Welcome, Images, Colours, Commercial Use, Screen Saver, The Musical Connection, The original Lissajous Curve, Moving into three dimensions Trouble shooting, How to purchase
Intro, Navigation, Support and updates
Jules Lissajous was a french scientist who lived in the nineteenth century. He generated the curves that later got his name by attaching mirrors to a pair of tuning forks. These beautiful curves are well known to any physicists who are particularly interested in the physics of sound, but not so widely known otherwise. To read a bit more about Lissajous see: Jules A Lissajous (at the Mac Tutor history of mathematics archive).
This application extends his idea to curves with three or more oscillations happening simultaneously. The idea of doing this was inspired by Barbara Hero's work on musical connections of the Lissajous curves, and it is a natural idea in music because musical chords are often triads rather than diads. The idea of showing these curves as 3D figures was suggested to me by the composer Charles Lucy after he saw some of the figures I'd made - then once that was done, I found out that such curves are already well known to knot theoreticians as Lissajous knots.
Lissajous 3D has tool tip help - hover the mouse over a button or text box, and in most cases a message pops up. Often however, the tip will end with a three dot ellipsis ... - this means that it has more help available. To show it at any time press F1 which will bring it up in a help window.
In this way you can find out most things by reading the extra help. Here, I will just mention a few things useful to know.
To find the example shapes try File | Open, or File | Files for a drop list.
To make yoiur own shapes, for a quick start try the Wizard. It is intended for young children especially, but anyone can use it. The wizard has several pages so once you have gone through the first page be sure to press the Next Page button to go on to the other things you can do.
You can turn the shape around by dragging on the
picture with the left mouse button held down- or vary the
spin position in the
Move window..
You can move it about with the right button held down - or vary the
across
and up fields in the
Move window.
Zoom in / out by holding both buttons down and moving the mouse
up or down, or vary the Depth in the Move window.
Hold down various combinations of the Ctrl, Alt or Shift keys as you drag with the mouse or click on the up / down arrows to get a faster zoom in / out or turn the shape more quickly. The Shift key down doubles the speed, the Ctrl key triples it and the Alt key quadruples it. You can use several of these at once - if you hold down all three then the speed multiplies by twenty four. It is easy to lose the shape out of the window with the fast movement options - if this happens you can get it back again using Shape | Reset Position.
This idea of holding down various combinations of Ctrl, Alt or Shift to change the numbers more quickly works with all the up / down arrows in Lissajous 3D.
You can set the shape spinning too. The technique here is to click and drag, and release the button while the shape is still turning (so while the mouse is moving). You can switch this feature off, or on again, from Move | Mouse up sets spinning.
You can set the shape swaying back and forth as well, from Move | Sway. It can spin and sway at the same time - swaying around as it spins - the sway gets added on top of the spin, so sometimes it might rather have the effect of speeding up and slowing down the spin periodically as it whirls.
Some of the examples change shape continually. This is set using Shape | Let the waves drift by. Again it can do all these things at the same time - spin, sway and change shape.
You can pause everything - movement and shape changes. from the Pause / Play button you see in various places such as the Shape or Move windows. You can pause just the shape changes from Shape | Pause wave drift.
If the shape is spinning, you can pause just the spin or let it continue again by selecting or unselecting. Move | Let shape spin.
You can get back to the help page your are reading now from Help | Intro and Overview window.
If Lissajous 3D doesn't work as expected, first see the Trouble shooting section. Supposing that doesn't solve it, contact Robert Walker (the developer) support@robertinventor.com. Be sure to contact support if you have any questions at all, or indeed suggestions for new features too.
For updates, visit the Lissajous 3D download page. At times my programs may get updated rather often with minor updates, so that keen users of the program can try out the newest features as soon as they are available. The version number doesn't change each time for those updates. So, to check to see if your copy is newer, go to Help | About and compare the date there with the date on the download page. To update your copy of Lissajous 3D, just run the setup program again. It is okay to install it on top of your existing copy.
The download site is robertinventor.com and there you will find Virtual Flower (another 3D program), and the music program Fractal Tune Smithy which can be used in conjunction with Lissajous 3D (as you will find out if you try the relevant options in the Opts and Scr Svr windows). You will also find some other apps, and new ones get added from time to time.
Several opt in lists are maintained for announcements of new releases, or updates. Contact support@robertinventor.com and let me know if you want to be informed of any updates of Lissajous 3D, also if you want to be informed about any new 3D programs released - I have in mind to develop more eventually over the next few years - or indeed you can opt to be informed about all new releases as they occur.
You can save the images, and set the size of the image to save or copy to clipboard, using Images | Image Size and Save Image As.You can choose from various file formats. The Bitmap (BMP) and PNG formats are lossless so are the best to use if you want to edit your image in another program. When you want a small file size, for instance to e-mail to someone or to use in a web page, select jpg or png. Most of the images work well either as Pngs or as jpgs.
You can also save the animation as animation frames which you can convert into an animation in other programs - for instance you could use a gif animator to convert the frames into an animated gif.
You can also make an animated jpeg in flash format directly from Lissajous 3D - or alternatively as an animated png. Generally pngs give a high quality losssless save, but when used in flash movies using the method used here then the size of the pngs approximately doubles so you will probably find the highest quality animated jpegs are a bit better in the sense that they give smaller files and are probably not noticeably degraded in quality..
You do all that from Images | Animation Frames.
