"Pitch, Pi, and Other Musical Paradoxes (a practical guide to natural microtonality)"
by Charles E. H. Lucy copyright 1986-2001
Recipe for homemade (or professionally manufactured) 3- dimensional Physical Modelling spiral for all meantone and similar tunings.
Length of flexible wire (8 to 10 metres) (25 to 30 feet) or 5-8 wire coat hangers.
50 (Elastic) rubber bands.
31 wooden clothes pegs (clothes pins) or 31 tags or sticky labels for pegs.
5 one metre (3 foot) lengths of different colored string or wool..
One cylindrical can.
(For optional coloured note positions) Palette of paints and brushes. Pitch to colour
Hamster Baskets in England produce and sell this spiral for educational,
demonstration, and experimental purposes.
Their version is made from steel and comes with copper Large and iron small fixable clip-on "intervals sticks", to join and stabilise the spiral and scales. Loops are welded onto the spiral, to which you can also attach colours, labels, and/or strings.
Although they are delivered adjusted for LucyTuning, by changing the radius of the spiral, any meantone or similar tuning and its scales can be represented (including Pythagorean).
Directions (for homemade version):
To prepare coat hangers:
Unbend coat hangers. Join ends of straightened wires end to end with tape or weld to provide a 8 to 10 metres continuous length of wire.
Construction from wire: Bend wire around a conveniently sized cylindrical tin (can) to form spiral spring shape of 8-12 inch (200-400mm) diameter.
Position, place or tie 31 elastic bands for markers at intervals of 208.65 degrees around the spiral spring using protractor.
Label thirty-one clothes pegs (pins). One for each note name: Fbb Cbb Gbb Dbb Abb Ebb Bbb Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# F## C## G##.
Attach labeled clothes pegs: Suspend the wire spiral with its elastic band markers by a loop bent from the wire after your last mark. Eg. From a lamp or ceiling. Attach the clothes pegs in the above sequence (at the marks, beginning at the end you started your angle measurements from. (the bottom of the spiral), and ascending by one fifth (208.65 degrees) each step.
There are many possible modifications for this basic design, eg. paint markings, color code pegs. blue tack, or chewing gum instead of elastic bands, although the angle and spiral shape should be precise.
Display various scales
1. Greek Modes and Indian equivalents using only naturals, attach a length of string to join the pegs: C-D-E-F-G-A-B and back to C.
You will notice that the Large intervals move upwards and one radian around the spiral. The small intervals move downwards and 68.45 degrees around the spiral.
Starting at each of the points and listing them in ascending order will give you the seven Greek modes (Indian names in brackets):
starting from C=Ionian (Indian Bilaval Scale); D=Dorian (Kafi); E=Phrygian (Bhairavi); F=Lydian (Kalyan); G=Mixolydian (Khamaj); A=Aeolian (Asavari; and B=Locrian Mode.
2. Sharps and flats Attach second and third strings of different colours in the following sequences:
You will see that you have generated the same shapes, but higher and lower on the spiral and with a small phase shift between them of (2s-L)=54.084" or 16.2252 degrees
(1) Up into the sharp tonalities [G# Major scale (Ionian) i.e. 8 sharps]
(2) Down into the flat tonalities [Ab Major (Ionian) Scale i.e. 4 flats]
3. Hindi Scale Tie strings to connect the following sequence: C-D-E-F-G-Ab-Bb-C
This produces the shape of one of the fundamental Hindi scales. You will notice that it follows the sequence of the Ionian mode up to G but diverges after that as though it was a flat Western scale having three flats. You could consider it as an Ab Major with the Eb sharpened to E in Western terms or as the root of five further modes following the pattern of four adjacent Large intervals and a single Large separated by two separate small intervals. (LLsLsLL).
4. The Mysterious Middle Eastern Interval. East of a line North/South from about where the Iron Curtain used to stand, there is a cultural area which uses an interval absent from Western music. This interval is called the sharp second and is the difference between two Large and one small interval (2L-s). Examples of this interval are found in many Indian, Hungarian, and Romanian scales. You can represent it on your model by joining Ab and B; Db and E; Eb and F#, or any other way which gives the same angles or phase.
I trust these instructions have given you a taste for the geometry of musical harmonics. This is just a beginning. You can build your own new musical scales by connecting the pegs or interval sticks in any way which makes sense to you, and produce new modes, scales, and harmonies. Have fun with it.
For Apple Mac users with OS X this link will download a (work in progress)
program which shows the spiral in 3D which can viewed from various angles.
Clicking on notes will sound the appropriate pitches.
Clicking on notes whilst holding SHIFT will add notes to a chord, which may be heard by hitting RETURN.