Lambdoma

=Lambdoma=

[#Welcome Welcome], [#lambdoma_pref Lambdoma keyboard preferences], [#lambdoma_opt Various options], [#lambdoma_rec Recording], [#lambdoma_comp Composing], [#lissajous_opt Lissajous Options], [#colour_code_opt Colour Code Options], [#rectangular_scale Rectangular Scale], [#purchase Purchase]

Welcome
This task is espeically for use with Barbara Hero's Lambdoma keyboard, as used in music therapy. The lambdoma is a rectangular scale consisting of all ratios from 1/1 to 16/16 and you can read about it on her site Lambdoma

You can also explore the lambdoma and other rectangular scales with the mouse or PC keyboard - though normally the p.c. keyboards have some limitations on chords you can play see [faq_mus.htm#play_triads How do I play triads from the PC keyboard? (some work and some don't)]

If this is your first time with Fractal Tune Smithy (FTS), glance at the short [index.htm#Intro Intro]. Particularly note that you can get help for any window by clicking on its ? icon.

Fist time users should check that FTS can retune the notes appropriately on your system. You can do this from within the program using. If you hear two notes identically tuned then it is tuning the notes correctly. If not, there are solutions you can use - such as to install a suitable soft synth to play your notes. See [testmymidiplayer.htm#test_synth Test my Sound card / synth], which also gives more details.

You can also check to see if the example midi clips in this help are played correctly tuned. See [testmymidiplayer.htm ] [testmymidiplayer.htm#qt_note ]

First use the button to set everything to their usual settings for the Lambdoma - however you don't need to do this if you installed FTS first time using the special Lambdoma installer as it is done already.

To show the keyboard layout in FTS, click the

You will see the first quadrant when you first run FTS - you make the other quadrants using the buttons in the main window. You can use the button to try other rectangular scales.

You can resize this picture, as with most windows, with a click and drag on the boundary of the window. Hover the mouse over the right or bottom edges of the window, or lower right corner, and find teh position where it shows a double arrow, then hold the mouse left button down and drag and it will change shape. If you lose the double arrow or the mouse goes of the edge of the mat, just click on the edge to show the double arrow again, and continue.

You can show a Lissajous curve as you play - click this icon in the FTS Lamdoma window:



The Lissajous curve is a curve that shows how two frequencies relate to each other. This is a generalisation of the idea to triads and chords of more than three notes. If you play a chord, you will see the curve change as you add notes to it.

For options for the Lissajous curve see [#lissajous_opt Lissajous Options] :

The standard setting is to have a drone. You change its settings using the. button.

The drone is set to switch off a second after you release the key - that is so that you can play a series of notes above a continuous drone, which probably will sound better than a lot of short drone notes - and yet when you stop playing, the drone will also stop after a short pause. If you prefer it to switch off immediately, you need to to go to the drone window and look for. Leave this selected, and set it to switch off after 0 seconds,. You unselect it if you want a continuous drone which never switches off.

You can have a chord as a drone too if you like - enter all the ratios for the desired chord as drone notes. For instance if you want an extra drone an octave lower, enter the drone notes as 1/1 1/2 which will play the note at the 1/1 frequency and another drone note at half that frequence, which means an octave below. You can add 1/4 to add a drone two octaves below or 1/8 for three octaves below and so on. Perhaps octave drones would be most useful for the Lambdoma, but if you wanted to have any of the other Lambdoma ratios as a drone for instance, for some reason you can just add it in.

When you use the standard settings, the drone plays in the second part in the window, while the keyboard plays the first part (that's what the Part field is about in the main window). This means you can select the instrument for the drone by varying the instrument for the second part - so you can set it to a different instrument from the keyboard if you like. To change the instrument for a part, first highlight the part you want to change, then select the new voice from the menu. You can also enter the name of the instrument into the text field below the list of parts.

You can use - to set the notes to play for an extra four seconds (or whatever time you like) after you release the key.