Finally, you can also copy the image to the clipboard - go to the main window so that it is your active window (e.g. click on its title bar) and use Ctrl + C. Alternatively, use Ctrl + Shift + C at any time when working with any of the Lissajous windows. The size of the image you get here is set using Images | Image Size and Save Image As.
You may be interested to know that my music composing program Fractal Tune Smithy can make e-cards. This only works wth Outlook Express or other programs that can read the raw "eml" format (most other e-mail programs can't read it). You can use any image on your computer so could use it to send your Lissajous 3D images as e-cards. They are musical e-cards so optionally you can also add music to the card - as an attachment, or indeed if your recipient also uses Outlook Express you may be able to include the music as background sound so that the recipient hears the tune playing as soon as they look at the card.
If you use another e-mail program, you can also use FTS to make an html page for the e-card which you may well find you can send "As HTML" or some such. Maybe as "stationery". In this case your program won't need to understand the ".eml" format.
There is no gif or animated gif save yet - the plan is to add this in summer 2004. The reason for this is that patent royalties are high (thousands of dollars) for any programmer who wants to include a save in the .gif format as part of a program. The patent is for the LZW compression method used to make the gifs small, so strictly speaking, it is acceptable to add a save in gif format without compression, but the images will be huge files without it so no-one usually bothers with that idea. The patent doesn't run out in Europe until summer 2004 though it has already run out in the US. Since the software author lives in the UK then the Europe patent law surely has to apply.
The File | Save As option saves your work in the Lissajous 3D format (*.LJ) which is only understood by the program itself - has data about things like the number of waves in every direction that wouldn't mean anything to any other program. Be sure to save your work in this format too so that you can come back to it later and edit it in Lissajous 3D.
The idea is that you save your work as Lissajous 3D files when you are working on them in Lissajous 3D - and then when you want an image or animation to use in a document, web page or other program, you can save those in these image and animation formats understood by other programs.
To change the colours click on the colour patch and then on the colour wheel. This shows the hue - pure colour - around the edge of the wheel. To make it more intense or more washed out and pale use the saturation bar, which mixes white in with the pure colour, and to make it darker or lighter use the brightness bar. Use the drop list of colour names at the top to find named colours, for instance, browns are dark reds or oranges. You get greys, white or black if you set the saturation level to zero.
You can also show a colour disk, clover or rectangle. The disk shows the colours shading to white at the centre, so saturation varies radially. The rectangle shows hue horizontally and saturation vertically - you are probably familiar with this from the conventional colour chooser dialog in windows. In either case, you click to choose the hue and saturation, then use the brightness bar to adjust the lightness or darkness. The clover is just included for fun, with saturation again varying radially.
The actual colours you can show on a monitor are limited because they have to be made by mixing red green and blue light. You can show all possible hues, but when you set the saturation to 100 percent, this just means to set it to the most saturated that the colour can be on a computer monitor. In fact, most spectral colours are shown slightly unsaturated on a monitor. The yellow may look near enough to pure yellow but it actually has a bit of white mixed in compared with pure spectral yellow that you could get in a monitor if it used six primary colours of light instead of three. One can use more than three primary colours of light - but this is only normally done in specialist equipment for colour researchers. So in fact there are other colours outside the rim of the colour disk which are more a bit more saturated, particularly in the intermediate positions like yellow, cyan and magenta, that can't be displayed on your monitor.
Also the brightness range is limited for most colours as none of the other colours can be shown quite as bright on your monitor as a pure white, the colour you get with all three of the primary colours of light shining simultaneously.
You might wonder why the primary colours for a computer monitor are red green and blue rather than the red yellow and blue of artists that we learn about at school. That is because it uses mixing of lights. Mixing of pigments does indeed use red, blue and yellow - well actually in printed images, it's more usually magenta (red-blue) cyan (sky blue) and yellow.
So for instance green on a printed picture is a mixture of yellow and cyan (sky blue), as you might expect perhaps. If you think of the sun's light as a mixture of red green and blue, as it would be represented on a computer monitor, then yellow pigment removes blue from incident light, and cyan pigment (sky blue) removes red, so when you mix those two pigments together you are left with only green, so that is how it works.
It isn't so hard to see that the green ink on your computer print outs is often a mixture of two colours - but its blue ink also may not be a pure pigment, rather a mixture of magenta and cyan, which may come as more of a surprise. This time the magenta pigment removes yellow from incident light, and cyan removes red, so when you mix those two pigments together you are left with only blue.
One should say that this is a simplified explanation. The suns light is made of a mixture of all colours of the spectrum from red to blue so one should really take that into account - each pigment removes a certain area of the spectrum, with different parts removed in different amounts, so this is just a first approximation to the true situation to treat it as if made of three colours of light red green and blue.
The red yellow and blue you learn about at school is another equaly valid choice for mixing of colour pigments, as good as the red cyan and magenta used for three colour printers. Which you use is a matter of choice.
The use of red green and blue as the primary colours of light for colour monitors is also a matter of choice, with other possibilities available, e.g. violet would be a good choice for some situations instead of blue. The colour sensitive cones in our eyes have response curves with fairly broad peaks, and each one in fact is sensitive to some degree to light throughout the colour spectrum. They don't even peak at red green and blue particularly. So the situation is complex and there is much room for choice in the primary colours of light.
