!  Well, it's hard to see very clearly, because the big variation
!  happens when the amplitude is very small. The real point is that
!  it's impossible to DEFINE this number lambda_tone except in
!  the isolated case when you have a trig identity (ie A=1);
!  and even then its true meaning is suspect. 
!  
!  If you have a complex valued function  z  (such as  e^{i(w_0)t} + e^{iwt} )
!  you can define the "instantaneous circular frequency" by
!  
!  Im(z-dot/z)
!  
!  where  z-dot   is the velocity function (also complex valued).
!  
!  If you work this out for  z = e^{(a+bi)t}  you will find the constant
!  frequency  b , which is good.
!  
!  If you do this for a sum of two sines of different frequencies,
!  you'll get something that varies with time, even if the amplitudes ARE
!  equal. 
!  
!  Sorry about the confusion! In any case, by varying  A  from  1   down to  0,
!  you can see that the formula I suggested for lambda_tone can't possibly
!  be right.

!  (1) the lambda_sum should not actually appear.  I was wrong
!  in thinking that it made sense. This formula is correct if  A = 1,
!  but not otherwise - certainly not if  A = 0,  for example.
!  In fact the "tone frequency" varies with t except in those cases.
!  I think we should leave it out.
!  
! I finally worked out what the envelope should be in our beats tool.
!
! I thought about what must happen as A is pushed from 1  to 0.
! At 1, the 'tone frequency' is the average of the two initial
! frequencies. But this can't be the sitation for other values of A.
! At A = 0, for example, the tone frequency is obviously omega_n.

! In fact, the best thing to say is that the pitch varies in a beat;
! it goes up when the beat is loud, and down when it's soft. 
! ??? is this true ???

! This made me annoyed enough to work out what the story actually is.
! There is a perfectly good envelope:
!
! g(t) = sqrt(1 + A^2 + 2*A*cos((omega-omega_0)*t + phi)).

!  The beat frequency fb = f2 - f1, where f2 > f1. The tone frequency ft = (f1 + f2)/2. 
!  beat period is  2pi/(w-w_n) and w_n=1


!! File: Beats
!! August 25, 2003 Hubert Hohn

PUBLIC PCFlag,Mac5Flag,M68KFlag,UnixFlag,xmax,ymax
PUBLIC toolLft,toolRgt,toolBas,toolTop,toolhdr,toolHgt,toolWid  ! tool boundaries
PUBLIC winLft,winRgt,winBas,winTop,winHgt,winWid      ! window
PUBLIC workLft,workRgt,workBas,workTop,workMidx  ! work area
PUBLIC backclr,black,drkgry,drkmid,midgry,litmid,litgry,white
PUBLIC red,yellow,green,cyan,blue,magenta,pink,colorscheme
PUBLIC planeclr,gridclr,rimclr,axisclr,axislabelclr,titleclr,rightsclr
PUBLIC numberlineclr,slotdrkclr,slotlgtclr,slideclr
PUBLIC largefonts, title$, SLUmode

LET toolHgt= 560
LET toolWid= 780
LET window$= "The d'Arbeloff Interactive Math Project"
LET colorscheme= 0

LET title$ = "Beats"

SUB ThisProgram
    CALL SineSumBeats
    CLEAR
END SUB


!! ---------------------------------------------------------
!! ------ Start TB4 Mac Header and Subs ------

!LET M68KFlag = 1
!LIBRARY "MacTools*", "HHLib.trc"
!ASK PIXELS winWid,winHgt
!LET winLft= 0
!LET winTop= 0
!LET winRgt= winWid-1
!LET winBas= winHgt-1
!SET WINDOW 0,winRgt,winBas,0
!CALL Palette
!CLEAR
!
!CALL ToolPanel
!CALL ThisProgram
!
!END
!EXTERNAL
!
!MODULE Mac4Parts
!    SUB SetTextFont(font,size,style$)
!        CALL MacTextFont(font)
!        CALL MacTextSize(size)
!        CALL MacTextFace(style$)
!    END SUB
!
!    SUB StringWidth(sw$,sl)
!        DECLARE DEF MacStringWidth
!        LET sl= MacStringWidth(sw$)
!    END SUB
!
!    SUB SetLineWeight(wgt)
!        CALL MacPenSize(wgt,wgt)
!    END SUB
!
!    SUB BoxDisk(Lft,Rgt,Bas,Top)
!        CALL MacPaintOval(Lft,Rgt,Bas,Top)
!    END SUB
!END MODULE

!! --- End TB4 Mac Header and Subs ---
!!---------------------------------------------------------
!!--- Start TB5 Cross-Platform header and subs ---

LIBRARY "c:\TB Gold\TBLibs\TrueCtrl.trc"     ! windows
LIBRARY "c:\TB Gold\TBLibs\HHLib.trc"

!LIBRARY ":TBLibs:TrueCtrl.trc"                   ! macintosh
!LIBRARY "HHLib.trc"
!
PUBLIC WinID
DECLARE PUBLIC OBJM_SET,OBJM_SYSINFO
LET winHgt= toolHgt
LET winWid= toolWid
DIM values(1)

CALL TC_Init

CALL Object(OBJM_SYSINFO,WinID,"MACHINE",system$,values())
IF system$="MAC" then
   LET Mac5Flag= 1
ELSE IF system$="WIN32" then
   LET PCFlag  = 1
END IF

CALL TC_SetUnitsToPixels                         ! 5.1 and up needs this
CALL TC_GetScreenSize(scrnLft,scrnRgt,scrnBas,scrnTop)
LET winLft= int((scrnRgt-scrnLft-winWid)/2)
LET winRgt= winLft+winWid-1
LET winTop= int((scrnBas-scrnTop-winHgt)/2) + 10
LET winBas= winTop+winHgt-1

CALL TC_Win_Create (WinID,"TITLE",winLft,winRgt,winBas,winTop)
LET values(1)= 2
CALL Object(OBJM_SET, WinID, "TYPE", "", values())

IF PCFlag=1 then                                 ! kill dithering
   LET  values(1)= 1
   CALL Object(OBJM_SET, WinID, "SOLID MIX", "", values())
END IF

LET  values(1)= 0
CALL TC_SetRect(WinID,winLft,winRgt,winBas,winTop)
CALL TC_Win_SetTitle(WinID,window$)
CALL TC_Show(WinID)

SET MODE "COLORSTANDARD"

ASK PIXELS winWid,winHgt                         ! must follow set mode
LET winLft= 0
LET winTop= 0
LET winRgt= winWid-1
LET winBas= winHgt-1
SET WINDOW 0,winRgt,winBas,0