If you use and play several notes one after another, with each one starting before the previous one finishes, they build up to make a chord. If each individual note played for four seonds from when it started, they would each switch off one after another too - but one is more likely to want a chord to end crisply, so that all the notes in the chord stop at once. So, to get the chord to end crisply, all the notes of the chord switch off at the same time - four seconds after the last note was played - all the notes apart from the drone (with the standard settings). The drone will stop immediately as well if you have it set to switch off after 0 seconds from the. window.

Lambdoma keyboard preferences
You need to make sure that FTS listens to the correct com port.

In FTS, go to, and select or and see which of those your keyboard is connected to - which one lets you play musical notes in FTS. The other one will give no sound at all.

If you use a USB to Serial port adaptor (e.g. for a laptop) then it may be set to use some other number for the com port. If so, select and set it to the appropriate number. If you need to find out the number of the com port it installed, you can find them all listed in - in XP you may need to select to see this. Then go to - and scroll down the list and you will see the com ports listed.

If that doesn't work you can try com ports 3 and 4. Or use the option which searches through all the ports up to 10 to see which are available - probably one of the ones you find will be the one you need.

Next, select (it will already be selected if you first installed FTS using the Lambdoma version of the FTS installer).

The value in this window specifies how abruptly to switch notes off when they end. Many synths or sound cards ignore this value, and it is preset to the usual standard setting for this parameter, 127 which means to switch them off as sharply as possible. If this parameter is recognised, a value of 0 will switch the notes off much more gently, so that the note gently trails away when it finishes.

As you play notes you will see scrolling text appear in this window. This is for diagnostic purposes, and shows that the key presses are being received from the Lambdoma keyboard. The scrolling text may possibly slow things down if you play notes in rapid succession - this is only while this window is open. When you close the window the notes will be played normally, and you only need to show it to set the keyboard up.

If you find the music is very quiet, be sure to visit the [troubleshooting.htm FAQ - Trouble Shooting] section [#why_quiet Why is everything so quiet?]. The first thing to check is the blue volume cone at the bottom of the window - it is preset to the middle position, but may be better at a far right position instead. You may also prefer to vary the volume from - this control sets a master volume for all music programs in Windows.

Perhaps there are particalar windows you would like to show in every session - if so, you can set FTS to show them using the F4 shortcut. For instance, maybe you want to always see the Lambdoma keyboard and the Lissajous window. Just show all the windows you want to see, and press the F4 key, and then every session from then on will start with those windows open.

You also need a file called Mscomm32.ocx in your Windows/System or System 32 directory - but the FTS Lambdoma installer installs it automatically if not present. FTS also automatically sets all the Com port settings appropriately whenever it opens them. It does this temporarily, whenever you run the program, without changing your existing Windows settings for the Com portss.

Various options
Use the window to select voices for drone and keyboard. To change the voice for a part, highlight it, and select a voice from the menu or enter the name into the text field. The Lambdoma keyboard (or p.c. keyboard) plays Part 1, and the drone plays part 2 with the standard settings. You can change these settings from the and windows.

If you have a plucked or percussive instrument for the drone then you will probably want to repeat it every note or every few notes, rather than sound it continuously. You can do this in the window - use - the other one is only for notes received from Midi In.

The standard setting pans the drone to the right and the lambdoma keyboard to the left. You can vary this in the window. For instance, the drone is panned hard right and the keyboard hard left. Perhaps you prefer them somewhat more to the center. To do that you would change the values 0 and 127 in the Pan window, to, say, 32 and 96, or whatever you like.

You can show the diamond shaped lambdoma from the Keyboard options window. That's not the preset just because the rectangular window gives a bit more space for the letters so they can be somewhat larger.

After you play a note or chord, you can move the entire chord around in the Lambdoma keyboard picture using the keys immediately to the right of the normal typing area of the keyboard (not the arrows on the num pad area). For this to work, the Lambdoma keyboard picture has to have its title bar highlighted to show that it is receiving input from the p.c. keyboard (the technical term for this is that it has the keyboard focus). Normally it gets highlighted in a darker colour than usual. This will also work if you have the Lissajous window as the window set to receive p.c. keyboard input. If it doesn't work right away, then click on the title bar of one of these windows, which will set it to receive keyboard input, and then you will be able to do it.