The primary colours of light and pigments have been standardised to these six colours for colour monitors, cameras, printers other equipment so that they would all be able to work together harmoniously to show colour pictures. So for instance, when you print out a picture from the monitor, its red green and blue values get converted into yellow, cyan and magenta pigments (usually with black as well) for the printer, but because everything has been standardised so that they work together, you see the same colours on the printer (close to the same anyway) as you see on the monitor.
The reason for the use of red green and blue lights as primary colours for light stems from the fact that our eyes have receptors sensitive to three colours, though the actual colours that our eyes are sensitive to aren't quite these ones exactly in fact. Here is a link that also goes into many technical details of how colours are shown on monitors and relationship of those to human perception: Colors and Colorimetry. Another interesting link is this one on Color Models, see also Computer Graphics Color models. The model for the colour disk here is the HSV (hue, saturation and value (brightness)) colour space which can be shown as a cone, hex-cone or cylinder- here is an on-line link to a page with a cut-away drawing of it as a single-ended cone: What is HSV.
For a short overview of Maxwell's colour triangle and the idealised red green and blue, see The Composition of Color.
Since these are demo shapes particularly included to help users get started, then it is fine to use them commercially. You can copyright your own shapes made with Lissajous 3D. You have my permission to use the ones that come with the program too.
The installer itself is freely distributable.You may include it on any program download site or CD compilation - for program details see the pad file: lj3d_pad_file.xml.You may not sell the evaluation copy. You may not charge for other users to use :Lissajous 3D. If you wish to use the program itself in a commercial fashion (rather than images or animations made with it), or use the program itself to promote other commercial products, please contact me first to discuss ideas or make arrangements.
I hope this is reasonably clear. If you have any questions or want anything particular clarified then let me know. Robert Walker support@robertinventor.com
(This is not to be considered as a legal document, it is just information to help Lissajous 3D users understand how I see the situation).
You can set Lissajous 3D as your current screen saver from the Display window - Start | Control Panel | Display. Choose the screen saver tab and select Lissajous 3D from the drop list. You configure it using the Settings button in that window. You can also configure the screen saver from within Lissajous 3D itself from the Scr Svr window. In Lissajous 3D you can play the screen saver shapes, and see the effect of your changes in the window by using the Play Screen Saver Shapes button.
Amongst other options for the screen saver, you can change how the shapes get randomised, or to choose the music you wish to play.
The screen saver gets installed when you run the setup program for Lissajous 3D. So, if you don't find it, run the installer again and make sure you have the option to install the screen saver selected.
When you run the setup program, you may also want to select the option to make a desktop shortcut to the screen saver. What this does is to make a shortcut that you can use to start the screen saver at any time from the desktop. It starts immediately, so you don't need to wait for your computer to go into idle mode, and it will run whether or not you have it selected as your current screen saver. The idea is that it lets you just start it up at any time if you want to see the Lissajous 3D shapes swirling around on your screen.
The music clips that come with the screen saver were made using the Fractal Tune Smithy app. You can also use it yourself to make new music for the screen saver. There is no need to be a composer to use it, or to be able to sing or read music or play an instrument - it is easy to use even if you have no musical background. Just run the program and you will find that you can make new "fractal" music with it instantly by varying the parameters. Or indeed you can run the program itself in the background to make background music for the screen saver, randomly varying the parameters itself.
You can also use any other music you have on your computer - just browse to them using the Open button for the music drop list in the screen saver settings window. You can also choose to play a random selection of tunes in the same folder. When you choose this random selection option, the music changes every time new shape gets shown.
It's also worth remarking that you can select a CD track for the screen saver too. If you choose to play a random selection from the CD then it accompanies the screen saver with a randomly selected track from whatever CD you have currently in you CD drive - change the CD and the music for the screen saver changes :-).
You can also use a play list, which can consist of any of your mp3s in any order. It is easy to make an m3u play list.
In its simplest form all it needs to be is a list of the names of the files you want to play one to a line in the order you want to play them - and start with #EXTM3U as the first line, like this:
#EXTM3U
first_file.mp3
second_file.mp3
...
where first_file.mp3, second_file.mp3 etc are the names of the music files you want to play. If they will be in the same folder as the m3u file, just give their names, otherwise their complete path c:\music\first_file.mp3 or whatever it is.
Then save this list of music clips as a file with the extension ".m3u", making sure to save it as plain text if you have a choice. That is your play list and you can now play it in Lissajous 3D or indeed in any program that can play m3u play lists (many music programs that can play mp3s can do this). It is a format originally developed for WinAmp which has now become the standard play list format for mp3s.
You can also specify the title to show for each clip, and indicate the length of each track. The format with those included as well is:
#EXTM3U
#EXTINF:120,first title
first_file.mp3
#EXTINF:120,second title
second_file.mp3
...
where 120 here is the length of the track in seconds - replace it by the actual length of the track so that your mp3 player can display the length of the track.
Getting started, Lissajous figures for pure harmonies, Exploring Lissajous patterns for pure harmonies with FTS and Lissajous 3D ,Just intonation harmony and tempered music, Listening to beats, Transforming tempered music to just intonation
The original Lissajous curves have a musical connection as they were generated using tuning forks. This application was inspired by a musical connection too, Barbara Hero's Lambdoma keyboard.
The musical connection between Lissajous curves and sound can be followed up using my Fractal Tune Smithy (FTS) program, which you can get from the Fractal Tune Smithy download page.