CALL Palette

IF PCFlag=1 then
   LET  values(1)= 0                             ! now force solid colors
   CALL Object(OBJM_SET, WinID, "SOLID MIX", "", values())
   CALL TC_Win_RealizePalette(WinID)             ! some PCs need this

   CALL TC_Win_SetFont(WinID,"arial",9,"plain")
   CALL StringWidth("0",sw)
   IF sw>7 then LET largefonts=1 else LET largefonts=0
END IF

CALL TC_Win_Switch(WinID)

CALL ToolPanel
CALL ThisProgram

CALL SetTextFont(1,12,"bold")                    ! now shut down and clean up
LET quit$= "click the mouse or press a key to close..."
CALL PlotTextCJ(workmidx,(workbas+worktop)/2,quit$,yellow)
CALL TC_CleanUp

END
EXTERNAL

MODULE TB5Parts
    SUB StringWidth(sw$,sl)
        DECLARE PUBLIC WinID
        LET sl= StrWidth(WinID,sw$)
    END SUB

    SUB SetLineWeight(wgt)
        DECLARE PUBLIC OBJM_SET
        DECLARE PUBLIC WinID
        DIM values(1)
        LET values(1)= wgt
        CALL Object(OBJM_SET,WinID, "WIDTH", "", values())
    END SUB

    SUB SetTextFont(font,size,style$)
        DECLARE PUBLIC WinID,Mac5Flag,PCFlag,largefonts
        IF Mac5Flag=1 then
           SELECT CASE font
           CASE 4
                LET font$= "Courier"
           CASE 16
                LET font$= "Times"
           CASE else
                LET font$= "Geneva"
           END SELECT
        ELSE IF PCFlag=1 then
           IF largefonts=1 then
              IF size<12 then
                 LET size=  6
              ELSE IF size=14 then
                 LET size= 10
              ELSE IF size=18 then
                 LET size= 12
              ELSE IF size=24 then
                 LET size= 14
              ELSE IF size=12 then
                 LET size=  8
              ELSE
                 LET size= round(72/96 * size * .8)
              END IF
           ELSE
              IF size<12 then
                 LET size=  7
              ELSE IF size=14 then
                 LET size= 12
              ELSE IF size=12 then
                 LET size=  9
              ELSE IF size=18 then
                 LET size= 14
              ELSE IF size=24 then
                 LET size= 18
              ELSE
                 LET size= round(72/96 * size)
              END IF
           END IF
           SELECT CASE font
           CASE 4
                LET font$= "Courier New"
           CASE 16
                LET font$= "Times New Roman"
           CASE else
                LET font$= "Verdana"
           END SELECT
        END IF
        IF style$= "normal" then LET style$= "plain"
        CALL TC_Win_SetFont(WinID,font$,size,style$)
    END SUB

    SUB BoxDisk(Lft,Rgt,Bas,Top)
        BOX DISK Lft,Rgt,Bas,Top
    END SUB
END MODULE

!  --- End TB5 Cross-platform header and subs ---
!! ---------------------------------------------------------

!! --- Start Unix Header and Subs ---

!library "HHLib.trc"
!LET UnixFlag= 1
!ASK PIXELS winWid,winHgt
!LET winLft= 0
!LET winTop= 0
!LET winRgt= winWid-1
!LET winBas= winHgt-1
!SET WINDOW 0,winRgt,winBas,0
!CALL Palette
!
!CALL ToolPanel
!CALL ThisProgram
!
!END
!EXTERNAL
!
!MODULE UnixParts
!    SHARE CharWidth
!
!    SUB SetTextFont(font,size,style$)
!        LET font$= "-adobe-courier-"
!        IF style$= "normal" then
!           LET style$= "medium-r-normal--"
!        ELSE
!           LET style$= "bold-r-normal--"
!        END IF
!        IF size=9 then
!           LET size$= str$(10)
!        ELSE
!           LET size$= str$(size)
!        END IF
!        LET test= SetFont(font$&style$&size$&"*")
!
!        IF size=9 then
!           LET CharWidth=  6
!        ELSE IF size=12 then  ! numeric output - axis labels
!           LET CharWidth=  7
!        ELSE IF size=14 then  ! rare
!           LET CharWidth=  8
!        ELSE IF size=18 then  ! rare
!           LET CharWidth= 10
!        END IF
!    END SUB
!
!    SUB StringWidth(sw$,sl)                      ! string width in pixels
!        ! LET sl= StrWidth(sw$)
!        LET chars= len(sw$)
!        LET sl   = chars*CharWidth
!    END SUB
!
!    SUB SetLineWeight(wgt)
!        ! CALL PenSize(wgt,wgt)
!    END SUB
!
!    SUB BoxDisk(Lft,Rgt,Bas,Top)
!        CALL Fill_Circle(Lft,Rgt,Bas,Top)
!    END SUB
!END MODULE

!! ------ End of TB Unix Header and Subs ------
! -----------------------------------------------------------


! ***

SUB SineSumBeats
    DECLARE PUBLIC black,drkgry,drkmid,midgry,litmid,litgry,white
    DECLARE PUBLIC red,yellow,green,cyan,blue,magenta,colorscheme
    DECLARE PUBLIC planeclr,gridclr,rimclr,axisclr,axislabelclr,slideclr
    DECLARE PUBLIC workLft,workRgt,workBas,workTop,workMidx
    DECLARE PUBLIC true,false

    DECLARE DEF quitWithin, infoWithin

    ! --- help screen array ---

    DIM  info$(1:15)
    MAT READ info$

    DATA "Beats"
    DATA ""
    DATA "This tool supports graphical exploration of the formation of beats in the sum of two sinusoidal waveforms."
    ! DATA "The formula for the period of the beats can be derived experimentally."
    DATA ""
    DATA "Drag the mouse cursor over the upper time series plane to display crosshairs and the values of t and x at the cursor position."
    DATA ""
    DATA "Click or drag the omega slider to change the frequency of g(t). Click among the omega slider hash marks to jump to the nearest tenth for omega."
    DATA "Click or drag the A slider to change the amplitude of g(t). Click among the A slider hash marks to jump to the nearest tenth for A."
    DATA "Click or drag the phi slider to change the phase shift of g(t). Click among the phi slider hash marks to jump to the nearest pi/2 for phi."
    DATA ""
    DATA "Click the [Envelope] button to toggle the graph of the envelope curve for the wave sum in the upper plane on or off."
    DATA "Click the [Zoom] button to toggle between time ranges of 20 pi and 60 pi."
    DATA ""
    DATA "Click the checkboxes under the upper plane to overlay any combination of"
    DATA "f(t),  g(t),  sin(((1+omega)/2)t),  or sin((w-1)t)"


    !  A "zoom" button toggles between a time range from  0  to  20 pi  and from 0  to  60 pi.  