You can move the displayed region of the lambdoma around in the infinite lambdoma using, again using the arrows immediately to the right of the typing area. For those exploring the 16 by 16 Lambdoma, you can use various shortcuts in the Keyboard Options window to move your keyboard in the larger region of the 16 by 16 Lambdoma. To do this, select in that window. The region of the 16 by 16 Lambdoma that you are currently playing will be highlighted. Then for instance with down then with the rectangular layout, the keys at the four corners of the num pad section will move your keyboard to the four corners of the 16 by 16 Lambdoma. If you have it set to a diamond layout then the four arrows at left right, top and bottom of the num pad section will move it to the four corners of the diamond.

You can play chords along with every note you play using the. There utonal means to play a chord along a horizontal row from every note played and otonal means to play a chord along a vertical row. Since a cluster of notes in the same row or all in the same column of the keyboard will go particularly well together then this method produces harmonious chords. If you set it to play broken chords there, then the notes get introduced one at a time in rapid succession, to build up to make the complete chord.

The standard setting is to show notes as unreduced ratios, such as 14/6 etc, according to the row and column in the Lambdoma. You can show reduced ratios instead such as 7/3 by using.

There's also an option to show note names, for instance solfeggio or the musical note names C, C# D etc if you are used to either of those systems. You can find this in You will want to select  note names here, so that the 1/1 is shown as, or as  in.

Some of the other notation systems are useful to display fine pitch distinctions - these are useful for the Lambdoma as many of its pitches can only be represented accurately by making fine pitch distinctions - you will only find a few of these notes on a piano for instance. The nineteen tone option makes a distinction between sharps and flats - and the notes in order of pitch for this notation are C C# Db D etc. Thirty one tone lets one display even finer pitch distinctions distinction with half flats and half sharpsSeventy two tone is a twelfth tone notation using a system of three accidentals up and down from any twelve tone pitch (by a sixth tone, third tone and quarter tone), with no distinction made between sharps and flats. It is known by many performers of microtonal music, and is compartively easy for a professional musician to learn as it takes the equally spaced twelve tone pitches that they are used to as its starting point, but again it doesn't quite have the fine distinctions you need to notate the Lambdoma pitches exactly.

Sagittal is a new notation which was developed in the last couple of years. It is particularly designed as a way of giving names and notations to many pure ratio pitches and to permit extremely fine pitch distinctions as well - while it requires the performer to learn relatively few new accidentals. It can be used to notate all the 16 by 16 lambdoma pitches exactly. The main interest of it here is that it uses a system of few accidentals which can be used by a performer to find the pitches by reference to the notes on a standard twelve tone staff.. It takes the Pythagorean tuning by pure fifths as its reference point, which makes the exact tuning of the intervals pure ratios as we need for the Lambdoma, and yet, since that tuning is close to the equal temperament system, again it is relatively easy for a professional musician to learn. It's implementation here is work in progress right now - more later.

Recording
You record using the .buttonfor waveform audio or the button to make midi clips.

The difference here is that waveform audio records the actual sounds played - the files are larger but if you send them to someone your recipient will hear exactly the same sounds that you heard. These tend to be large files, but you can record to mp3s if you want a smaller file to e-mail - mp3s are about a tenth of the size of waveform audio or so, depending on the amount of compression - smaller but they are still fairly large to send by e-mail if they are of any length.

The Midi files are far smaller and play the same musical notes - you can send a midi recording of even several hours playing and it will still be a tiny file - but the actual sounds heard will depend on your recipient's sound card synth. Midi recordings basically record notes for playback on the midi synth on the sound card (or a soft synth or any other hardware or software that recognises the format). The GM Midi sound set has been standardised so if your recording is for say an oboe or a reed organ or whatever, the same instrument will normally be used on every sound card, but the actual realisation of it will differ so your recipient will only hear exactly the same realisation of an oboe that you heard if they use the same soundcard or synth.

Then another nice thing about the Midi file format is that your notes can be played again on any device or indeed any instrument so a method that's quite often used is to record to midi when you actually play the notes - then do various audio realisations of your midi file on different instruments later. You can do that by playing the midi file in a player that will let you vary the instrument from the one saved in the midi file, so you can do the recording to audio at that point. Indeed, the Tune Smithy retuning midi player can be used to vary the instrument on playback in this way. The waveform audio recordings can't be played back again on different instruments in the same way.

