Tunes

Kirkwood gaps scale

Midi Clip

NWC file (for retuning and to see the score in NWC)

This is in a scale based on the positions of the Kirkwood gaps in the asteroid belt, due to Gene Ward Smith.

9/8 7/6 5/4 4/3 3/2 5/3 7/4 2/1

It has many just intonation harmonies, for instance if you play any set of three alternate notes in the scale, then the resulting triad is harmoious, making it rather easy to find consonant chords. It also has two pentads (five note chords).

It is an eight note scale with tne notes played from successive white notes on the score, so an octave in this score corresponds to less than a played octave.

Twelve Equal

This is in this unusual ( :-) ) modern scale with twelve equally spaced notes to an octave. The effect is to destabilise the scale so that no key is a preferred root, so you tend to get music that modulates rapidly. So this tune explores that feature of twelve equal. :-) .

(This is a joke entry as the smileys :-) are supposed to indicate, for instance poking fun at the idea that one can explore any tuning in a significant way with a single short piece like the ones on this page)

NWC file (doesn't need retuning)

Historical note

This scale is the basis for nearly all Western music since the early twentieth century. You can also find it in early music - because of its practicality for fretted instruments and perhaps because it sounds more harmonious on lute than on harpsichord, it was used early on for lute music. However most keyboard music up to the early twentieth century was played in tunings with some variation in key colour.

Bach's Well Tempered Clavier is generally thought to have been written to show off well temperaments rather than twelve equal. Well temperaments were used for keyboard instruments in his time. Twelve equal was only used for fretted instruments as far as we know. At least no-one seems to have found any evidence yet of its use for keyboard instruments, though it has been pointed out that it would have been practical to tune it in those days by tuning a lute say, then tune the keyboard to match the tuning of the lute.

Well temperaments are unequal tunings that have some fifths sufficiently detuned from the pure frequency ratio of the harmonic series to allow modulation to any key. Since the notes are slightly unequally spaced, they still have variations in key colour.

Well temperaments superceded mean tones which were used before then because they had many pure harmonic series thirds which permits especially harmonious major third diads. However, in mean tones (which are tuned with the fifths even flatter than twelve equal) one of the keys always has a very sharp wolf fifth combined with a sharp third making its major chord quite unplayable - well least that depends on your composing perspective - anyway it certainly makes it into a dissonance rather than a consonance. So as composers began to modulate to remote keys, meantones were no longer practical. Meantones were popular until quite late for organs because they work particularly well in long sustained chords. Before meantones you had music that treated thirds as a dissonance, only the fifths and fourths were considered pure - so then in those very early days they normally tuned to the Pythagorean tuning (which is fairly close to twelve equal but with pure rather than tempered fifths).

(added 9th December 2004).

Pigmie Scale

The scale is 8/7 8/7 21/16 3/2 7/4 7/4 2/1

The Pigmies have an amazingly rich musical culture. They use unusual musical intervals - and rhythms, and the music they make incorporates songs of animals and birds in the forest.

They can't see far in their jungle world and find their way by hearing - those who have travelled with them report how every now and then they stop completely motionless. Then set off again assuredly in some direction or other - they have heard the particular sound of the wind in the leaves of a tree, or a distant stream - and know where they are. So theirs is a world of sound, and their music is part of the world.

Anyway this uses a Pigmie scale, and is inspired by what little I have heard of their music. I imagine a group of Pigmies singing this and vanishing off into the forest. But of course it's not genuine Pigmie music, just a Western composition inspired by what I've heard of their music.

NWC file (for retuning and to see the score in NWC)

(added 24th November 2004).

Squaring the Octave

The scale this time is 9/8 8/7 7/6 6/5 5/4 4/3 3/2 8/5 5/3 12/7 7/4 16/9 2/1

As steps: 9/8 64/63 49/48 36/35 25/24 16/15 9/8 16/15 25/24 36/35 49/48 64/63 9/8

So all the scale steps are of the form: square / (square -1).

NWC file (for retuning and to see the score in NWC)

The instrument is the vibraphone.

This scale arose from a discussion at yahoogroups tuning, 20th August 2004 subject "Pedagogical question",

Gene Ward Smith pointed out that the sum of the reciprocals of the triangle numbers is 2/1 because

1/tri(1) + 1/tri(2) + .. + 1/tri(n) = 2 - 2/n

where tri(n) = (1 + 2 + ...+ n)

You can prove that easily using the formula tri(n) = n(n+1)/2. You do it by showing that it is true for n = 1 (easy) then from any n you get to n+1 by showing that 2 - 2/n + 1/tri(n+1) = 2 - 2/(n+1), from which the result follows for all n.

So you get an infinite scale which gets closer and closer to 2 as n gets smaller - and as a mathematician would say, the series sums to 2 - gets as close as you like to 2 and never exceeds it.

So then the idea arises of making a scale from these partial sums and you get

4/3 6/4 8/5 10/6 12/7 14/8 16/9 ...

Then, as Gene also pointed out, the scale steps in this scale are all square numbers

4/3 9/8 16/15 25/24 36/35 49/48...

which is easy to show from the 2-2/n formula for the scale.

Just for now, to save many words, let's call a scale step a square step if it is of the form square / (square -1).

So here you have infinitely many square steps mking the octave. But how about a smaller scale that is made entirely of square steps?