    !  The bottom plane shows the graphs of  f(t) = sin(t)  in red and  g(t) = A sin(omega t - phi) in yellow.

    !  The amplitude  A ,  the circular frequency  omega ,  and the phase lag  theta,  can each be controlled by the sliders at the bottom. 
    !  Hash marks on sliders are active, and arrow keys on the  omega  slider allow for fine adjustments.
    !  
    !  Rolling the cursor over the top window to produces crosshairs and readouts of  t  and  x.
    !  
    !  An "envelope" button displays the graph of an enveloping curve in the top window.  
    !  Toggles allow any combination of  f(t),  g(t), or  sin(((1+omega)/2)t)  to be displayed in the top window.


    ! ---------- Utility functions ---

    DECLARE DEF clamp,roundn,e

    ! --- functions and parameters

    LET f3= 2*3
    LET f5= 2*3*4*5

    DEF pi2 = 2*pi

    DEF f(x)= sin(x)                             ! fixed wave

    DEF g(x)= a * sin(w*x - p*pi)                ! adjustable wave

    ! the DE? simple harmonic - x" + x' + x = 0
    ! s^2 + s + 1 = 0
    ! -1 +- sqrt(1 - 4) / 2 ???

    ! g(t) = sqrt(1 + A^2 + 2*A*cos((omega-omega_0)*t - phi)).

    ! I've switched from omega_n  to  omega_0 , because I don't want to 
    ! encourage the notion that there's some system out there with 
    ! undamped natural frequency omega_n. 

    ! Is there a difference between undamped natural frequency and
    ! natural frequency?
    ! in this equation, I assume omega is the value from the slider.
    ! is the value of omega_0 just 1?

    ! I renamed the natural frequency from omega_0 to omega_n in damping ratio
    ! after we talked to Trumper. Do I need to change it back to omega_0?

    DEF notanum= .0123454321

    !    SUB Amplitudes
    !        BOX CLEAR 0,500,90,0
    !        SET CURSOR 1,1
    !        PRINT a,w
    !
    !        !   A (wt - (wt)^3/3! + (wt)^5/5!) + t - t^3/3! + t^5/5! = 0
    !        !   Awt - A(wt)^3/3! + A(wt)^5/5! + t - t^3/3! + t^5/5! = 0
    !        !  (Aw + 1)t - (Aw^3+ 1)t^3/3! + (Aw^5+ 1)t^5/5! = 0
    !        !   divide by t
    !        !  (Aw + 1) - (Aw^3+ 1)t^2/3! + (Aw^5+ 1)t^4/5! = 0
    !        !   let t^2 = k
    !        !  (Aw + 1) - (Aw^3+ 1)k/3! + (Aw^5+ 1)k^2/5! = 0
    !        !   now its a quadratic in k
    !        !   k = { (Aw^3+1)/3! + sqrt[((Aw^3+1)/3!)^2 - 4 (Aw +1)(Aw^5+1)/5!] } / 2(Aw +1)
    !        !   k = { (Aw^3+1)/3!  - sqrt[((Aw^3+1)/3!)^2 - 4 (Aw +1)(Aw^5+1)/5!] } / 2(Aw +1)
    !
    !        LET t1= (A*w^3+1)/3
    !        LET t2= ((A*w^3+1)/f3)^2
    !        LET t3= 4*(A*w +1)*(A*w^5+1)/f5
    !        LET t4= 2*(A*w+1)
    !
    !        WHEN error in
    !             LET k1 = 4*pi * (t1 + sqr(abs(t2-t3)) / t4)
    !        USE
    !             LET k1 = notanum
    !        END WHEN
    !        PRINT k1
    !        WHEN error in
    !             LET k2 = 4*pi * (t1 - sqr(abs(t2-t3)) / t4)
    !        USE
    !             LET k2 = notanum
    !        END WHEN
    !        PRINT k2
    !    END SUB

    DEF env(x)
        LET env = sqr(1 + a^2 + 2*a*cos((w-1)*x - p*pi))
    END DEF

    ! ---------- Graphing plane parameters and methods ----------

    LET wwid  = 600
    LET whgt  = 128
    LET frgt  = 200
    LET orange=   9

    ! ------------------------------------------
    ! --- w1 plane data ---

    DECLARE PUBLIC w1Lft,w1Rgt,w1Bas,w1Top,w1Midx,w1Midy
    DECLARE PUBLIC w1fLft,w1fRgt,w1fBas,w1fTop,w1x0,w1y0
    DECLARE PUBLIC w1xFirst, w1xSTik, w1xLTik, w1xLabel, w1xGridstep
    DECLARE PUBLIC w1yFirst, w1ySTik, w1yLTik, w1yLabel, w1yGridstep
    DECLARE PUBLIC w1xPiFlag, w1xMult, w1yPiFlag, w1yMult
    DECLARE PUBLIC w1wWid, w1fWid, w1wHgt, w1fHgt, w1Aspect

    LET w1Flag   = 1
    LET w1xPiFlag= 1
    LET w1xMult  = pi
    LET w1yPiFlag= 0
    LET w1yMult  = 1

    LET w1clr    = green

    LET w1Lft = worklft+90                       ! pixel window bounds
    LET w1Rgt = w1Lft+wwid
    LET w1Top = workTop+35
    LET w1Bas = w1Top+whgt

    LET w1fLft=  0                               ! function bounds * pi
    LET w1fRgt= 30
    LET w1fTop=  2                               ! function bounds
    LET w1fBas= -2

    LET w1xGridstep= 0                           ! horizontal grid intervals
    LET w1yGridstep= 0                           ! vertical grid intervals

    LET w1xAx$= "t"                              ! axis labels
    LET w1yAx$= "x = f(t) + g(t)"

    LET w1xSTik  =  0                            ! horizontal axis Tik marks
    LET w1xLTik  =  2
    LET w1xLabel = 10
    LET w1xFirst = w1fLft

    LET w1ySTik  = 0                             ! vertical axis Tik marks
    LET w1yLTik  = 1
    LET w1yLabel = 1
    LET w1yFirst = w1fBas

    ! --- w1 plane methods ---

    DECLARE DEF w1fncx, w1fncy, w1wndx, w1wndy   ! window/function transforms
    DECLARE DEF w1wWithin

    CALL w1Variables

    SUB w1InitPlane
        CALL w1SetBounds
        CALL w1DrawPlane(1,1,1)                  ! xaxis, yaxis, zeroaxes
        CALL SetTextFont(1,12,"bold")
        CALL PlotTextLJ(w1Rgt+8,w1y0+3,w1xAx$,axislabelclr)     ! axis labels
        CALL PlotTextLJ(w1x0-3,w1Top-12,w1yAx$,w1clr)
        CALL w1KeepGridLayer
    END SUB