To set the file name for the recording, click on the tied quaver (eighth note) button. This brings up the window, and you can set everything you want to set for the recording from here..

You can play back your recording using the button in the window. You can record using the button in that window - the main window record buttons have the same effect as selecting the suitable file format from the drop list and then pressing the button.

For details, and to find out how to use the other formats such as mp3s, see [faq.htm FAQ - general.] | [faq_gen.htm#audio_formats How do I record and play back recordings?]

Composing
The window is for display only - it shows the notes as you play them. I'd love to make an editable version of it too but if I do, then it is for rather a way into the future, as there is lot's of work involved. It will also probably be pretty basic in its capabilities for some time after that, if I do do anything of that sort. I may possibly develop it as a separate program instead - to interface with FTS to play the notes perhaps.

Meanwhile, if you want to use FTS for composing, you need to use it in conjunction with a notation program or sequencer to retune its notes. This is pretty easy to do, but you need to install a little app. called Midi Yoke first in order to do it..

Then, you can enter notes on the score and get your score retuned so that the pitches you hear are Lambdoma pitches rather than the usual ones that you hear for notes on a musical stave. For some general details on how this works, see [faq_mus.htm#compose_microtonally How do I use FTS with my music notation software or sequencer to compose microtonally?] Be sure to ask me if you want more details.

There is a bit of a logistical problem here as there are so very many Lambdoma notes and they are arranged in a two dimensional way rather than a one dimensional way, so most ways of mapping them onto a standard stave will be a bit awkward. Anyway contact me if you are interested to explore this more and want help with setting things up for composing in the Lambdoma via your notation software or have ideas that could be interesting to explore that may require more programming for the FTS interface in the future.

Lissajous Options
[#lissajous_intro Intro], [#lissajous_options Options], [#start_curve_at Start Curve at,] [#actual_and_ideal_lissajous About the ideal and actual Lissajous curves].

Intro
First, of all, what is a Lissajous curve?

First, let's suppose you have a point oscillating 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. Now, suppose 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 type shape 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 Lissajous curves were inspired by sound - as they were made by attaching mirrors to a tuning fork and using those to reflect light to make the patterns.

The connection with sound is that of course sound waves are frequencies. If you play a two note chord in stereo, and play one note in the left channel and one in the right channel then you can use these as the two oscillations to make a Lissajous curve - and this is exactly what Barbara Hero has been exploring. The curves will meet up eventually to make a closed figure if the frequencies of the two notes are in pure ratios with each other. The more complex the ratio, then the longer it will take to join up, so a complex ratio like the Pythagorean major third 81/64 will take a fair while to join up, while the nearby pure ratio 5/4 (the pure, or just intonation major third) will do so quickly. The Pythagorean major third takes 64 turns to join up, while the 5/4 does it in 4 turns.

Barbara Hero has been using actual recordings to make her Lissajous curves. The ones in FTS though are theoretical ones - because it turned out that they were the easiest to program.

What's really nice about this musical connection is that whenever you play the same chord you will see the same Lissajous pattern - for instance a pure minor third which involves the fifth and sixth partials in the harmonic series, and so the ratio 6/5 has one pattern, the septimal minor third (with 7/6) has another, and so on. The pattern is the same whatever frequency you choose for the root of the chord. Let's use the rectangular layout to describe this, just because it makes the description easier to follow by using rows and columns rather than diagonal rows.