Well that's easy because our scale sometimes ends with a square step below the octave, for instance at 16/9. So you can just have

4/3 6/4 8/5 10/6 12/7 14/8 16/9 2/1

Or for a very small one:

4/3 3/2 2/1

But that large 4/3 at the start is awkward, so then the idea was to just invert the rest of the scale to get a symmetric scale including the 4/3 and 3/2, with all the steps square again.

So you get

9/8 8/7 7/6 6/5 5/4 4/3 3/2 8/5 5/3 12/7 7/4 16/9 2/1

As steps: 9/8 64/63 49/48 36/35 25/24 16/15 9/8 16/15 25/24 36/35 49/48 64/63 9/8

where the pattern of steps repeats in reverse order after the central 9/8 step between the 4/3 and the 3/2.

So that''s the scale of this piece.

(added 21st August 2004)

Gene has just pointed out that this is a square scale in another sense.

The nine limit tonality diamond is a way of showing all the simplest pentads (five note chords) in the harmonic and subharmonic series in a single compact diagram..

Then you can find this scale as a square within the diamond:

Here it is

 
                 5/5

             5/6      6/5
               \    /
        5/7     6/6       7/5
               /    \
    5/8     6/7      7/6      8/5
      \    /            \    /
 5/9    6/8      7/7      8/6      9/5
      /    \            /   \
    6/9      7/8      8/7      9/6
               \    /
        7/9      8/8      9/7
                /   \
             8/9      9/8

                 9/9

There the diagonal rows slanting down to the right are overtonal (otonal), e.g. the top row is 5/5 6/5 7/5 8/5 9/5 - pitches 5 to 9 of the harmonic (overtone) series for 1/5.

The rows down to the left are undertonal (utonal), e.g. the top row is 5/5 5/6 5/7 5/8 5/9 - pitches 1/5 to 1/9 of the subharmonic (undertone) series for 5/1.

The other rows show the overtone and undertone series for hamonics 5 to 9, (1/5 to 1/9) based on 1/5, 1/6, 1//7, 1/8 and 1/9 as the root (5/1, 6/1, 7/1, 8/1 and 9/1 for the sub harmonic series)

The same pitches up to octave equivalence in the range 1/1 to 2/1

                 1/1

             5/3      6/5
               \    /
       10/7      1/1       7/5
               /    \
    5/4     12/7      7/6      8/5
      \    /            \    /
10/9    3/2      7/7      4/3      9/5
      /    \            /   \
    4/3      7/4      8/7      3/2
               \    /
       14/9      1/1      9/7
                /   \
            16/9      9/8

                 1/1

- where I've shown all the notes in the range 1/1 to 2/1 up to octave equivalence to make it easier to compare ith the scale. Then the four pentads which Gene mentioned are shown by the lines in that picture - and as you see, they also form a square in the middle. This way of showing the ntoes has repeated pitches so they actually share more than the 1/1, 3/2 and 4/3 in the middle - for instance the top-most one sloping diagonally down to right shares the 1/1 4/3 and 3/2 with the top-most one one sloping down to the left.

(added 23rd Aug 2004).

Very Septimal Dorian echoes

Another piece in Al Farabi's tuning, for Pan pipes and harpsichord this time - the harpsichord unusually just playing a slow melodic line on note at a time.

It is in G septimal minor, and the scale I'm using is 1/1 8/7 7/6 4/3 3/2 12/7 7/4 2/1 - a symmetrical very septimal Dorian scale

NWC file (for retuning and to see the score in NWC)

It exploits the symmetry of the scale by playing the pan pipes tune in inversion - you hear it played forwards, then backwards at double the speed, and this is played twice, then you hear the same thing again in inversion (turned upside down). The tune consists of a single bar of five beats which is what gives it the unusual rhythm - when speeded up to double speed that makes the length of the call plus reversed answer section seven and a half beats.

The harpsichord however goes its own way and doesn't try to follow the symmetry of the rest of the piece at all.

It then ends with a recapitulation of the original unreversed tune with some extra notes added to fill in (or in some cases, blur) the harmony - and the harpsichord eccentrically playing something new once again so it isn't an exact recapitulation - then ends with a short coda of a six beat phrase.

The scale has two slendro diesis scale steps of 49/48, one of which is prominent in the very first phrase you hear. These are fairly small steps, so it fits in with my recent explorations of scales with tiny scale steps on the improvisations page. I'll try composing a piece with syntonic comma steps too.

(added 19th August 2004)

Very Septimal Dorian

Another piece in Al Farabi's tuning, for Pan pipes and acoustic guitar.

It is in G septimal minor, and the scale I'm using is 1/1 8/7 7/6 4/3 3/2 12/7 7/4 2/1 - a symmetrical very septimal Dorian scale

NWC file (for retuning and to see the score in NWC)

Here is another one I did a while back - but I can't seem to find the NWC file for it any more - perhaps I forgot to save it under a new name when making one of the other tunes. However here is a score and the retuned midi clip. Sorry the clip is a bit quiet - put the volume up to hear it.

(added 19th August 2004)

Sad very septimal Dorian

The septimal minor tuning with 7/6 instead of 6/5 for the minor tune is more kind of sad and also more laid back. Here it is in a symmetric version of the minor scale which also has septimal seconds (as are also used in the septimal dominant seventh chord). Those are large seconds so the step from the second to the third is a rather tiny one for a semitone - from 8/7 to 7/6, or 49/48.