Then once you have installed FTS, you need to go to Lissajous 3D and select Opts | Change shape in response to Fractal Tune Smithy tunes. Then start both programs up. Whenever you play a chord in FTS or play a fractal tune with chords in it, your Lissajous 3D figure will change in synchrony. A good choice of figure for this experiment will be one with a continuous ribbon or tube rather than one made up of facets and fragmentary lines - such as the African shawl, polished gold rope, simple 3D Lissajous, etc etc.
You will find a Lissajous button in FTS already , in View | Player and View | Lambdoma. This shows 2D Lissajous curves. When you run it with Lissajous 3D you can explore the same curves in 3D :-).
This only works with FTS as of writing this, and it works because FTS communicates with Lissajous 3D directly while it is running to send it information about the chords played. Lissajous 3D doesn't attempt to analyse the music played directly (a much harder thing to do), so it needs this information to be provided. However, if you are a programmer yourself, and wish your program to interact with Lissajous 3D in the same way, contact support@robertinventor.com for details and I will explain how it is done.
The special thing about the Lissajous curves is that they join back to the starting point neatly when you play music based on small whole number ratios. This is musically meaningful as pitches related to each other by pure ratios form the most harmonious chords possible on most musical instruments. This is an observation that is originally attributed to Pythagorous, though as little of his work survives, only fragmentary quotes, this is more of a matter of legend really.
For instance the warmest and most melodious major chord on most instruments is played using the intervals 1/1 5/4 3/2. That's the same as 4/4 5/4 6/4, so if you enter the numbers 4, 5 and 6 in Lissajous 3D you get this curve.
Add a 7 as a fourth wave : 4 5 6 7, and the result is once more a very harmonious chord (known as the harmonic dominant seventh). Other numbers in small ratios to each other such as 7/6 are also particularly harmonious (the 7/6 one is the so called septimal minor - a dark rich mysterious minor chord much loved by microtonal composers).
The reason these chords are so harmonious is that most instruments have a timbre made out of an overtone series of frequencies based on whole numbers. So if you play pitches on these instruments in small number ratios, many overtones meet together and re-inforce each other and they sound nice together.
Note - there are exceptions to this rule including bells, and gongs. These have "inharmonic timbres". But most instruments like voice, strings, wind instruments etc have "harmonic timbres" in which the constituent frequencies that make up the note are all whole number multiples of the fundamental. Some are in between the two - the piano has a basically harmonic timbre, so generally small number ratios sound pretty good on piano too - but the frequencies of the overtone series are somewhat flatter than you would expect - and get flatter the higher up you go up the series - a consequence of the high tension applied to piano wires.
To explore the widest range of possibilities for pure harmony chords in FTS, look for the Lambdoma shortcut in your purple FTS Extra Shortcuts folder. You will find it on your desktop after you install FTS.
Then use the coloured squares button, which will bring up a coloured keyboard showing a musical pattern of ratios, known as the Lambdoma Keyboard. This is actually just a small section of an infinite keyboard which includes all possible pure ratios arranged in a musically meaningful fashion.
In this keyboard, any notes all in the same row will sound together to play particularly harmonious chords. Also notes all in the same column play harmonious chords - of another type. So it is easy to find nice sounding chords with this layout.
You can play the Lambdoma in Fractal Tune Smithy using the mouse. Just click on notes, and use the space bar as a sustain pedal to sustain overlapping notes. The preset eight by eight board gives plenty to explore as it is, however you can also move it around in the infinite ratios array if you so wish as explained in the help for that window in FTS.
You can also play these notes from the PC keyboard, which will play the top four rows of the Lambdoma. That is a bit limiting, so go to Keyboard Picture Options and select Moveable P.c. keyboard region, and you will find that you can move the playing area up and down using the num pad arrow keys (with num lock switched on).
Note that the PC keyboard isn't particularly designed as a musical instrument and may drop notes when you play some chords. It is suprisingly good and normally all single notes work fine, also most often, two note chords work fine but some three note chords may not work. The details depend on your hardware, it seems (probably the physical keyboard itself). Anyway if you can't get the chord you want this way by holding down the keys simultaneously, you can play the notes one at a time, and use the space bar sustain to get them to overlap to make chords.
So, play a chord in the Lambdoma, and you will make Lissajous curves which are based on small number ratios. They will close around back to the starting point, sooner if you play simpler ratios like 3/2, 5/4 etc, or after many twists and turns if you play more complex ratios.
You will find you get a drone in this window, because it is needed for music therapy work - it may be quite useful for making the chords too (one less note to sound) - anyway when you wish to, you can switch this off using the Drone dialog.
The idea of exploring this musical connection originates with Barbara Hero, creator of the Lambdoma keyboard, used in music therapy. Much thanks to her :-) - if it weren't for her I would never have written this program. She has been using actual recordings of chords to make her Lissajous curves in conjunction with a laser scanner,
What's really nice about this musical connection is that whenever you play the same chord you will see the same Lissajous pattern - this is because we hear musical intervals in terms of the ratios between the frequencies of the pitches. So for the interval between the fifth and sixth harmonics in the harmonic series is a pure minor third, which means that the pure minor third uses the ratio 6/5. You will find that 6/5 has one pattern, and the septimal minor between the sixth and seventh harmonics (which uses 7/6) has another, and so on. The pattern is the same no mater what frequency you choose for the fundamental of the chord - or to put it another way - you see the same figure for the same chord whatever its pitch. For instance all minor chords 1/1 6/5 3/2 (e.g. C Eb G) have the same pattern, whether it is C minor or A minor or G minor or F# minor or whatever and whether it is near middle c or several octaves above or below. They all have exactly the same Lissajous pattern in 3D.