Then for instance if you switch off the drone, and then play a chord in any row of the Lambdoma, then move it vertically (e.g. using the arrow keys to right of the typing area), you see the same Lissajous pattern - that's because it uses the same relative pitches. Pllay notes in a column and move that column of notes horizontally and again the Lissajous pattern stays the same. For instance if you play 4/4 5/4, then that's a major third. Move it across to 4/5 5/5 and you will again get a major third again, this time based on 4/5 instead of 5/4 as the root note of the chord. With the drone switched off, youl see the same Lissajous pattern in both these positions, and in all the other places you can move it to horizontally. You can find the minor third in one of the rows as 6/6 6/5, and this has the same pattern as the various vertical translations of it such as 5/6 5/5. So does the setpimal minor third such as 7/7 7/6, and so on. The pure fourth is 4/3 and the fifth is 3/2 and the whole tone comes in many flavours including 10/9 and 9/8. You will find many other flavours of chord in the Lambdoma, each with its own pattern. The intervals between notes in the same column all remain the same under horizontal displacements, and the intervals between notes in the same row remain the same under vertical displacements.

You need the drone switched off to see this way the patterns stay the same as you shift the pitches. With the drone switched on it works a bit differently, because the drone note stays in the same place as you shift all the other notes in the chord, so with the drone on, shifting the chord does change the pattern. The pattern would stay the same if the drone itself shifted along with the other notes in the chord.

The pattern will also vary depending on the place you start each oscillation, which you can vary in the Lissajous Options window. All those ways of looking at it when you vary the start positions though could be considered as the same higher dimensinoal curve seen in different ways. See the help for [#start_curve_at Start curve at].

On line links to pages about Lissajous figures: A java applet to explore Lissajous curves - the Lissajous Lab. To find out more about Lissajous: Jules A Lissajous (at the Mac Tutor history of mathematics archive ). For many other links about Lissajous see the Geometry.net > Scientists > Lissajous Jules.

So far, that is the conventional Lissajous curve for a diad or two note chord.

But, what about three oscillations, in three directions. That would trace a more complex curve, in fact a curve in three or more dimensions (thanks to Charles Lucy for pointing out to me that these are higher dimensional curves, which I hadn't noticed).

Here, I've kept to two directions still for the drawing, horizontal and vertical - but you can have several oscillations at once along either of those directions. This means it is like a shadow or projection of a higher dimensional curve. (I will explore possibilities for drawing them as higher dimensional curves at a later date - with perspective, and maybe ones you can rotate in 3D).

In three dimensions the result is a Lissajous knot. You can see a picture of one here: Lissajous Knots and make them indeed yourself using Knot Plot. To make them type a command into the command window such as lissajous 1 3 5 for the major chord or lissajous 1 3 7 for the septimal minor etc. It has two extra parameters which change the shape of the curve, e.g. lissajous 1 3 5 0.5 0.5 or whatever. The command is case sensitive so be sure to type lissajous rather than Lissajous which it will see as an "unknown command".

You use the scale slider in Knot Plot to zoom in / out and use left mouse drag on the figure to drag to turn it around. Right click on the Knot Plot control panel for a menu of options.

You can also make them in 3D in my Lissajous 3Dprogram, which can also be set to respond to changes in the chords as you play the notes from the lambdoma keyboard. Details are given in the help for that program.

Options
First a note about the title bar for the Lissajous figure. Sometimes it says that the curve continues - in that case, that means that the calculation went on for a fair while and the curve hadn't met up yet at the point that the calculation stopped - it probably continued a fair way beyond the part that actually gets drawn. With some tuning systems, this could mean that it does continue endlessly without ever meeting up - but in the case of the Lambdoma, all the numbers involved are pure ratios, so the lines will meet up and it just means that the ratio is a complex one that will take many turns to join up.

At other times it says that it is incomplete. This means that the calculation went somewhat ahead of the drawing, enough to show that it does join up, but there hasn't been enough time available to draw the complete figure. FTS stops drawing it after 1 second with the standard settings, because you don't want it to spend all its time drawing just the one curve when you are playing.

In either case, to see more of the curve, try pressing, and increase the time out there if necessary, and you may be able to show the complete figure..

- adds a title on the picture itself (rather than the title bar) with information about the curve - the notes played and the number of turns it took to make it.

Show curve as it is drawn - you can use this to watch the curve grow slowly - especially if you also select, or.