It is in G septimal minor, and the scale I'm using is 1/1 8/7 7/6 4/3 3/2 12/7 7/4 2/1 - a symmetrical very septimal Dorian scale

sad_septimal_minor.mid

sad_septimal_minor_src.mid

NWC file (for retuning and to see the score in NWC)

I've made it into a twelve tone scale by adding tunings of the other notes - this is just to help with retuning the unretuned midi. I used 1/1 135/128 9/8 7/6 9/7 21/16 45/32 3/2 14/9 12/7 7/4 15/8 2/1 but you can use any twelve tone tuning with the desired tuning for symmetric G minor.

Unretuned midi clip for retuning to the scale:

(13th May 2004)

My Improvisation in Al Farabi's very septimal Dorian is in this same tuning.

Paul Erlich has just told me about David Canright's harmonic diagram of the tuning (e-mail to MakeMicroMusic at yahoogroups):

See fig 5 for his paper on Harmonic-Melodic diagrams

Lines show the 7/4s and the 3/2 fifths. The septimal minor thirds 7/6 can also be picked out easily diagonally up and to left in the diagram and if you look closely you can also see where the two 9/7s are. This small scale is so rich in harmonic possibilities :-).

Also with the semitones so small, it is sort of between a pentatonic with decorated notes and a septatonic.

Kraig Grady gave a reference to Al Farabi who studied this tuning too - see page 4 here:

http://www.anaphoria.com/xen10pur.PDF

Here is a rather nice musical story I found about him:

:http://www.dovesong.com/positive_music/Turkey.asp

(14th May 2004)

Major and Super Major

Well this is a crazy piece. See what you make of it!

major_supermajor.mid

NWC file (for retuning to the scale)

Scale: 9/8 5/4 9/7 3/2 15/8 27/14 2/1

the 9/7 here is the super major. 1/1 9/7 3/2 is what you get if you turn the sub minor chord 1/1 7/6 3/2 on its head, in similar fashion to the transformation of a minor chord 1/1 6/5 3/2 into a major chord 1/1 5/4 3/2.

Incidentally the repeated little microtonal phrase high on the celeste reminds me of the song of the Chiff Chaff - a common sound in woods in the UK in spring and summer. You may never see it - it is retiring in habit and dull in its colourings, and next to impossible to spot once the trees have leaves on them, but you hear it often - this rather loud and very distinctive song.

While at it, here is a new version of the Minor Subminor - just added a few more notes and made a more definite sounding conclusion to it. Also added a Contra Bass doubling the oboe line to give it a bit of support.

minor_subminor2.mid

NWC file (for retuning to the scale)

Quarter tones best approximation to seven equal

This is in the mode 0 4 7 10 14 17 21 24 of 24 equal. Its the best approximation to seven equal in 24 equal. The inspiration behind it is that the 24 equal neutral third is

also close to 11/9, as is the seven equal neutral third. So maybe this is a way to make a scale which shares the harmonic languages of seven equal and twelve equal.

24_eq_7_approx.mid (played on viola).

NWC file (for retuning to the scale)

Coach drone

I was travelling in a coach from Edinburgh to Glasgow in the UK. I often enjoy the "coach music" that a coach makes as it goes along - I mean just the engine sounds and the wind blowing past it and wheel sounds etc.

This particular one was playing a beautiful drone like an Indian Tanpura.

These are the notes I heard, and I've tried to get some of the quality of the sound - but really it was much subtler and gentler than this. I've added a tune on top of the drone - at the time I felt it would be wonderful to improvise on top of it if one had the coach to oneself :-).

It's in just intonation.

coach_drone.mid

coach_drone.mp3 (32-bit, 478 Kb)

You can find a higher quality recording of it on my mp3.com.au site

NWC file (for retuning to the scale and to see the score)

Minor Subminor

This is a little piece in a seven tone scale made of stacked just minor and subminor triads. A just minor triad is one using the 6/5 which is a little sharp compared with the equal tempered minor third, and a subminor triad just means one involving the "lazy" 7/6 minor third - that's my own pet name for it because it is very relaxed and you kind of ease into it.

minor_subminor.mid

Score (or open in new window)

NWC file (for retuning to the scale)

boffons_with_zany_accompaniment.mid

Jacob Van Eycks Boffons + accompaniment as exercise for just intonation major third

This is a piece by Jacob Van Eyck, the dutch recorder player, bell tuner, and carillion player from the 17th century. He is a firm favourite with recorder players because of his wonderful pieces for the recorder that fit the instrument so well. Also famous in the history of bell making for his part in the development of the modern church bell timbre. At his time he was much famed for his virtuoso playing, which he played to entertain passers by in the churchyard - people came from far afield to hear him.

You can read more about him here: http://www.danlaurin.com/vaneyck.preface.english.html

This piece is based around a repeating sequence of major chords: I, IV, I, V, I, IV, V, I, in the key of G.

The score is actually shown in the key of C, with accidentals for the F# where it occurs. Might be a bit of an anachronism to call it in G major, but that will give an idea. Has a fair number of F naturals in scale passages, with F sharps for leading tone type notes and the major thirds of the II chord.

As a result, it is a wonderful exercise in playing the just intonation major third.

