that connection, and a suggestion by the composer Charles Lucy to try 3D patterns for chords of three or more notes.

It's quite a natural thought - if diads lead to 2D lissajous patterns - then triads need 3D Lissajous patterns to show them in full detail - and tetrads, 4D patterns and so on.

So I'm sure it has been thought of many times independently. A search shows that they have also been studied by mathematicians e.g. as "Lissajous knots".

The connection with musical intervals is quite a close one.

Pure interval ratios sound pleasant in music

Frequencies in exact small integer ratios tend to sound pleasantly "in tune" together.

This may be partly because we tend to play music on harmonic timbres (voice, strings etc). When you play a single note on e.g. a violin, we would identifiy it as a particular pitch. But if you analyse the rich texture you find it consists of not just a single pitch, but many frequencies played simultaneously that are whole number multiples of the "heard" pitch of the note. These are called the "partials" of the note. So if you play two harmonic timbre notes pitched at small integer ratios with each other, then many of the partials in the two notes coincide.

One can also train oneself to hear those component frequencies of a played note - for instance bell tuners may need to listen to the component frequencies of a bell as they adjust it to make a nice sound.

There are many other factors involved probably. This is an area of much discussion and controversy, what exactly makes a musical interval sound pleasant or harmonious to the ear.

Close to pure but not quite pure musical intervals beat

What is sure anyway is that when you play notes that are slightly "out of tune" then the slightly mis-aligned partials beat together making a wah wah type effect. This effect is known as "beats". This is something that you may perhaps listen out for when you tune a stringed or keyboard instrument by ear - you may tune to eliminate the beats. Or, as a piano tuner does, you may tune intervals carefully to achieve a particular number of beats per second.

This effect can sometimes be rather attractive, even very attractive. You often hear this beating sound in the sound of even a single bell as the timbre of a bell often includes some extra close together partials, due to the complex shape, and it is part of the distinctive timbre of some old church bells.

However, at other times it can get discordant and even be painful to the ear. It depends on the context, the instrument timbre, etc. Often you get many close pairs of partials in the chord beating simultaneously at different rates for instance. In some contexts the result may merge to give an interesting pleasantly rough texture. In other contexts they may jar and jangle.

Not unlike mixing spices to season food perhaps?

So - when you want harmonious chords - why not just make all your chords out of simple pure ratios?

Well - you can do that in some situations. If all the notes are part of a harmonic series - so they are all multiples of a single base frequency - then all the intervals will be pure ratios.

But unfortunately this only works with very special scales. For instance - the penatatonic scale, used all over the world e.g. for folk melodies, in many cultures - can never be justly tuned in this way. You can get all the fifths in tune for instance - but then find that the major and minor thirds are very sharp or very flat. Or if you try to get the major and minor thirds in tune, the fifths and fourths go out of tune. It just can't be done, it's mathematically impossible to do it.

Singers manage fine because they automatically adjust and "fudge" it - or if classically trained, may sing in approximately twelve equal which has the thirds a bit sharp but not wildly so.

You can never stack pure intervals all the same size together to make an octave

So anyway - you can never make pure ratio type musical intervals (e.g. 3/2) all of the same size that stack together to make an octave or whole number of octaves.

That's because of the way we hear musical intervals. A musical interval sounds the same if the ratio of the two pitches is the same. When you stack musical intervals, ratios multiply, and there is no way to multiply e.g. 3/2s to achieve the octave at 2/1, or multiples of an octave at 4/1, 8/1 etc.

So, no equal division of the octave like twelve equal can ever achieve pure ratio tunings. And approximately equal divisions will always favour some intervals at the cost of putting other intervals more out of tune.

But musicians find equal divisions convenient to play in - especially for fretted instruments like guitar, or for music with a lot of transposition to distant keys (as it is easier to transpose like that and have the tune sound the same after the transposition if all the semitones are the same size or at least similar in size).

So over the years various compromises have developed, the most famous and widespread of which is undoubtedly the twelve equal tuning of a modern piano, guitar, orchestra etc.

The modern twelve equal tuning uses approximations to pure intervals

This illustrates our point about the impossibility to get pure ratios in an equal temperament - the modern twelve equal tuning requires all the fifths to be very slightly flat. Then it requires all thirds to be quite a bit sharp, about a sixth of a semitone sharp compared with the pure tuning. The minor thirds similarly, are all quite a bit flat.

Just to be clear -the musicians who play this music (usually anyway) don't think of the intervals as flat or sharp, since they have been brought up to play hem as close to twelve equal as they can make them, have trained for years to do that - and nearly all the music they have heard has been tuned in that way too.

So to them the intervals may very well sound beautifully in tune. Indeed, the music they play is beautifully in tune in the twelve equal system.

A microtonalist who plays music in, say, seventeen equal will similarly try to be in tune in that system, and may well play music beautifully in tune in the seventeen equal system.