The standard setting in FTS is that you see the curve change as you release the notes of a chord of three or more notes, until you reach a two note chord. Even if you release all the notes almost instantaneously, maybe one will linger for a few milliseconds, which may be enough so that FTS will redraw it. Maybe you prefer to continue to see the curve for the complete chord as you release its notes - if so select this option.

and - this only affects the way the curve is labeled in the title or title bar. You can choose whether to show the ratios from the 1/1 in the title bar, or the ratios from the lowest note of the chord. The standard setting is from the Lambdoma 1/1 because then the ratios there match the notes actually played. You may prefer to show it as ratios from the lowest note, particularly if you play the Lambdoma without the drone, because that means that the same pattern always gets shown with the same ratio values. For instance the major third would always be shown as 1/1 : 5/4.whatever column you play it in, whether you play it as 4/4 5/4 or as 4/5 5/5 or whatever.

- click this button to copy it to the clipboard. You can then switch to your word processing or image processing program, and use the Paste option (e.g. right click and paste or Edit / paste) to copy it there for editing / printing.

Note for those who want to upload these images to web pages or send them via e-mail
Normally you would save an image of this type as a, as it has large areas of uniform colour which compress exceedingly well in gifs. Images in this format are saved along with a custom palette of up to 256 colours which lists the colours needed for your particular image. The palette for the gif will be made automatically by the program you use to save it, but you may want to do it yourself in order to set how it should be done.By way of example, in Paintshop Pro you go to and then for the method for choosing those 256 colours for the palette, choose and. That may work better than the preset if it is preset to do it using methods suitable for photographs.

You can also save it as a, which is a newer format - it is now undertsood by all modern web browsers,. Pngs don't have the restriction to 256 distinct colours. The other web page format often used is the - that format is less suitable for this type of image, as it will tend to create larger files than the gifs for large flat areas of colour than this, and the edges of the lines may be a bit fuzzy. A gif will only lose some of the colour info in the reduction to 256 colours and otherwise will be exactly as you saved it. Pngs are lossless (well I think you lose some colour information if the original image is at a very high colour resolution). Jpegs compress the data and this is a lossy format and can save with fuzzy lines depending on the level of compression you use..

- makes lines that shade from one colour to another. The preset here is red to blue - but you can change the colours to use if you click.

The colours window has help to explain how you use it - basically you work left to right - select the item you want to change, click on the new colour for it in the Colours list, then click Choose from custom colours if you want to fine tune it - then you may find it easiest to use the Wheel, or one of the named colours. See the help for the [#Colours Colours] window.

- you will understand this best if you look at the colour wheel:. Try setting the colour to red say, which you do by setting the hue 0 and saturation and brightness both 100; Then using the up / down arrows to the right of it of the hue field, move the hue around to blue. If you hold the arrow button down with the mouse, the hue will change continuously. So now, if you go up from 0 to 600, that's the long way round, while if you go in the other direction,from 0 to 599 then from there on down to 400, that's the short way round because it is a shorter distance around the circumpherence of the wheel..

- that corresponds to the way of presenting things in terms of red, green and blue values - and you just go along a straight line there from one colour to the next. So for instance, red is 255, 0, 0 in this system (all the numbers are in the range 0 to 255) and blue is 0, 0, 255.. Then the straight line between these two in RGB coords goes 255 0 0, 254 0 1, 253 0 2, etc, until you reach blue at 0, 0, 255. In this particular case it is somewhat like the short way round the hue circle in terms of the colours you get - but at the mid point you have 127 0 128 which is a dark magenta - while on the hue circle you would have a pure magenta, which is 255 0 255 in RGB units. General effect is a like the hue arc, but will have darker colours for the middle of the curve.

- Following the example of Lissajous Lab, this sets the number of line segments to draw per turn. So for example if you set this to 4 or 6, then show the Lissajou circle for 1/1 :1/1 then it will turn into a square or hexagon. Preset to 100, and small numbers here will mean that the picture can refresh more quickly.

, and - use to open and save your Lissajou figures. After they are saved, they get added to the drop list so you can open them again by selecting them. The save nicludes the colouring of the figure, so that when you open it again in the future it will look identical - except of course that the size depends on the sizeof window you show it in.

- this takes the ratios from the title bar and makes them into a file name by replacing '/' which is unacceptable for windows (or indeed Unix) file names into 'o', etc. If you use this system, you can figure out from the file name what the ratios or intervals were in the Lissajou fifure.