To make it easier, I've added an accompaniment as a series of just major chords. Just for fun I've also added a percussion track in 3/4 played against the 4/4 of the melody. Not the usual 3 against 4, but a part in 3/4 just going its own merry way ignoring the 4/4 bar lines, i.e. 3 bars against 4 (which won't make it easier of course, but fun). That is just for starters, - the percussion gradually gets totally zany and crazy and prob. humanly pretty unplayable, but sort of thing that is easy in a midi file.

boffons_zany_acc_only.mid

Score for the recorder parts only

boffons.htm

NWC file (for retuning to the scale)

Whenever B, E, and F# are played as major thirds, which is most of the time, they need to be flatter than normal to be in tune with the accompaniment. On the recorder, this can be done by a technique that seems to involve varying the amount of turbulence in the breath (see the Recordings page), in which case, they are nearly as flat as you can get them while keeping the volume steady. One can also use finger shading techniques.

Obviously suitable for other instruments as well.

One needs to have I, IV and V pure in key of G, so that corresponds to I,V and II in key of C, so one can use this just intonation scale for the piece:

135/128 9/8 6/5 5/4 27/20 45/32 3/2 8/5 27/16 9/5 15/8 2

One can't have all of I,V, IV and II pure in G in just intonation, but one can have I, V and II pure if there are no IVs.

(sorry, mistake here in previous upload of this page).

hexany_phrase_transformations.mid

Hexany phrase transformations

To find out about the hexany, see Into the third dimensison - musical sculpture on the Musical Geometry pagge.

If you look at the original piece Hexany phrase, you'll see that I say that this may be seed for larger piece later on.

Well, here it is. Gene Ward Smith has developed a technique for transforming the tuning of a tune while keeping the melodic line intact.

When applied to the hexany, it gives 48 variations.

One can change major chords to a minor chords and vice versa by inverting everything. Well in the hexany, there are four types of "major chord", each with its minor chord inversion. This technique transforms all these types of major and minor chords into each other in all possible ways. Some transformations invert the melody, others leave it the same way up, but transformed into a different type of major chord, or whatever.

Here it is played slowly at a steady pace:

Score This shows just the original untransformed tune.

NWC file (for retuning to the scale).

Here it is with continual variations in speed

hexany_phrase_transformations_varying_speed.mid

Here it is again, played faster, and with some tempo variation, breath marks and fermata:

hexany_phrase_transformations_with_tempo_variance.mid

NWC file (for retuning to the scale).

I've made some changes to the original phrase - completed some of the chords as triads, added a bit more counterpoint (which is more interesting when the phrase gets turned upside down), and made sure that all the harmonies are triads of the hexany. The original phrase has some impure harmonies that sound quite nice for unresolved chords in the hexany. These also could be interesting transformed (e.g. it can be nice to play two triads in the hexany together, with an edge shared), but for this first try out, seemed best to keep to the pure triads.

For more about the hexany and other CPS sets, with some models, see the beginning of the section Exploring chords in the Wilson CPS sets.

How to do this yourself

You can transform a tune in this way by making a midi_remap file with same extension as the midi clip. Then retune in FTS using a just intonation scale such as the hexany, (if twelve tone scale, you can try playing simple diatonic tunes in the just intonation scale, but will prob. need to use root control to get the chords right before transforming, for the II chords at least).

Midi Remap file

Original untransformed midi clip for retuning - as saved from NWC. (Not intended for listening to as it is).

Original untransformed midi clip for retuning (faster version with tempo changes)

The midi remap file needs to have same name as the source midi clip, with extension .midi_remap (or alternatively, you can make a file called midi_remap - no extension - to transform any midi clips in the same folder when played by FTS).

So to transform the faster version, here is the same file as before, but saved under same name as the source midi clip for the faster version, with extension .midi_remap.

Midi Remap file for faster version

I'll explain more presently in the help for the FTS beta (not done yet).

7 limit adaptive puzzle

7_lim_adaptive_puzz.mid

The soprano and mezzo soprano parts split into chords occasionally (three soprano parts and two mezzo sopranos - i.e. like a small choir with the sopranos splitting into separate voices at that point).

NWC file (for retuning to the scale). Uses new option for root control in Fractal Tune Smithy, with the cross-set for David Canright's 7 limit just intonation twelve tone scale 1/1 28/27 9/8 7/6 5/4 4/3 45/32 3/2 14/9 5/3 7/4 15/8 2/1, and channel 1 as the root control channel.

5 limit, soft vertical springs, 7-limit tuning, soft vertical springs , 5 limit, rigid vertical springs, 7 limit, rigid vertical springs.

This is a piece that would puzzle a real-time adaptive retuning program. It is exploring various shades of sharps and flats.

It starts off in C major, then suddenly jumps to G# / Ab without any preparation. A series of chords follow leading back to C. Only when that sequence is finished can one work out what the original exact pitch of the G# or Ab should be if one is to keep to the purest possible chords, and end back at the original pitch for the C.

This then happens a second time, with another sequence of chords.

In fact, the first time, the note is an F# at 49/32, in other words, two 7/4s above the C at 1/1. The second time it is an Ab at 49/32, two wide 8/7 whole tones below 1/1 (at least, that is one way to interpret it). So in this one, first time it is an F#, second time it is an Ab, but both times at the same pitch!

A real time adaptive tuning program couldn't know in advance if it was going to be a 49/32 or an 8/5 (the more usual ratio for an Ab), or some of the other possiblities for F# or Ab. The difference in pitch between 49/32 and 8/5 is 76.0345 cents, or three quarters of a semitone.