So - it depends what your aim is. The intervals are out of tune if you compare them with pure ratio intervals. They will always depart at least very slightly from pure ratios in any system that requires players to play using equal divisions of the octave.

The resulting music is very bright sounding if you come to it after a period of listening to justly tuned music, because of all the strongly beating partials in the chords.

Only octaves, and open fifths sound fairly calm - the octaves are completely in tune and the open fifths only very slightly flat. The other chords are very bright and they all vie for your attention. So in twelve equal tuned music - there are few moments of true rest and really mellow chords. They may seem so if you are used to nothing else. But even the final chords of a piece normally have those major or minor thirds which are so bright in twelve equal.

Sometimes this is just what one wants and very suitable for the style. If well atuned to it, the intervals may sometimes not even seem so very bright. But it can also sometimes be quite jarring or even painful if you come to it after a break listening to other music.

I remember once hearing a twelve equal piece on church organ on the radio, immediately after a piece in a tuning with pure ratio major thirds, and it was so painful I wanted to stop listening - because of the abrupt change in the tuning, and the long sutained chords of the organ. But when used to it, then it sounds just fine - after maybe just a few minutes sometimes.

Some types of music that use pure ratios

In choral music, singers often automatically adjust chords to pure ratios, but in keyboard and orchestral music, then the chords are pretty close to twelve equal.

In "barber shop quartets" then the singers may sing very pure harmonies, often using the characteristic seventh harmonic (which you can't approximate at all closely in twelve equal).

Classical Indian music (based on a drone) also uses pure ratios extensively - well it did until recently. Now many Indian instrumentalists use the twelve equal system.

Music with drones often uses pure ratios, or has some pure ratio, or close to pure ratio intervals in it. That's because you really hear the beats strongly with a drone. Though if it blends too much with the drone, there may be a tendency to play e.g. slightly sharp to bring the note out above the drone.

Pure ratios in music therapy

So anyway - if what one wants are restful and calming chords, mellow and warm sounding, then probably pure ratios based music is what you want.

It is no wonder then that pure ratios based music has been found useful in music therapy. Barbara Hero has developed a music therapy system based on her revival of an ancient system from Pythagoras called the Lambdoma. It is known also to microtonalists as the Tonality Diamond, and is based on pure ratios. In this system, a diamond layout of keys is used, and all the notes in a diagonal row are members of the harmonic series so are in perfect tuning with each other. Notes in a diagonal row running the other way are in the subharmonic series, so still in small number ratio intervals with each other, and also sound good together.

She also uses the Lissajous patterns as well as part of the therapy, and it is helpful for her clients to see the patterns of the chords as they play them, and the Lissajous patterns are also somehow calming and restful to look at, kind of harmonious looking.

To explore the Lambdoma in Fractal Tune Smithy, see Lambdoma Music Therapy

Music with impure intervals

Music with impure intervals also have interesting Lissajous patterns. This time, they never quite join up. But if you set the curves to fade as they go around, particularly if you show a large number of cycles, you get interesting effects.

There are many tuning systems with impure intervals apart from twelve equal. For instance the historical tunings of keyboard instruments, and modern systems with other numbers of notes to an octave, for instance nineteen equal, and thirty one equal etc to explore.

Then, the Gamelan music of Indonesia uses a system not based on ratios at all, not in any obvious way, doesn't even try to approximate them. The seven equal tuning system of Thailand is another system that goes its own way and isn't particularly ratios based.

Exploring Lissajous patterns for particular musical intervals in the companion program Tune Smithy

To find this feature in Lissajous 3D, go to Pattern |Shape, and you can set it to show the numbers as cents, or as ratios or cents.

So if you want to explore the Lissajous Pattern for a particular musical interval, you can set it up there very easily. For instance to explore the pure ratios fourth (e.g. from C to F), you would use 1/1 and 4/3, or to explore the twelve equal interval, you would enter 1/1 and 500.0 cents. There are 100 cents to a semitone, and the fourth is made up of 5 semitones on the piano.

However, it is even easier than that if you get and download Fractal Tune Smithy. You can play the chords in FTS and Lissajous 3D can immediately respond by displaying the same chord as a 3D pattern.

FTS also has its own window for showing the Lissajous patterns. It shows them as lines in 2D, which creates a different kind of effect and is well worth taking a look at if you are interested in them. The ones that never quite join up can be followed much further in 2D than they can in 3D because all you have to draw is a simple line rather than a 2D curve.

To find this feature in Lissajous 3D, go to Opts | Music Options for FTS. If you start up Lissajous 3D using one of the Lissajous 3D icons in Fractal Tune Smithy then it will be automatically set up to respond to FTS, but you can use this window to configure how it works.

To read more about Lissajous 3D, go on to: Triads, harmonic polyrhythms and 3D patterns

To get the program and the screen saver, download and install Lissajous 3D.

The program comes with a Free Test drive with all the features completely unlocked (start the test drive at any time):

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