The resulting pattern does vary depending on the place in the cycle where you start each oscillation - the phases of the waves. The phase of a wave is something one doesn't hear readily in ordinary circumstances, so all the patterns you get as you vary the wave shifts correspond to the same chord. So there is something to be said for trying these out with the phase of the waves set to drift in Lissajous 3D, as all those patterns correspond to the same chord.
Just intonation music refers to music based on small numbered ratios like 5/4, 3/2, 7/4 and so on. It is traditionally most often encountered in music with a drone such as Indian music, because with a long sustained drones rich in harmonics, any departure from pure just intonation will be much more noticeable. These typically don't have chord progressions or key changes - they depend instead on rhythmic and melodic subtleties..
"Western Music" is often played in tunings tempered to enable chord progressions to occur smoothly, and transitions to remote keys to occur easily. Originally western music was tuned in systems with the semitones unequal in size, with a distinction between two sizes, "diatonic" and "chromatic" semitones in some systems, and a great variety of semitone sizes in others. Gradually over the years since the seventeenth century or so, the semitones have become more and more equal in size, perhaps because of the need to modulate more and more rapidly to new keys. The tuning system most often used now is the twelve equal temperament which simply spreads all the notes out evenly so that each semitione in the twelve tone scale is the same size, and all keys are tuned identically. So we have reached the ultimate end point of this particular trend :-).
The so called "equal temperaments" used by Bach were actually what we nowadays call well temperaments - ones that favour some keys such as C major over others so each key has its own colour, so that is what Bach explored in his "Well Tempered Clavier - they are approximately equally spaced with some variation from semitone to semitone. Composers and musicians knew the twelve equal tuning in those days too - some people are under the impression that it is a new tuning but that isn't so - it was used early on for lutes. It is particularly practical for fretted instruments as it means you can use the same pattern of frets for all the strings. Equal temperaments with various numbers of notes to an octave are favoured by many microtonal guitarists today as well. So, partly this would have been for practical reasons, however another possible factor is that the beats are much less apparent on lute than they are on harpsichord with its rich timbre and bright third partials, or the long sustained notes of the church organ.
Tempered chords make Lissajous figures that don't join up at all, but go on endlessly - obviously one can't draw such a figure in its entirety, so in Lissajous 3D the ribbon will go on for a few cycles, and you set the number of cycles from the Ribbons window. When chords like this are listened to carefully on intruments with harmonic timbres then you hear patterns of beating intervals in the chords - a wah wah effect which indicates that the notes aren't in perfect just intonation harmony. Indeed, piano tuners often listen out for these beats and count them in order to help them tune pianos to twelve equal.
Unfortunately, many of the instruments on your p.c.'s midi synth may be far from ideal to use to hear beats as they often have other beating type effects, which obscure the ones we are listening out for. For instance, sound cards nowadays often use actual samples of recorded sound for the notes, which are great for improved realism, but can cause beats by the looping of the sound clips used to generate the notes - here the noticeable thing is that you get them even with a single note played on its own, and not just in chords. A single note may also beat with sounds made by the PC fan or any loud background hum in your room. Strong beating type effects can also be caused by instruments on your soundcard if they have a little residual vibrato in them. They are often recorded with some vibrato - intentionally so indeed and with great care to get the vibrato and the underlying wave to both join together smoothly simultaneously at the end of the loop - this is done because it sounds "more natural", but unfortunately when such notes are played together in a chord, the vibratos then may not be synchronised. This can cause a jarring wah wah type effect from the way the different vibratos interact, which is extremely noticeable in just intonation chords - and the noticeable feature this time is that you get this effect particularly strongly when you the several unison notes (all the same pitch) on the same midi instrument.
However with care you may find an instrument on your PC that works. Try and find a pure voice with little vibrato - but rich in harmonics. Experiment in FTS and see which sound best. The reed organ for instance may be a good choice. If you have a variety of sound devices, try them all as some may be better for this than others. You can install a soft synth to get yet more variety.
The best ones for hearing just intonation ones well are full and rich sounding as they will be rich in harmonics. Strings are amongst the best in this respect but may well have much residual vibrato depending on your system.
Pure flute like voices may seem a good idea, but they are low in numbers of harmonics. The flute itself has a few, but not many. The ocarina voice is often the purest one you have, and sine waves are lowest of all, no harmonics at all apart from the fundamental. Even if low in vibrato, these pure tones aren't the best choice for learning to hear just intonation chords - because they have few or no overtones, they will sound fairly good together if played at any wide enough interval, not just pure ratios. There is far less in the way of distinction between the small number intervals and the tempered ones on pure sine wave like instruments.
It is a natural thought, since the pure ratio intervals sound most harmonious of all, to wonder if one can play music in which all the chords are in this form. However, this only really works if you have very limited harmonic movement. Even if you keep to a pentatonic five note scale, you can't keep the pitches of all the chords one might want to use in that scale in pure just intonation ratios with each other. The numbers simply don't work out to let you do that - always one of the chords gets out of tune if you try. (In the case of the pentatonic scale on C, you can get all the fifths in tune, ratio 3/2, but if you do then the major third becomes a bit sharper at 81/64 instead of the pure ratio 5/4. So some compromise is needed in fixed pitch music, either restrict ones choice of chords to the pure harmonies possible with the given pitches, or adjust the pitches to play approximations to pure harmony only, but with a wider range of chords that can be played (this is called tempering the notes).