- here gives the original Lissajou curves. You can use the same shorcuts here as for the, so is short for. The ones with any number of cos or sin functions in a row such as (shorthand ) will draw fastest as a bit of extra code has been added in to make those especially fast.

'''Technical note. '''These curves such as actually no longer range from -1 to 1 in the x and y coords, - but because it looks nicer, they get centred in the screen and drawn large enough to fit. Values here are in radians - which means that an angle of 2*PI corresponds to a whole turn. If you want the forumla to give -1 to 1 range for x and y, you need to use for cosines, because multiplying the rest of the formula will change its range of -1 to 1 into a range of -PI/2 to PI/2 which is enough for the cos of the value to span the complete range of -1 to 1. With a sin, if its argument has a range of -1 to 1, you need to multiply it by PI/4 to get. As it is, will always be positive in value - the curve you see for this one would be in the bottom right corner of the window if it didn't get recentred. To take an example, you could redo as:. for instance, to get all the values in the range -1 to 1. It's not the same fomula any more - so the shape will change when you do this. as:.

Start Curve at
- Here you can set the angle, in whole turns to start the waveform at. Try changing this, and you will find that the curve changes its shape. This corresponds to phase changes in the waveform. Since human ears can't hear the phase of a sound in a steady sustained note, then it is fair to say that all of these correspond to the same chord as far as hearing is concerned. The animations in the Lissajous Lab will give an idea about how this works.

If you try two simultaneous fundamental notes 1/1 for instance, you will get a circle if the point is moving to the right or left at it's fastest speed when it is stationary in the up / down direction - and a diagonal line if it moves at its fastest speed in both directions at the same time.

I've done this parameter so that the position the curve starts at horizontally, and vertically, depends on the pan position - the position of the note as it is heard, whether left or right. All the notes panned to the left play across the x axis, and the ones to the right affect the y axis. Then the position each oscillation starts at depends on the pan position too. The ones to the left start at 0 turns, the ones to hard right start at 0.25 turns, and notes in between start at in between positions.

So, though one would expect to get a straight line for 1/1 1/1, in fact, you get a circle after taking account of the this shift for the pan position. Add or subtract 0.25 to one of the turns and you will get the expected straight line.

More details:
Sometimes you may have all notes are in the same position, say in the centre. Then one of them gets put into the horizontal (x) axis anyway - the lowest note in the chord in fact, and the others get put into the vertical axis, and they also get drawn as if panned hard left and hard right. All that is to ensure you get an intersting Lissajous figure even with no panned notes.

About the ideal and actual Lissajous figures
The curves you see are the ones you would get for a pure frequency (sine wave). Tthe nearest approximation to a pure frequency in Midi is often the ocarina voice.

They show the ideal curves - these are the ones you would get if the far left-most or right-most notes sounded only on one speaker and not at all in the other - as is the case with Barbara's original Lissajous figures. Actually, in Midi, it is likely that even when the notes are hard left or hard right there will be a bit of sound played in the other speaker for the note, because that corresponds to what happens in a recording of a performance of a note - even if the player is at the far right, a bit of sound will get played over the left speaker too.

So, in summary, this is just to say tht these are ideal curves. Some time I hope to program FTS to show Lissajous figures for the actual sounds you hear too, but that is more of a technical challenge, at least if it gets done as you play, so I probably won't do it for some time if I do It is a matter of the latency - it probably requires programming in Direct Sound or the like to make it so the curve can change in real time as you play.

Actually, as well as the latency, you want as little going on as possible while recording in the Windows API. You can get artefacts (clicks and gaps in the recording), especially on slower machines if you overload it too much, so it mightn't work so well to record to a temporary waveform in memory and do its Lissajous curve at the same time while you play. Doing it for a section of a pre-recorded waveform would be far easier so that may come first.

Colour Code Options
You can select various colour codings from the drop list at the head of this window.

The standard setting here is to use the Lambdoma one - which shows a spectrum with middle C (a little below concert pitch) at 256 Herz as green, as Barbara Hero has it. It's the absolute pitch that counts here - though if you want to make the 1/1 green whatever its tuning, choose instead.