    SUB w1SetBounds
        SELECT CASE zoomState
        CASE 0
             LET w1fLft  =  0                    ! function bounds * pi
             LET w1fRgt  = 60

             LET w1xSTik =  0                    ! horizontal axis Tik marks
             LET w1xLTik =  2
             LET w1xLabel= 10
             LET w1xFirst= w1fLft
        CASE 1
             LET w1fLft  =  0                    ! function bounds * pi
             LET w1fRgt  = 20

             LET w1xSTik =  0.5                  ! horizontal axis Tik marks
             LET w1xLTik =  1
             LET w1xLabel=  5
             LET w1xFirst= w1fLft
        END SELECT
        CALL w1Variables
    END SUB


    ! --- w2 plane data ---

    DECLARE PUBLIC w2Lft, w2Rgt, w2Bas, w2Top, w2Midx, w2Midy
    DECLARE PUBLIC w2fLft, w2fRgt, w2fBas, w2fTop, w2x0, w2y0
    DECLARE PUBLIC w2xFirst, w2xSTik, w2xLTik, w2xLabel, w2xGridstep
    DECLARE PUBLIC w2yFirst, w2ySTik, w2yLTik, w2yLabel, w2yGridstep
    DECLARE PUBLIC w2xPiFlag, w2xMult, w2yPiFlag, w2yMult
    DECLARE PUBLIC w2wWid, w2fWid, w2wHgt, w2fHgt, w2Aspect

    LET w2Flag   = 1                             ! pixel window visibility
    LET w2xPiFlag= 1                             ! pi switch for x axis
    LET w2xMult  = pi
    LET w2yPiFlag= 0                             ! pi switch for y axis
    LET w2yMult  = 1

    LET w2clr = red
    LET w2clr = yellow

    LET w2Lft = workLft+90                       ! pixel window bounds
    LET w2Rgt = w2Lft+wwid
    LET w2Top = w1Bas + 115
    LET w2Bas = w2Top + 128

    LET w2fLft=  0                               ! function bounds
    LET w2fRgt= w1fRgt
    LET w2fTop=  2
    LET w2fBas= -2

    LET w2xAx$= "t"                              ! axis labels
    LET w2yAx$= "f(t) = sin(t)"
    LET w2yAx$= "g(t) = A sin(wt+phi)"

    LET w2xGridstep= 0                           ! horizontal grid intervals
    LET w2yGridstep= 0                           ! vertical grid intervals

    LET w2xSTik =  0                             ! horizontal axis Tik marks
    LET w2xLTik =  2
    LET w2xLabel= 10
    LET w2xFirst= w2fLft

    LET w2ySTik =  0                             ! vertical axis Tik marks
    LET w2yLTik =  1
    LET w2yLabel=  1
    LET w2yFirst= -2

    ! --- w2 Plane methods ---

    DECLARE DEF w2fncx,w2fncy,w2wndx,w2wndy      ! window/function transforms
    DECLARE DEF w2wWithin

    CALL w2Variables

    SUB w2InitPlane
        CALL w2SetBounds
        CALL w2DrawPlane(1,1,1)                  ! x axis, y axis, zeroaxes
        CALL SetTextFont(1,12,"bold")
        CALL PlotTextLJ(w2Rgt+8,w2y0+3,w2xAx$,axislabelclr)     ! axis labels

        LET bas1 = w2Top - 30
        LET bas2 = w2Top - 12
        LET w2Eqx= w2x0 + 28

        CALL PlotTextRJ(w2Eqx,bas2,"f(t) = ",red)
        CALL PlotTextLJ(w2Eqx,bas2,"sin(t)",red)

        CALL PlotTextRJ(w2Eqx,bas1,"g(t) = ",w2Clr)
        CALL StringWidth("A sin(w",sw)
        CALL SwapOmega(w2Eqx,bas1,"A sin(w","w",w2Clr)
        CALL SwapPhi(w2Eqx+sw+2,bas1,"t - p)","p",w2Clr)

        CALL w2KeepGridLayer
    END SUB

    SUB w2SetBounds
        LET w2fLft  = w1fLft                     ! function bounds * pi
        LET w2fRgt  = w1fRgt

        LET w2xSTik = w1xSTik                    ! horizontal axis Tik marks
        LET w2xLTik = w1xLTik
        LET w2xLabel= w1xLabel
        LET w2xFirst= w1xFirst
        CALL w2Variables
    END SUB

    ! ----------- horizontal sliders ------------

    LET slideWidth= 100

    ! --- horizontal slider 1 ---

    DECLARE PUBLIC h1axis,h1wLft,h1wRgt,h1wBas,h1wTop,h1fLft,h1fRgt
    DECLARE PUBLIC h1name$,h1form$,h1clr,h1First,h1STik,h1LTik,h1Label
    DECLARE PUBLIC h1PiAxis,h1Mult,h1fMin,h1fMax

    DECLARE DEF  h1Within                        ! window/function transforms

    LET h1PiAxis= 0
    LET h1Mult  = 1

    LET h1clr   = yellow
    LET h1name$ = "A"
    LET h1form$ = "%.##"
    LET h1Places= 2
    LET h1Click = 0.1

    LET h1axis  = workbas - 25
    LET h1wLft  = w2Lft
    LET h1wRgt  = h1wLft+100

    LET h1fLft  = 0
    LET h1fRgt  = 1

    LET h1STik  = 0.1                            ! short tick marks
    LET h1LTik  = 0.5                            ! long tick marks
    LET h1Label = 0.5                            ! labels
    LET h1First = h1fLft                         ! first tick mark

    CALL h1SliderVariables

    SUB h1InitSlider
        CALL h1DrawSlider(h1name$,a)
    END SUB

    ! --- horizontal slider 2 ---

    DECLARE PUBLIC h2axis,h2wLft,h2wRgt,h2wBas,h2wTop,h2fLft,h2fRgt
    DECLARE PUBLIC h2name$,h2form$,h2clr,h2First,h2STik,h2LTik,h2Label
    DECLARE PUBLIC h2PiAxis,h2Mult,h2fMin,h2fMax

    DECLARE DEF h2Within                         ! window/function transforms

    LET h2PiAxis= 0
    LET h2Mult  = 1

    LET h2clr   = yellow
    LET h2name$ = ""                             ! omega
    LET h2form$ = "%.###"
    LET h2Places= 3
    LET h2Click = 0.05

    LET h2axis  = h1axis - 45
    LET h2wLft  = w2Lft
    LET h2wRgt  = w2Rgt

    LET h2fLft  = 0
    LET h2fRgt  = 2

    LET h2STik  = 0.05                           ! short tick marks
    LET h2LTik  = 0.1                            ! long tick marks
    LET h2Label = 0.5                            ! labels
    LET h2First = h2fLft                         ! first tick mark