However, performers using variable pitch instruments - voice, strings etc - are able to vary the pitch of a note depending on the harmonic context. So pure ratio harmonies are possible for such performers - it has been found that unaccompanied (a Capella) choirs for instance often gravitate to pure ratio harmonies in long sustained chords.
There's some interest in attempts to do this sort of adjustment automatically on a computer. Various approaches are tried. FTS explores one particular approach, which is of interest here as it provides a way to get as many pure ratio chords as one can in the music, so good Lissajous figures - though it has some musical drawbacks too.
If you want to transform tempered music such as some of the fractal tunes in FTS or recorded midi clips to just intonation harmony try FTS | Out | Retune to j.i. harmony (with diesis pitch shifts). This makes the chords just, but it may cause some shifting up and down in pitch from time to time - on occasion these small pitch shifts can sound strange to unaccustomed ears.
The reason pitch shifts are inevitable with pure ratios is that for instance three major thirds such as c e g# c' will reach c' at (5/4)*(5/4)*(5/4) - when you combine musical intervals you multiply the numbers. This reaches the interval 125/64 instead of the expected octave 2/1 which would be 128/64. That's quite a large discrepancy as these things go. So if you go up three pure major thirds then quite a large pitch shift is needed at some point to reach the expected octave.
The reason pitch shifts are inevitable with pure ratios is that for instance three major thirds such as c e g# c' will reach c' at (5/4)*(5/4)*(5/4) (when you combine musical intervals you multiply the numbers). This makes the interval 125/64 instead of the expected octave 2/1 (or 128/64) which is quite a large interval as these things go, so if you go up three pure major thirds then quite a large pitch shift is needed.
Progression by minor thirds in a pure ratio diminished seventh (ascending or descending) also introduces pitch shifts - this is an even larger and very noticeable shift. Then there is a common chord progression I iii vi ii V I which goes up by a major third (5/4) then down by four fifths (3/2). The result this time is a rather smaller pitch shift of 80/81, - however, easily noticeable to musicians if you play two such notes one after another. Do this a few times, maybe in a repeating section of the piece, and the music will gradually spiral down in pitch until it is very noticeably flatter. This progression I iii vi ii V I is known as a "Comma pump" because the 81/80 interval that it shifts the music by is known as a comma.
The strong dovetailing options in the FTS just intonation Opts window work to reduce the number of pitch shifts from a note to the next note or chord in the piece. This is quite effective for some styles of music if they require few of these comma pumps or diesis shifts, but will increase the potential for a comma pump or diesis shifts over larger sections of the music. It can be quite effective for some chord sequences such as some early baroque, for instance, if they don't happen to have comma pump problems.
The original curves use just two oscillations. The idea is to start with a point oscillating up and down - and then add another oscillation left and right on top of that, and run both of them simultaneously.
If our point oscillates up and down in the middle of a screen, it will trace a vertical line. If instead it oscillates left and right it will trace a horizonatal line. If it oscillates in both directions at once then it will trace a curve - and the curve depends on how frequently it is oscillating in each direction. It draws a circle if both oscillations are the same frequency, a figure of eight if one is twice the frequency of the other, and so on.
So that basically is the idea of the original Lissajous curve, which uses two oscillations at once to trace the curve. The original version used mirrors attached to tuning forks to reflect light, and the moving point then traced a curve through persistence of vision
Physicists can make them on oscilloscopes nowadays. They tweak their oscilloscopes so that the horizontal scan oscillates instead of moving continuously from left to right. They set that to a frequency which is in some pure ratio connection with the oscillation shown vertically - then the oscilloscope will trace a Lissajous pattern. You can also draw them using two pendulums - one pendulum has a drawing board mounted above it which oscillates with one frequency, and the other pendulum moves a pen which draws on the drawing board for the first pendulum. If you adjust the length of the pendulums accordingly, you get a Lissajous figure again - at least, you would in the ideal situation with no friction involved. If you try out the experiment, actually you will get friction effects, and gradual decay of the swings, leading to a more complex pattern.
On line links to pages about Lissajous figures: A java applet to explore Lissajous curves - the Lissajous Lab. For many other links about Lissajous see the Geometry.net > Scientists > Lissajous Jules.
You can make two dimensional Lissajous curves in Lissajous 3D by just setting the third frequency in the Shapes window to 0 - the figure will then lie entirely within a plane - though of course you can rotate it to see it at different angles - and using Ribbon | Tube you can make it from a 3D tube.
How the curves are made - tutorial, Features of the curves and options, The magnet and planet, Techy note about the magnet and planet options
So now, what about three oscillations, in three directions. That would trace a more complex curve, in fact a curve in three or more dimensions. The result is a Lissajous knot. You can see a picture of one on-line here: Lissajous Knots.
So that is what Lissajous 3D does. From the shapes window you can set the numbers of waves per cycle for each direction in space. These are the scan frequencies for our Lissajous figure. What matters are the ratios between them - so if you had the numbers 2 3 5 here, you could just as well have them as 4 6 10 (all doubled), or 1 1.5 2.5 (all halved) or any other multiple of those three numbers. All will give the same figure.