To vary the colour of the fundamental, just click on the, etc buttons. The colour repeats around the spectrum in every octave.

The Lambdoma colouring is based on the frequency of light - if you divide the frequency of light by 2 many times to reduce it to the same frequency as sound, then the spectrum spans a bit under an octave. I know that sound isn't radiation - but this is fun to do and it's how it is done in this field.

You can read about it on Barbara's site of course, and also take a look at Charles Lucy's explanations - the Scriabin and Rosicrucian schemes came from there. See: Pitch, Colour, Scriabin, and others. Charles Lucy's colour scheme, which is also based on the colour spectrum, is in the presets drop list for this window too. It comes from a book by Guy Murchie called "Music Of The Spheres". For another site on colour in music see Colour music in Australia - de-mystifying De Maistre.

- this lets you make a colour for each degree of the scale independently. You can vary the number of colours - preset to 6 but you may well want to change that to 12 for twelve tone scales. Also you can set the - most will want to repeat at octaves because notes an octave apart sound "the same" - but some composers explore the similarity in sound of notes at other intervals such as 3/1 or 3/2, so may want to set another repeat interval here. For instance a 3/1 repeat interval would be appropriate when working in the Bohlen Pierce scales, and a 3/2 for work in Wendy Carlos's beta or gamma scales.

You change the coloures using the Colours window which you bring up using the Change Colours box. See the help for the [#Colours Colours] window.

If you would like to start with a colour spectrum, and modify that, then use the button, which will make a specturm with whatever your desired number of colours.

- here if you set the saturation for the 1/1 to 100 and the brightness to 100 then the background colour for the 1/1 is white and the letters are at maximum brightness. The preset is to make both 80 percent, which means the background is 20 percent of the colour, and the foreground is 80 percent of it in terms of both saturation and brightness. Then as you go up by octaves the brightness andsaturation change - the preset here is that the letters get brighter as you go up in pitch, but less saturated (more washed out) - while the background changes in the other direction. This is quite a subtle change - if you want to make it more dramatic to see what happens, increase the number of units to change by per octave.

,, and - dark background here means that if you have it without the octave coding, then the background is black instead of white. For the custom background, it uses whatever colour you set in - the preset value here is white.

- the preset here does the background at its maximum brightness and the text in a darker shade of the same colour. You can vary the brightness of the background and the numbers.

Finally the check boxes at the bottom of this window set all the things in FTS that you can choose to colour code. The standard setting here is to colour code the Lambdoma only. To see what the other ones are, click their buttons.

Rectangular Scale
You can give your scale a name at the top of this window - this appears in the lambdoma title bar, and in the title bars for some of the other tasks. In most of the other tasks, apart from the Lambdoma, it also gets shown as the description of the scale in the main window, above the data area for the scale itself.

This window is used to make rectangular scales such as the one used with the Lambdoma keyboard - see if not already there.

The scale goes like this:

The pattern there is that the denumerator (number at the top of the fraction) starts as 1 and increases to the right and downwards, and similarly for the denominator.

Then for the lambdoma, the scale gets flipped horizontally and vertically to match the standard Lambdoma keyboard layout. Also numbers like 2/6 in this array will be made instead as 1/3 which is the same fraction but in simpler form.

So, what you do here is just to set what you want for the first and last denumerator and denominator - and what size of step you want between them. The scale gets made or remade automatically whenever you change the values.

As you chanage the first denominator or denumerator, you will see that the last one moves in synchrony - that is to make it easy to move between rectangular scales of the same size starting at various positions. For instance users of the eight by eight Lambdoma keyboard will normally want to keep to eight rows and columns, or possibly sixteen by sixteen, and vary the start numbers only. If you want to explore some other size of region, again maybe you want a fixed size region and will proably want to vary the start numbers most of the time, after you have set the region size by varying the end numbers.

To make the Lambdoma keyboard presets, go to and use the buttons there to make the four quadrants.

That's about it - quite a simple window this one :-).

Purchase
Please buy it from this order page so that Barbara Hero can be credited: Purchase Fractal Tune Smithy for the Lambdoma.

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