    DECLARE DEF h2LstpWithin , h2RstpWithin
    DECLARE DEF h2LmoveWithin, h2RmoveWithin

    LET h2AnimStep= 0.001
    LET h2MoveStep= 0.002

    SUB h2InitSlider
        CALL h2SliderVariables
        CALL h2DrawSlider(h2name$,w)
        CALL SwapOmega(h2wLft-18,h2wbas-2,"w","w",h2Clr)
        ! CALL h2MoveButtons
    END SUB

    ! --- horizontal slider 3 ---

    DECLARE DEF h3Within                         ! window/function transforms

    DECLARE PUBLIC h3axis,h3wLft,h3wRgt,h3wBas,h3wTop,h3fLft,h3fRgt
    DECLARE PUBLIC h3name$,h3form$,h3clr,h3First,h3STik,h3LTik,h3Label
    DECLARE PUBLIC h3PiAxis,h3Mult,h3fMin,h3fMax

    LET h3piAxis= 1
    LET h3Mult  = pi

    LET h3clr   = yellow
    LET h3name$ = ""
    LET h3form$ = "%.##"
    LET h3Places= 2
    LET h3Click = pi/2

    LET h3axis  = h1axis
    LET h3wLft  = h1wRgt + 100
    LET h3wRgt  = h3wLft + 100

    LET h3fLft  = 0
    LET h3fRgt  = 2

    LET h3STik  = 0                              ! short tick marks
    LET h3LTik  = 0.5                            ! long tick marks
    LET h3Label = 1                              ! labels
    LET h3First = h3fLft                         ! first tick mark

    CALL h3SliderVariables

    DECLARE DEF h3Fncx

    SUB h3InitSlider
        CALL h3SliderVariables
        CALL h3DrawSlider(h3name$,p)
        CALL SwapPhi(h3wLft-15,h3wbas-2,"p","p",h3clr)
        LET h3piAxis= 0
        LET h3Mult  = 1
        BOX SHOW h3pi$ at h3wRgt+42,h3wBas-3
    END SUB


    ! ---------- Text Output Rects ----------

    ! --- text rectangle 1 ---

    !    LET t1BasLn = w1Bas + 38
    !    LET t1Lft   = w1Lft + 160
    !    LET t1Rgt   = t1Lft + 150
    !    LET t1Bas   = t1BasLn +  5
    !    LET t1Top   = t1BasLn - 15
    !    LET t1Label$= "_[sum] = "
    !    LET t1Eqx   = t1lft + 55
    !    LET t1vBasLn= t1BasLn + 20
    !    LET t1vBas  = t1vBasLn +  5
    !    LET t1vTop  = t1vBasLn - 15
    !    LET t1clr   = green
    !
    !    SUB t1Label
    !        CALL SetTextFont(1,9,"bold")
    !        CALL StringWidth("tone",sw9)
    !        CALL SetTextFont(1,12,"bold")
    !        CALL StringWidth(" = ",sw22)
    !        LET sw= sw9+sw22
    !
    !        CALL SuperSubScriptRJ(t1Eqx-sw22-2,t1BasLn,"_[sum]",t1Clr)
    !        CALL DrawLam(t1Eqx-sw-10,t1BasLn,t1Clr)
    !        CALL PlotTextRJ(t1Eqx,t1BasLn,"= ",t1Clr)
    !        CALL PlotTextRJ(t1Eqx,t1vBasLn,"= ",t1Clr)
    !    END SUB
    !
    !    SUB t1Set
    !        CALL PlotTextLJ(t1Eqx,t1BasLn,"2",t1Clr)
    !        CALL StringWidth("2",sw)
    !        LET lft = t1Eqx + sw + 1
    !        BOX SHOW t1pi$ at lft,t1BasLn
    !        LET lft = lft + 12
    !        LET txt$= " / ((w_[n]+w)/2)"
    !        CALL SuperSubScriptOmegaLJ(lft,t1BasLn,txt$,"w",t1Clr)
    !    END SUB
    !
    !    SUB t1vSet
    !        CALL t1vClear
    !        IF (1+w)<>0 then
    !           LET t1$= trim$(using$("---%.##",4/(1+a*w)))
    !           CALL PlotTextLJ(t1Eqx,t1vBasLn,t1$,t1Clr)
    !           CALL StringWidth(t1$,sw)
    !           BOX SHOW t1pi$ at t1Eqx+sw+4,t1vBasLn
    !        ELSE
    !           CALL SwapInf(t1Eqx+2,t1vBasLn,"8","8",t1Clr)
    !        END IF
    !    END SUB
    !
    !    SUB t1Clear
    !        BOX CLEAR t1Lft-2,t1Rgt,t1Bas,t1Top
    !    END SUB
    !
    !    SUB t1vClear
    !        BOX CLEAR t1Eqx-2,t1Rgt,t1vBas,t1vTop
    !    END SUB
    !
    !    SUB t1Init
    !        CALL t1Label
    !        CALL t1Set
    !    END SUB
    !
    !    ! --- text rectangle 2 ---
    !
    !    LET t2BasLn = t1BasLn
    !    LET t2Lft   = w1Rgt - 165
    !    LET t2Rgt   = w1Rgt + 20
    !    LET t2Bas   = t2BasLn +  5
    !    LET t2Top   = t2BasLn - 15
    !    LET t2Label$= "_[packet] = "
    !    LET t2Eqx   = t2Lft + 75
    !    LET t2vBasLn= t2BasLn + 20
    !    LET t2vBas  = t2vBasLn +  5
    !    LET t2vTop  = t2vBasLn - 15
    !    LET t2Clr   = cyan
    !
    !    SUB t2Label
    !        CALL SetTextFont(1,9,"bold")
    !        CALL StringWidth("packet",sw9)
    !        CALL SetTextFont(1,12,"bold")
    !        CALL StringWidth(" = ",sw22)
    !        LET sw= sw9+sw22
    !
    !        CALL SuperSubScriptRJ(t2Eqx,t2BasLn,t2Label$,t2Clr)
    !        CALL DrawLam(t2Eqx-sw-10,t2BasLn,t2Clr)
    !        CALL PlotTextRJ(t2Eqx,t2vBasLn,"= ",t2Clr)
    !    END SUB
    !
    !    SUB t2Set
    !        LOCAL z$
    !        IF a-w<>0 then
    !           CALL SetTextFont(1,12,"bold")
    !           LET t2$= "(2/|w_[n]-w|)"
    !           CALL SuperSubScriptOmegaLJ(t2Eqx,t2BasLn,t2$,"w",t2Clr)
    !           LET t2$= "(2/|wn-w|)"
    !           CALL StringWidth(t2$,sw)
    !           BOX SHOW t2pi$ at t2Eqx+sw+4,t2BasLn
    !        ELSE
    !
    !        END IF
    !        CALL t2vSet
    !    END SUB
    !
    !    SUB t2vSet
    !        CALL t2vClear
    !        CALL SetTextFont(1,12,"bold")
    !        LET t2a= 1
    !        IF t2a-w<>0 then
    !           LET t2$= trim$(using$("----%.##",2/abs(t2a-w)))
    !           CALL PlotTextLJ(t2Eqx,t2vBasLn,t2$,t2Clr)
    !           CALL StringWidth(t2$,sw)
    !           BOX SHOW t2pi$ at t2Eqx+sw+4,t2vBasLn
    !        ELSE
    !           CALL SwapInf(t2Eqx+2,t2vBasLn,"8","8",t2Clr)
    !        END IF
    !    END SUB
    !
    !    SUB t2vClear
    !        BOX CLEAR t2Eqx-2,t2Rgt,t2vBas,t2vTop
    !    END SUB
    !
    !    SUB t2Init
    !        CALL t2Label
    !        CALL t2Set
    !    END SUB
    !
    !    ! --- t3 Natural Frequency ---
    !
    !    LET t3NatLft= h3wRgt + 100
    !    LET t3NatBas= h3wBas - 3
    !
    !    SUB t3Init
    !        CALL SuperSubScriptOmegaLJ(t3NatLft,t3NatBas,"w_[n] = 1","w",litgry)
    !    END SUB