The wave shift sets the position that each wave starts in in its cycle. To understand this, let's start with two synchronised oscillations. This may not work too well with a narrow ribbon as it may get so thin as to be almost invisible in 2D, so for this experiment, either use a wide ribbon, or use a tube - a good idea might be to start with the Simple 3D Lissajous or one of the other example figures based on a tube.
So first go to the Move window and stop all movements and all shape changes for your chosen figure. Then in the Shapes window, set the first two (x and y) numbers of waves per cycle both to one. Set all the other numbers of waves per cycle to zero, to make a two dimensional Lissajous figure consisting of generated by just those two frequencies.
Be sure to set the wave shifts for both your waves to zero.
With these settings, the point moves back and forth left to right, and simultaneously moves from the top to the bottom, and so it should trace the diagonal of a square - a straight line. Note - the actual direction it moves depends on what position you turn the figure to. The main thing is that you should see a straight line at this point.
Now, instead, start the first wave a quarter of a way through its cycle - set the first wave shift to 90. You will then find that it traces a perfect circle. The point tracing this curve actually moves back and forth left to right and up to down as before - but it starts the left right oscillation at its left-most postion when the up down one is in the middle of its movement. So it starts at middle left. Then if you work out where it should go, you will find that as it continues it visits top centre, middle right, and bottom centre, and in fact, it traces a circle.
If you choose other values for the wave shift between 0 and 90, it will trace an ellipse. You can watch the motion around the curve using Shapes | More Options | Swoop. If you don't see anything then increase the swoop amount - five or ten percent for instance would be about right for most figures.
Try switching the swoop on and off to get a feel for how the curve gets generated. So anyway that introduces the idea. The 3D curves just work the same way but with a third oscillation in and out superimposed on top of the left / right and up / down oscillations we already have.
You will find that with the flat ribbon (triangle or quadrilateral strip) the width often varies as you go around the curve. What happens is that it draws the original Lissajous curve, then shifts all the waves forward a little by the same amount, and then it draws it again. So, when it is wide, this is a place where the shape you get is particularly sensitive to changes in the phases of the waves. If you prefer a constant width, choose to show the ribbon as a tube instead.
The basic idea of the Lissajous figure gets transformed in various ways in Lissajous 3D. Some of the shapes aren't particularly smoothly curved. They are made using the option in the Ribbons window to make the curve out of very few segments to get an effect that looks a bit like those string figures you can make with a loop of string - for instance the Hazy sea creature. Others use wide ribbons as well, so wide that they make large flat surfaces, for instance for Tipsy Poly.
The smooth curves are also sometimes so wide that they curve around and across through the middle of the figure as in the Morphing CD. You can also twist the ribbon as in the Twisted Brass figure for instance. Others have a shape that is just a fragment of the figure swooping around the curve as in the sky dragon. Also some of them use another idea - to set the number of steps per cycle to just 1, do maybe one cycle only, or many, and to use tubes, and vary the ribbon width and the number of steps across the ribbon. Couple this with the star polygon cross section and you get the twisted star visitors, and various other such ones. You wouldn't 'be expected to be able to recognise the original Lissjous figure in these, rather, it affects how they move about.
The bead effect in some of the figures such as the Pearl necklace at dawn is obtained by using a tube with the option to twist the tube, set to a large number of degrees in Ribbon | Twist tube by. Then you set the number of steps per cycle to a suitable number too. If the number of steps per cycle is 100 for instance and the twist is by 18000 degrees (50 whole turns) you will get beads - gets twisted by a half turn for each step in the ribbon. So the general rule is to multiply the number of steps per cycle by 180. Vary the numbers a bit around that point and you will get less distinct beads, or twisted tubes with a rope like effect.
The Shapes | More Options window also lets you do things like add decay or friction or a magnet (or planet) - these are inspired by the actual physical experiments - but are intended purely as a source for interesting visual effects. Don't expect the modelling to be so very exact here.
The impulses get added to the original motion of the curve. One way to think of it is to suppose that the Lissajous curve is traced by a rocket pre-programmed with the impulses needed to trace the curve in free space - and not programmed to compensate for the presense of any other forces. The magnet or planet perturb its motion, and so affect the curve it traces. If you switch off the gravity or magnet then it will follow the Lissajous curve as usual. If you switch it on then the curve gets perturbed.
However, it won't go into a normal elliptical orbit around the planet for instance, because of its original motions (the pre-programmed rocket impulses) which are still going on in addition to the gravitational pull. Instead it continues to trace the Lissajous figure but transformed as a result of the perturbations of the planet.
Technically all I did (for those mathematically inclined) is to look at the Lissajous figure as if generated by a moving point, and find the velocities needed to trace the curve. Then I perturb those velocities by the forces regarded as a series of impulses acting on the moving point. Then I find the curve that results, which is what you see.
If you want to test to see what happens with the rocket engines switched off as it were, try the test in Shape | More Options | Set up for Test.
Thanks to Christopher Taylor for his account of experiments he did making real Lissajous curves using pendulums, with friction, decay and a magnet. It was as a result of thinking about these that I added these features. He is a mathematician and amateur astronomer - his web site is the Hanwell Community Observatory.