If the program makes the wrong decision, the pitch will drift, and not just slighlty, but by three quarters of a semitone each time it does it. (Or alternatively, it will have to make some rapid adjustments of pitch of notes after they are played, which prob. won't sound that good). Since this unprepared Ab / G# occurs three times in all, the program could end up over a tone sharp at the end of the tune, if it chooses to play them all as 8/5s!

It's 7 limit, which means it uses seventh harmonic notes (such as 7/4 in ratio notation). These tunings are used in jazz / blues, and occur in many scales from around the world, but are rather rare in Western classical music as normally played / sung.

The root control voice shows which note to use as the root of the scale. E.g. if it shows a C then going up by semitones from C will give David Canright's 7 limit twelve tone scale. If it shows an F#, then going up by semitones from F# will give this scale, and so on.

Here is a log of all the intervals played, including all intervals between pairs of notes:

7_lim_adaptive_puzz.txt

Plain ratios are the notes played, and the ratios between ~s are the intervals of the chords. Can have two or more ratios, e.g. before top note of a triad, shows the intervals between that note and both the lower notes.

E.g.:

49/48 ~3/2~ 49/32 ~12/7, 8/7~ 7/4

A leisure time adaptive retuning program has a better chance of making a sensible retuning of it.

Here are some solutions as found by John deLaubenfels program:

Zip of them all

Soft vertical springs let the intervals of the chords vary while keeping notes comparatively steady in pitch throughout the piece. Rigid vertical springs work the other way - notes vary in pitch, while intervals of chords remain steady.

Here is the original 12-tet, just to show how much a difference the adaptive retuning makes. This piece was originally written in 7-limit rather than 12-tet, and I think it doesn't work too well in 12-tet, especially at the point when all the black notes are sounded together to make a chord.

More about the scale:

The scale is:

1/1 28/27 9/8 7/6 5/4 4/3 45/32 3/2 14/9 5/3 7/4 15/8 2/1

David Canright writes:

"This offers the chance to contrast, for example, a standard minor scale (on 5/4) with a septimal minor on 1/1: 1:1 9:8 7:6 4:3 3:2 14:9 7:4."

The septimal minor also gives a nice septimal pentatonic scale, which I use in this piece:

C Eb F Ab Bb C
=
1/1 7/6 4/3 14/9 7/4 2/1

steps
7/6 8/7 7/6 9/8 8/7

At the end of bar 11, all these notes are sounding together.

The intervals they make are: 9/8 8/7 7/6 9/7 21/16 4/3 and their inversions.

How it gets back to the C from the G#, and from the Ab

Here is the SCALA info for the septimal pentatonic:

SCALA info:   Interval class, Number of incidences, Size:
  1:  1  9/8               203.910 cents  major whole tone
  1:  2  8/7               231.174 cents  septimal whole tone
  1:  2  7/6               266.871 cents  septimal minor third
  2:  1  9/7               435.084 cents  septimal major third, BP third
  2:  1  21/16             470.781 cents  narrow fourth
  2:  3  4/3               498.045 cents  perfect fourth
  3:  3  3/2               701.955 cents  perfect fifth
  3:  1  32/21             729.219 cents  wide fifth
  3:  1  14/9              764.916 cents  septimal minor sixth
  4:  2  12/7              933.129 cents  septimal major sixth
  4:  2  7/4               968.826 cents  harmonic seventh
  4:  1  16/9              996.090 cents  Pythagorean minor seventh
top

Adaptive puzzle

adaptive_puzzle.mid

NWC file (for retuning to the scale). Uses new option for root control inFractal Tune Smithy, with cross-set for just intonation twelve tone scale 1/1 16/15 9/8 6/5 5/4 4/3 45/32 3/2 8/5 5/3 9/5 15/8 2/1, and channel 1 as the root control channel.

adaptive_puzzle_continued.mid, NWC file

Also here it is with an extra section I've added at the end:

This is a piece that would puzzle a real-time adaptive retuning program. It has three versions of the Ab / G#, and the only way to figure out which is to be which is to look several bars ahead (see below).

Starts with pentatonic / diatonic melody in C major. The melody is somewhat inspired by the modal feel of old folk melodies, and the melody could be thought of as in A minor / mixolydian equally well.

Then you suddenly hear a high Ab on the violin. Or is it a G#?

The other instruments join in, and after a while it resolves down in the progression G# (8/9) F# (5/4) D (4/3) G (4/3) C.

So in fact it was a G# at 3^4*5/2^7 (405/128).

(if you look at the score, then at that point the root was F#, which in the twelve tone j.i. scale used is 45/32, and it is 9/8 above that).

Then the pentatonic / diatonic melody repeats, then you hear the high Ab /G# again. This time it resolves in progression A# E C. So it was a 25/16 this time (two major thirds up from the 1/1).

While resolving to the C, it goes down to Ab for a moment to take in a Ab major chord, so we get the 8/5 Ab here as well. Returns to C with a couple of bars in C minor.

Back to the original melody again, then for coda, you hear the high Ab / G# again. If you look at the score, the root at this point is E, so it is the 25/16 G# again. This time the key signature is E major, so the melody gets changed a little, starting from the third degree of the major scale rather than the second (the A# becomes an A natural). It starts resolving back via the E as before. However on the way back, it changes direction twice in quick succession via a couple of diesis shifts, pure minor thirds changing in pitch as they are played. These are highly audible (and meant to be) as melodic shifts with small steps in the melody.