    ! ------ t4 text output - initial values -------

    CALL SetTextFont(1,12,"bold")
    LET t4BasLn1 = w1Bas + 38
    LET t4BasLn2 = t4BasLn1 + 20
    CALL StringWidth("x = 55.55",sw)
    LET t4Lft    = w1Rgt - sw - 12
    LET t4Rgt    = w1Rgt - 12
    LET t4Bas    = t4BasLn2 +  5
    LET t4Top    = t4BasLn1 - 15
    LET t4clr1   = litgry
    LET t4clr2   = litgry
    LET t4Label1$= "t = "
    LET t4Label2$= "x = "
    CALL StringWidth(t4Label2$,sw)
    LET t4Eqx    = t4Lft + sw

    SUB t4Init
        CALL t4Clear
        CALL PlotTextRJ(t4Eqx,t4BasLn1,t4Label1$,t4clr1)
        CALL PlotTextRJ(t4Eqx,t4BasLn2,t4Label2$,t4clr2)
        ! CALL t4Values
    END SUB

    SUB t4Clear
        BOX CLEAR t4Lft-2,t4Rgt+2,t4Bas,t4Top
    END SUB

    SUB t4ClearValues
        BOX CLEAR t4Eqx-2,t4Rgt+15,t4Bas,t4Top
    END SUB

    SUB t4Values
        CALL t4ClearValues
        LET n1$= using$("--%.##",t)
        LET n2$= using$("-%.##",x)
        CALL PlotTextRJ(t4Rgt,t4BasLn1,n1$,t4clr1)
        CALL PlotTextRJ(t4Rgt,t4BasLn2,n2$,t4clr2)
        BOX SHOW w2pi$ at t4Rgt+2,t4BasLn1
    END SUB


    ! --- Rollover parameters and cleanup ---

    LET rolft  = w1Lft - 25
    LET rorgt  = w1Rgt + 40
    LET robasln= w1Top - 27
    LET robas  = robasln+ 5
    LET rotop  = robasln-10

    SUB ClearRollText
        BOX CLEAR rolft,rorgt,robas,rotop
    END SUB

    SUB ClearRollover
        CALL w2ShowGraphLayer
        CALL w1ShowGraphLayer
        CALL t4ClearValues
        !CALL ClearRollText
        LET clearflag= 0
    END SUB


    ! --- Buttons ---

    LET zlft= w1Lft
    LET zrgt= zlft  + 50
    LET ztop= w1Bas + 25
    LET zbas= ztop  + 18

    SUB ZoomButton(state)
        CALL SetTextFont(1,12,"Bold")
        CALL DrawButton(zlft,zrgt,zbas,ztop,5,"zoom")
    END SUB

    LET elft= zRgt + 2
    LET ergt= elft  + 78
    LET etop= w1Bas + 25
    LET ebas= etop  + 18

    SUB EnvButton(state)
        CALL SetTextFont(1,12,"Bold")
        CALL DrawButton(elft,ergt,ebas,etop,5,"envelope")
    END SUB

    SUB Buttons
        CALL SetTextFont(1,12,"Bold")
        CALL DrawButton(zlft,zrgt,zbas,ztop,5,"zoom")
        CALL DrawButton(elft,ergt,ebas,etop,5,"envelope")
    END SUB


    ! --- checkbox - overlay ---

    DECLARE PUBLIC cb1Lft,cb1Rgt,cb1Bas,cb1Top,cb1Txt$,cb1Clr,cb1State
    DECLARE DEF cb1Within

    LET cb1Lft = ergt+40
    LET cb1Bas = w1Bas + 41
    LET cb1Txt$= "sin(((1+w)/2)t)"
    LET cb1Clr = cyan

    CALL cb1Variables


    ! --- check boxes ---

    DECLARE PUBLIC c1cnt, c1stp, c1siz
    DECLARE PUBLIC c1lft, c1rgt, c1bas, c1top
    DECLARE PUBLIC c1NameList$(),c1ColorList(),c1Switch()

    LET c1cnt= 4                                 ! box count
    LET c1top= w1Bas + 28
    LET c1stp= 18                                ! box interval
    LET c1siz= 12                                ! box size
    LET c1bas= c1top + (c1cnt-1)*c1stp + c1siz
    LET c1lft= ergt + 40
    LET c1rgt= c1lft + c1siz

    MAT redim c1NameList$(1:c1cnt)
    !    DATA "g(t)","f(t)","sin(((1+w)/2)t)","sin((w-1)t)"
    DATA "","","",""
    MAT READ c1NameList$

    MAT redim c1ColorList(1:c1cnt)
    DATA 24,23,26,28
    MAT READ c1ColorList

    MAT redim c1Switch(1:c1cnt)
    DATA 0,0,0,0
    MAT READ c1Switch

    SUB c1Init
        CALL c1DrawCheckBoxes
        CALL c1SetCheckBox

        LET basy= c1Top+10+0*18
        CALL PlotTextLJ(c1Rgt+8,basy,"g(t)",c1ColorList(1))
        LET basy= c1Top+10+1*18
        CALL PlotTextLJ(c1Rgt+8,basy,"f(t)",c1ColorList(2))
        LET basy= c1Top+10+2*18
        CALL SwapOmega(c1Rgt+8,basy,"sin(((1+w)/2)t)","w",c1ColorList(3))
        LET basy= c1Top+10+3*18
        CALL SwapOmega(c1Rgt+8,basy,"sin((w-1)t)","w",c1ColorList(4))
    END SUB