The magnet and planet options model the force as a series of impulses following the inverse cube and inverse square law respectively - so the more steps you have per cycle then the closer the result is to the exact curve you would get for a real magnet or planet. However as a precaution, if the ribbon has very few steps, it won't reduce the number of calculation steps to any less than about three thousand steps or so in order to make sure that it is at least reasonably accurate. In this case it calculates the intermediate positions, but only joins together the ones of those that you need for the ribbon.
Lissajous 3D requires OpenGL to function. This is included as standard with all Windows operating systems since Windows 9x so you probably have it.
Most graphics cards support OpenGL. However your card may support it for particular resolutions or colour depths only. Lissajous 3D will show a dialog when it starts if that happens, listing all the available resolutions and colour depths on your graphics card. The list will show which ones support Open GL - change to one of those, or choose to run only in full screen mode in a supported resolution if you don't want to change your current display resolution. If it still doesn't work, try another one from the list, until you find one that is okay.
If it still won't run, check for an updated Open GL driver for your graphics card from the manufacturer's web site.
If by chance you are missing the OpenGL dlls that normally get installed with Windows, you can get them from the Microsoft site. For details of what you need, and the link see the OpenGL page Downloads for Gamers and 3D Pro users. The direct link for the download for these currently is http://download.microsoft.com/download/win95upg/info/1/W95/EN-US/Opengl95.exe
Three dimensional animations like this are computing intensive. If you find that your computer gets slow to respond, check the settings under Opts | Animation timings. These are designed to make sure the graphic rendering pauses frequently for a few milliseconds at a time, which will normally make sure that other processes can continue to work at the same time. The settings here will be suitable for most, but if necessary you can increase the length of the pauses and reduce the maximum time for each frame.
Also, the 3D tubes are slower to render than the other ribbon shapes, so you can try Ribbons | Disable tubes - which will show all the tubes and twisted tubes etc as flat ribbons (triangle strips).
If you have a zippy computer and graphics card, you may wish to unselect the options in Opts | Animation timing. With the sleeps during tube or ribbon construction, you may get pauses in the animation from time to time - switch this option off to get a smoother animation, but one that is more cpu intensive.
Increase the time for each frame if you want to render a particularly complex lissajous figure without the time out cutting in after two seconds. See Opts | Animation timings
It doesn't matter how fast your computer is, as you will always be able to make figures so complex that they require seconds to render if you specify enough cycles to draw and use figures that don't close round quickly to complete the curve. There is no practical limit to the complexity of the figures you can make.
Some of the drawing is done in the OpenGL library itself. This may cause a pause at the end of each frame during which your computer may become slow to respond - but so short as to be barely noticable in the faster animations. This is a large and complex library and you should have it already on your computer - and I haven't attempted to modify it to reduce the amount of this pause.
I have modified the gle.dll library to add time outs to it, as that is a tiny dll the download size is of no significance. You will find that you have a version in your Lissajous 3D folder called gle_with_time_out.dll. This makes it easier to interact with FTS and other programs while it is making shapes which look like tubes or twisted tubes or beads rather than ribbons - the ones that have Ribbon | Tube selected.
If you have other OpenGL programs, none of them will be affected by this change as the dll name is changed too - they won't recognise it as the same dll. I have also added an option to change the material along the length of the tube which isn't in the original version of the library. The source code for this mod is available if anyone wants it, at http://robertinventor.com/gle_with_time_out/index.htm
Purchase on-line, Other order methods, Product Levels,
See also, Will my program be hardware locked when I purchase it?
Choose the order type, then click Order Now and Continue after that. The order gets processed when you fill in your details and click the Place your Order button on the third page.
Secure order - for details - see the Privacy Statement.
You are welcome to try the process out first, as far as the order page to see how it works. If you have any questions about the order process, or any problems occur, be sure to ask :-). support@tunesmithy.co.uk.
You will receive an unlock key code instantly when you complete the order. It is valid for all future releases of Lissajous 3D.
You use this key to unlock the program you already have - if you haven't downloaded it yet, see Download Now. This means you have the opportunity to evaluate it first and make sure it is suitable for your needs.
When you get the key, enter it into the Unlock window you see at the start of the session, or use Help | Unlock .
You can also pay in your own currency, if you are in any of the supported countries. To do that, go to the order page, then in the next page, select either bank/wire transfer, invoice, or giro payment. Global Collect will then send you an invoice in your own currency, which you can pay by your chosen method, after which you will be sent the unlock code.
You can also pay by Telephone or Fax
Alternatively, fill in one of these forms (available on-line) Post, Purchase order
You can also pay in GB pounds and send a cheque directly: Post UK.
Be sure to contact me with any questions about the order process. support@tunesmithy.co.uk
For an extra few dollars, you can buy Virtual Flower as well.
Buy Virtual Flower and Lissajous 3D
(you will also find this as an option in the order form for Lissajous 3D).
If you aren't able to view the on-line forms, then use the off-line versions of these pages - but send me an e-mail or letter first to check the current prices.
Telephone or Fax, Post, Purchase order, Post UK
Individual user and Home Use. Home users are free to use Lissajous 3D on any of their machines - you don't need to buy a site license for home use.
Combined license for Virtual Flower, and Lissajous 3D, This license covers both programs, and will also cover some of the other 3D programs I plan to develop in the future.
Multi User license - You only need this if you are getting Lissajous 3D for an organisation. It allows use of Lissajous 3D on any number of computers belonging to your organisation, and any number of users. In the case of educational institutions, the teachers may also use the program at home.