C.f. Margo Schulter's post to MakeMicroMusic "This can be startling to listeners in the 16th century or 21st century, definitely "xenharmonic," and something that got Vicentino very mixed reviews." http://groups.yahoo.com/group/MakeMicroMusic/message/166.

Then returns to the original melody once more, to end the piece.

The new section plays around with the tunes (a phrase from each), in A minor / mixolydian, to bring out this aspect of the original melody. It then ends with the original tune once more, but on this last return, it ends on a single note A, which leaves the whole piece ambiguous between A minor and C major.

Here is a text log of all the notes and intervals played (apart from the new section):

adaptive_puzzle_log.txt

Or, without the repeats:

adaptive_puzzle_log_no_repeats.txt

The plain ratios are the notes, and the ratios between ~s are the intervals of the chords.

So for example:

(24)
5/6 ~6/5~ 1/1 ~5/4~ 5/4 ~2/1~ 5/2
is the A minor chord that opens bar 24, with the 5/6 on the 'cello, the 1/1 and 5/4 on second violin, and the 5/2 on the first violin.

(I've added the bar numbers in brackets before each bar).

The puzzle for an adaptive retuning program

The adaptive tunign challenge is to find the optimal solution that has no overall shift of pitch (and no fudging of the pitches), which a real-time adaptive tuning program couldn't possibly do without looking ahead several bars (i.e. so that you play it with a delay of several bars before the notes sound). A leisure time adaptive tunign program can look ahead no problem, so could do it in principle, but I wonder if it might find it somewhat of a puzzle too?

This puzzle needn't have a unique solution. It would depend on which note one was keeping steady - in this piece the C as the 1/1 is same pitch at the end as at the beginning, but one could choose another note for this. Also some of the chords could be tuned in various ways depending on which pairs of notes one wanted to have as low ratio just intonation.

Answer: John de Laubenfels leisure time adaptive tuning program can cope with it (apart from the diesis shifts of course). Result sounds sweet too, but interestingly, it comes up with another solution.

A comma pump is a sequence of chords that can only be tuned to pure intervals if the pitch of the melody drifts from beginning to end of the note. E.g. might start at C = 1/1 and end up at C = 80/81.

For an example using the first three bars of "God save the King" see RENAISSANCE "JUST INTONATION" - scroll down to The myth of drifting pitch.

This isn't a comma pump, because there is a solution that has no pitch drift, but if the adaptive tuning program makes the "wrong decisions" it turns into a comma pump.

Graham_Breed_blues_scale.mid

Graham Breed's blues scale

NWC file (for retuning to the scale).

Instruments: Oboe and Kalimba (African "thumb piano").

This is in a twelve tone scale Graham Breed recently posted to crazy music.

0.0 133.6 182.4 386.3 449.3 498.0 653.2 680.4 835.6 884.4 947.3 1151.2 1200.0

It's part of a discussion we are having about the best melodic scales.

See post 532 to Crazy music, in the thread "Melodic Scale Design & Golden Flutes".

It's twelve tone in sense of having twelve notes, but with a huge range of step sizes from 27.2 cents (a bit over an eighth of a tone) to 203.9 cents (over a tone). The smallest step is from G to Gb. This piece doesn't actually use the tiniest step from one note to the next anywhere, but you can find it in the Kalimba accompaniment for successive off beat quavers in the triplet that brings one back round to the start of the repeat, and in that pattern it can perhaps be heard as a kind of melodic step, passing by very fast.

Jacky_Ligon_golden_meantone.mid

Jacky Ligon's golden meantone non octave scale

NWC file (for retuning to the scale).

This is in a scale Jacky Ligon recently posted to crazy music.

0.0 75.12 121.54 196.666 318.212 393.332 439.758 514.878 636.424 711.544 757.971 833.090 954.637 1029.756 1076.183 1151.302 1272.849

See post 517 to Crazy music "Melodic Scale Design & Golden Flutes".

piece uses the mode 0 1 5 6 9 12 13 16.

Jacky Ligon has a reputation for making beautiful scales. Music in them more or less writes itself!

2001 A MOS Odyssey is in another of his scales.

Chopi scale

Midi format:

chopi_scale.mid

Played on Kalimba - African thumb piano. Tuned to the exact pitches and range of the original xylophone.

It's in 5/4.

NWC file (for retuning to the scale).

This scale may have come from Thailand via Madagascar. The tuning is from a Xylophone tuned by Venancio Mbembe of the Chopi people.

You can hear him playing in it here: TIMBILA TE VENANCIO

and read about him here, and his instrument the timbala xylophone: FEATURE ON THE TIMBILA XYLOPHONE

Exact seven equal has very flat fifths at 685 cents.

The stretched octaves make the fifths purer, though they aren't stretched as much as would be needed to make them completely pure all the way, (which would make the octave repeat at 1228.42 cents); instead, the double octave is about 2428 cents, and the scale is interestingly uneven with some of the fifths purer than others.

For the original measurements in Herz, see Chopi Scale (pdf file), from the World Scale Depository maintained by Kraig Grady - the scale was collected by Hugh Tracey in the 1940s - for some pictures of Chopi Land, and xylophones there, see the Chopi Recording Trip, Sept / Oct 1998.

11 and 13 flavour 7 note scale

Midi format:

1st movement, 2nd movement.

NWC files (for retuning to the scale).

1st movement, 2nd movement.