    !    ! --- radio buttons ---
    !
    !    DECLARE PUBLIC r1cnt,r1stp,r1siz
    !    DECLARE PUBLIC r1lft,r1rgt,r1bas,r1top,r1Name$,r1NameClr
    !    DECLARE PUBLIC r1NameList$(),r1ColorList()
    !
    !    MAT redim r1NameList$(1:4)
    !    MAT READ r1NameList$
    !    DATA "g(t)","f(t)","sin(((1+w)/2)t)","sin((w-1)t)"
    !
    !    MAT redim r1ColorList(1:4)
    !    MAT READ r1ColorList
    !    DATA 24,23,26,28
    !
    !    LET r1cnt= 4
    !    LET r1Lft= ergt + 40
    !    LET r1Top= w1Bas + 28
    !    LET r1Name$= ""
    !    LET r1NameClr= 21
    !
    !    DECLARE DEF r1Within
    !    
    !    CALL r1SetVars


    ! --- Default Parameters ---

    LET a,olda=  1
    LET w,oldw=  1.05
    LET p,oldp=  0
    LET r1num = -1
    LET AverageState = 0

    CALL InitPi(worklft+1,worktop+10,yellow,h3pi$)
    CALL InitPi(worklft+1,worktop+10,litgry,w2pi$)
    LET envelopeFlag= 0
    IF a=1 then LET EnvBtnFlag= 1

    ! --- Draw the screen ---

    CALL InitScreen

    SUB InitScreen
        BOX CLEAR worklft,workrgt,workbas,worktop
        CALL w1InitPlane
        CALL w2InitPlane

        CALL h1InitSlider
        CALL h2InitSlider
        CALL h3InitSlider

        CALL Buttons
        CALL w2DrawGraph
        CALL w2KeepGridLayer
        CALL DrawGraphs(1)

        CALL t4Init

        ! CALL c1Init
        CALL cb1Init

    END SUB


    ! ----------------- Event manager -----------------

    DO
       LET clearflag=  0
       LET oldmx    = -99999
       CALL SetTextFont(1,12,"Bold")
       DO
          GET MOUSE: mx,my,ms
          IF w1wWithin(mx,my)=true then
             CALL Rollover
          ELSE IF clearflag=true then
             CALL ClearRollover
          END IF
       LOOP until ms=2
       IF clearflag=1 then CALL ClearRollover

       IF h1Within(mx,my)=true then
          IF my<h1axis then
             CALL h1MouseClick
          ELSE
             CALL h1MouseDrag
          END IF
       ELSE IF h2Within(mx,my)=true then
          IF my<h2axis then
             CALL h2MouseClick
          ELSE
             CALL h2MouseDrag
          END IF
       ELSE IF h3Within(mx,my)=true then
          IF my<h3axis then
             CALL h3MouseClick
          ELSE
             CALL h3MouseDrag
          END IF

       ELSE IF h2LstpWithin(mx,my)=true then
          CALL h2LstpButtonUp(ms)
          CALL h2LeftStep(w,h2AnimStep)
          CALL h2MouseAction(1)
       ELSE IF h2RstpWithin(mx,my)=true then
          CALL h2RstpButtonUp(ms)
          CALL h2RightStep(w,h2AnimStep)
          CALL h2MouseAction(1)

          !       ELSE IF h2LmoveWithin(mx,my)=true then
          !          CALL h2LmoveButtonDown
          !          DO
          !             GET MOUSE: mx,my,ms
          !             CALL h2LeftStep(w,h2MoveStep)
          !             IF w>h2fLft and w<h2fRgt then
          !                CALL h2MouseAction(1)
          !             END IF
          !          LOOP until ms=3
          !          CALL h2LmoveButtonUp
          !       ELSE IF h2RmoveWithin(mx,my)=true then
          !          CALL h2RmoveButtonDown
          !          DO
          !             GET MOUSE: mx,my,ms
          !             CALL h2RightStep(w,h2MoveStep)
          !             IF w>h2fLft and w<h2fRgt then
          !                CALL h2MouseAction(1)
          !             END IF
          !          LOOP until ms=3
          !          CALL h2RmoveButtonUp
          !
          !       ELSE IF mx>c1Lft and mx<c1Rgt and my>c1Top and my<c1Bas then
          !          CALL MouseUp(mx,my,ms)
          !          LET ChkBox= int((my-c1top)/c1stp) + 1
          !          LET c1Switch(ChkBox)= 1-c1Switch(ChkBox)
          !          CALL c1SetCheckBox
          !          SELECT CASE ChkBox
          !          CASE 1
          !               LET gtFlag= c1Switch(ChkBox)
          !          CASE 2
          !               LET ftFlag= c1Switch(ChkBox)
          !          CASE 3
          !               LET atFlag= c1Switch(ChkBox)
          !          CASE 4
          !               LET btFlag= c1Switch(ChkBox)
          !          END SELECT
          !          CALL DrawGraphs(1)

       ELSE IF cb1Within(mx,my)=true then
          CALL cb1MouseUp(ms)
          LET atFlag= cb1State
          CALL DrawGraphs(1)

       ELSE IF EnvBtnFlag=true and mx>eLft and mx<eRgt and my>eTop and my<eBas then
          CALL MouseButtonUp(eLft,eRgt,eBas,eTop,ms)
          LET envelopeFlag= 1-envelopeFlag
          CALL DrawGraphs(1)

       ELSE IF mx>zlft and mx<zRgt and my>zTop and my<zBas then
          CALL MouseButtonUp(zLft,zRgt,zBas,zTop,ms)
          LET zoomState= 1-zoomState
          CALL InitScreen

       ELSE IF infoWithin(mx,my,ms)=true then
          CALL infoButtonUp(ms)
          CALL InfoPage(Info$)
          CALL InitScreen
       ELSE IF quitWithin(mx,my,ms)=true then
          CALL quitButtonUp(ms)
          EXIT SUB
       ELSE
          CALL MouseUp(mx,my,ms)
       END IF
    LOOP


    ! --- graph window rollover events ---

    SUB Rollover
        IF mx<>oldmx or my<>oldmy then
           LET oldmx= mx
           LET oldmy= my
           SET COLOR litgry