Score:

1st movement, 2nd movement.

13_limit_7_tone_mvt1.nwc

Here is a piece in a scale with pure 11/9s. 11/9s are neutral thirds, mid way between major and minor thirds.

scale is 1/1 12/11 11/9 4/3 3/2 13/8 11/6 2/1.

I find the 11/9s kind of very beautiful, but a little kind of cold beauty somehow. While 13/8 I find very warm.

So adding that in helps to make a balanced scale I think.

The 11/9 is very close to mid point between 1/1 and 3/2 as 11/9 * 11/8 = 121/81. This is the 243/242 comma.

So this scale has the 11/9 "diminished seventh" as 1/1 11/9 3/2 11/6.

Ratio from 13/8 to 12/11 is also close to 3/2 - this time, by the 144/143 comma.

Score as zip, Score as Word 7 document.

7 equal trio

Midi format:

1st movement, 2nd movement, 3rd movement, 4th movement

Dynamic range for playback of midi clips seems to vary a fair amount depending what you play them on, and the quiet voices are too quiet when they are played in Quicktime (commonly used midi player in web browsers). Actually, I think there is supposed to be a standard for the decibel range of a midi clip, but if so, it doesn't seem to work or be adhered to too closely.

You can hear the mp3s at http://www.mp3.com.au/RobertWalker

Note on mp3s

NWC files (for retuning to the scale).

1st movement, 2nd movement, 3rd movement, 4th movement

Score:

1st movement, 2nd movement, 3rd movement, 4th movement

mvt 1, p2, p3, p4, p5, p6, p7, p8, mvt 2, p2, p3, mvt 3, p2, mvt 4, p2, p3.

Notes

Instruments: Violin, viola and glockenspiel, joined by cello for second movement.

First movement is in 3/8, has a rather independent glockenspiel part in the middle following its own way with waves of sound, gradually getting louder and louder a bit like the sea, which the other instruments basically ignore, but it sounds okay somehow.

They are joined for the second movement (in 7/4) by a cellist who clearly is a very individual character, and takes a little while to get into the spirit of things. However, has a very interesting idea to contribute, which is repeated over and over, and the others eventually take up on it.

Third movement is a very conventional seeming 4/4 somewhat after style of Haydn. You'd hardly think that the parallel thirds are actually 11/9s (pretty close) and the cadences are III to I (11/9 taking place of 3/2) rather than V to I.

Last movement is in a lyrical 11/4.

About the scale:

7 tone equal temperament (7-tet) has seven equally spaced notes to the octave. Near seven equal scales are traditional in Thailand and Mozambique (see the Chopi scale on the improvisations page).

7-tet is an interesting scale to compose in because it turns all ones expectations on their head.

The fifth is a dissonance with strong beating, as is the fourth. The consonances in this scale are the third, and sixth.

The third is actually a neutral third, halfway between the pure minor third at 6/5 and the pure major third at 5/4. In fact it is almost exactly equal to 11/9, which is why it sounds so nice - 11/9 is a pleasant diad.

I find that in 7-tet one uses a kind of cadence involving movement by 11/9 where normally one would use movment by 3/2. I.e. III to I instead of V to I.

The 11/9 diad is a more complex interval than a 6/5 or 5/4, and I find it is interesting enough to the ear to take the place of a triad in cadences.

The triads in 7-tet are very harsh sounding, if one thinks of them as triads, because of the dissonant fifth with the relatively pure third. However, when you add a fourth note on the top you get a nice kind of diminished seventh type chord, with neutral thirds for the intervals instead of minor thirds. I think of the 7-tet triads as a kind of incomplete neutral diminished seventh.

7-tet is a very easy scale to compose in because all the steps are the same size. So, if you transpose a melodic phrase up or down by one step, or two steps (as for the cadences), or whatever, it will be identical, and then you can repeat that transposition as often as you like. It's not so easy to improvise in I find! However you'll find an improvisation in it on the improvisations page.

Of course, this is just one "take" on 7 equal.

dan_stearns_20tet_diat.mid

Dan Stearn's diatonic in 20-tet

NWC file (for retuning to the scale).

Scale:

steps 3 3 2 3 4 1 4 in 20 tone equal temperament (i.e. 20 notes equally spaced to an octave).

= Dan Stearn's diatonic. This is a scale with a mixture of fairly ordinary type chords and really far out ones.

Instruments: Guitar and Church organ.

For more about it: see his post 598 to Crazy music in the thread "Melodic Scale Design & Golden Flutes".

Andante in Quarter comma meantone

Andante in C Major for Flute and Obbligato keyboard

Here is a movement somewhat in style of C.P.E. Bach and J.C. Bach (sons of J.S. Bach, who wrote music in a simpler style, easier for those at the time to listen to), with a bit of a surprise ending.

Quarter comma meantone has pure major thirds, and reasonable fifths, but one major third in every three is sharp, and one of the twelve fifths is a wolf fifth, which is very sharp and unplayable.

It is usually tuned so that the wolf fifth is in a remote key, so that it is seldom used. Also tuned so that scales like C major and A major have only pure major thirds. At the time, it was thought a reasonalbe trade off, to have one unplayable scale, in order to have pure major thirds in most of the other scales. For this piece, I chose the wolf fifth between D# and Bb.

A bit of history: at the time of J.S. Bach, quarter comma meantone was still a very popular scale, but used less often than before (though still a very prevalent tuning for church organs for long after). Other scales were used that let one modulate more freely to remote scales, such as Werckmeister III; these have somewhat sharper major thirds, closer to the ones we are used to today.