           CALL w1ShowGraphLayer
           SET COLOR litgry
           PLOT mx,w1Bas-1; mx,w1Top+1
           ! SET COLOR green
           PLOT w1Lft+1,my; w1Rgt-1,my

           SET COLOR litgry
           CALL w2ShowGraphLayer
           PLOT mx,w2Bas-1; mx,w2Top+1

           LET t = w1Fncx(mx)/pi
           LET x = w1Fncy(my)
           CALL t4Values
           LET clearflag= 1
        END IF
    END SUB

    ! --- h1 slider events ---

    SUB h1MouseClick
        CALL h1GetClickVal(ms,h1Click,a)
        CALL h1MouseAction(1)
    END SUB
    SUB h1MouseDrag
        DO
           CALL h1GetDragVal(ms,h1Places,a)
           CALL h1MouseAction(2)
        LOOP until ms=3
        CALL DrawGraphs(1)
    END SUB
    SUB h1MouseAction(drawstep)
        IF a<>olda then
           LET olda= a
           CALL DrawGraphs(drawstep)
        END IF
    END SUB

    ! --- h2 slider events ---

    SUB h2MouseClick
        CALL h2GetClickVal(ms,h2Click,w)
        CALL h2MouseAction(1)
    END SUB
    SUB h2MouseDrag
        DO
           CALL h2GetDragVal(ms,h2Places,w)
           CALL h2MouseAction(2)
        LOOP until ms=3
        CALL DrawGraphs(1)
    END SUB
    SUB h2MouseAction(drawstep)
        IF w<>oldw then
           CALL DrawGraphs(drawstep)
           LET oldw= w
        END IF
    END SUB

    ! --- h3 slider events - multiply p by pi in calculations

    SUB h3MouseClick
        CALL MouseUp(mx,my,ms)
        LET mx= Clamp(mx,h3wLft,h3wRgt)
        LET p = h3Fncx(mx)
        LET p = roundn(p,.5)
        CALL h3MouseAction(2)
        CALL h3MouseAction(1)
    END SUB
    SUB h3MouseDrag
        DO
           GET MOUSE: mx,my,ms
           LET mx= Clamp(mx,h3wLft,h3wRgt)
           LET p = h3Fncx(mx)
           CALL h3MouseAction(2)
        LOOP until ms=3
        CALL DrawGraphs(1)
    END SUB
    SUB h3MouseAction(drawstep)
        IF p<>oldp then
           LET oldp= p
           CALL h3Mark(p)
           CALL DrawGraphs(drawstep)
           BOX SHOW h3pi$ at h3wRgt+42,h3wBas-3
        END IF
    END SUB

    ! --- graph drawing routines ---

    SUB w2DrawGraph                              ! addend waves
        CALL w2ShowGridLayer
        SET COLOR red
        FOR wx= w2lft to w2rgt
            LET x = w2Fncx(wx)
            LET y1= f(x)
            LET wy= w2Wndy(y1)
            PLOT wx,wy;
        NEXT wx
        PLOT
        CALL w2KeepGraphLayer
    END SUB

    SUB DrawEnvelope(estp)
        IF w<>1 then
           SET COLOR litmid
           FOR wx= w1lft to w1rgt step estp
               LET x = w1Fncx(wx)
               LET y1= env(x)
               LET wy= w1Wndy(y1)
               IF wy>w1Top and wy<w1Bas then
                  PLOT wx,wy;
               ELSE
                  PLOT
               END IF
           NEXT wx
           PLOT

           FOR wx= w1lft to w1rgt step estp
               LET x = w1Fncx(wx)
               LET y1= -env(x)
               LET wy= w1Wndy(y1)
               IF wy>w1Top and wy<w1Bas then
                  PLOT wx,wy;
               ELSE
                  PLOT
               END IF
           NEXT wx
           PLOT
        END IF
    END SUB

    SUB DrawGraphs(stp)
        ! stp>1 means animation

        CALL w1ShowGridLayer                     ! sum curve (beats- upper plane)
        IF envelopeFlag=1 then CALL DrawEnvelope(stp/2)

        SET COLOR w1clr
        FOR wx= w1lft to w1rgt step stp
            LET x = w1Fncx(wx)
            LET y1= g(x)+f(x)
            LET wy= w1Wndy(y1)
            PLOT wx,wy;
        NEXT wx
        PLOT

        CALL w2ShowGridLayer                     ! addend curve (lower plane)
        SET COLOR w2clr
        FOR wx= w2lft to w2rgt step stp
            LET x = w2Fncx(wx)
            LET y2= g(x)
            LET wy= w2Wndy(y2)
            PLOT wx,wy;
        NEXT wx
        PLOT

        ! ---

        ! when a=1 we need (w+1)/2, average frequency as in the books
        ! when a=0 we need (1+1)/2, 1 cycle per 2pi

        ! when a=1, (w^a+1)/2 = (w+1)/2
        ! when a=0, (w^a+1)/2 = (1+1)/2

        ! LET ft= (w^a+1)/2

        ! LET dif= a * abs(1-((w+1)/2))
        ! LET omega_tone = 1 + a*abs(1-((w+1)/2))
        ! LET ft= 1 + dif

        LET ft= (1+w)/2
        LET fb= abs(w-1)

        IF atFlag=1 then
           SET COLOR cyan                        ! tone frequency
           FOR wx= w1lft to w1rgt step stp
               LET x = w1Fncx(wx)
               LET y1= sin(ft*x)
               LET wy= w1Wndy(y1)
               PLOT wx,wy;
           NEXT wx
           PLOT
        END IF
        IF btFlag=1 then
           SET COLOR magenta                     ! beat frequency
           FOR wx= w1lft to w1rgt step stp
               LET x = w1Fncx(wx)
               LET y1= sin(fb*x)
               LET wy= w1Wndy(y1)
               PLOT wx,wy;
           NEXT wx
           PLOT
        END IF
        IF ftFlag=1 then
           SET COLOR red                         ! simple sine
           FOR wx= w1lft to w1rgt
               LET x = w1Fncx(wx)
               LET y1= f(x)
               LET wy= w1Wndy(y1)
               PLOT wx,wy;
           NEXT wx
           PLOT
        END IF
        IF gtFlag=1 then
           SET COLOR w2clr                       ! addend curve (lower plane)
           FOR wx= w1lft to w1rgt step stp
               LET x = w1Fncx(wx)
               LET y2= g(x)
               LET wy= w1Wndy(y2)
               PLOT wx,wy;
           NEXT wx
           PLOT
        END IF

        IF stp=1 then                            ! keep for rollover
           CALL w2KeepGraphLayer
           CALL w1KeepGraphLayer
        END IF
    END SUB
END SUB

! --- end of Sine sum beats code -------------------