The modern twelve tone equal temperament was used regularly for lutes as it was fairly easy to lay out the frets for it, and it sounds good on a lute. See Margo Schulter's post 25000 to the Tuning List.

However, it was not usually used for keyboards (and doesn't sound so good on a harpsichord because of the prominent fifth partial in the harpsichord timbre), and Bach's well tempered clavier was probably intended for a well tempered scale such as perhaps WIII. Twelve tone equal temperament may be an early development as one can calculate 2^(1/12) as the cube root of the square root of the square root of 2. (Alternatively, one can use the method for finding a^b using logarithms, but this was only available at end of C17).

NWC file (for retuning to the scale).

Score. This score is completely conventional, except for the lyric line instruction to retune to QC meantone - other tunings can be tried as well.

17-tet_hurdy_gurdy_player.mid

17-tet hurdy gurdy player

I've been imagining what it would be like to walk round a street corner and run into a 17-tet hurdy gurdy player.

score (Octave no longer corr. to notated octave on score).

NWC file (for retuning to the scale).

Scale is 17-tet major pentatonic. Here 17-tet stands for seventeen tone equal temperament - in other words you have seventeen equally spaced notes to an octave. The scale most pianos are tuned to nowadays, and for most of the last century, has twelve equally spaced notes to an octave. Before that, pianos were tuned in a variety of ways. The 17 notes are C, C#, Db, D, D#, Eb, E, F, F#, Gb, G, G#, Ab, A, A#, Bb, B, all equally spaced.

This is an ordinary pentatonic scale, but has a very sharp major third, so that one gets a lot of fast beating if one uses that. Even more if one adds in the fifth as well. For this reason, somewhat rare to use it, but it has beautiful interval steps, and depending how it is used, the beating can be rather attractive.

Notes are C D E G A in 17-tet.

The drone is E A d, or if one uses the notation with notes numbered by the octave, so that E4 is an octave above E3, it is E3 A3 D4.

Here I use a drone in consecutive fourths.

There's only one major third in the major pentatonic, between the C and the E. Listen out for the beating whenever the melody plays a C, for instance, in the sustained C of second half of third bar. The beating is fast, so may well sound just like a somewhat rough or slightly sour note, which may appeal if one likes the rough sound of some folk instruments and the hurdy gurdy.

The melody starts and ends on A, and A minor is the home key of the piece, so I suppose it is really in the minor pentatonic scale on A.

Found this interesting page by Ivor Darreg about moods of n-tet:

http://www.furious.com/perfect/xenharmonics.html

"I could explain here that the seventeen-tone system turns certain common rules of harmony upside-down: major thirds are dissonances which resolve into fourths instead of the other way round: certain other intervals resolve into major seconds; the pentatonic scale takes on a very exciting mood when mapped onto the 17 equally-spaced tones, and so on; but I can't expect you to believe me until you hear all this yourself. If you try to play these pieces in another system, it just doesn't work; they lose their punch; the magic is all gone. "

Found it while following up a link from the Huygens-Fokker Foundation biography of him:

http://www.xs4all.nl/~huygensf/english/darreg.html

Day's end song

Here's a little tune I wrote at the end of a rather nice day.

days_end_song.mid

You can hear the mp3 at http://www.mp3.com.au/RobertWalker

NWC file (for retuning to the scale).

Scale:

1/1 9/8 5/4 3/2 9/5 2/1 | Dominant Pentatonic, Shang: China

Instruments: Oboe and Cor Anglais.

At the moment I'm exploring the idea of letting the consonances of a just intonation scale (i.e. one with completely pure intervals) guide ones composition.

Twilight Bells

Here is a little piece in Dan Stearn's 3-7-11 lydian hybrid scale:
1/1 8/7 9/7 7/5 32/21 12/7 27/14 2/1

See post 18474 to the Tuning list.

No idea if this is anything like the way the scale is intended to be used, but for some reason suggests bell like harmonics to me.

twilight_bells.mid

You can hear the mp3 at http://www.mp3.com.au/RobertWalker/TwilightBells

NWC file (for retuning to the scale).

For Rhodes piano.

Hexany recorder trio

Recorder trio playing in the hexany - a musical geometry scale with six notes at vertices of an octahedron.

hexany_recorder_trio.mid

NWC file (for retuning to the scale).

This may be the first of several movements.

Hexany phrase

hexany_phrase.mid

NWC file (for retuning to the scale).

This may be seed for larger piece later on.

2001_a_MOS_Odyssy_scale10.mid

2001 A MOS Odyssey

You can hear the mp3 at http://www.mp3.com.au/RobertWalker

NWC file (for retuning to the scale).

Scale: tenth of ten scales Jacky Ligon posted to the Yahoo groups Tuning List at the start of this year - see post 17744.

Instruments: Guitar patch and Cor Anglais patch - patch = midi synth instrument; the Cor Anglais patch goes too low for a real Cor Anglais.

7 Limit octony lullaby

Here is a little lullaby I did in the 7-limit octony - a musical geometry scale with eight notes at vertices of a cube.

7lim_octony_lullaby.mid

NWC file (for retuning to the scale).

String Quartet

This is a string quartet minature in 12-tet - i.e. twelve equally spaced notes per octave, the standard piano tuning.

Because it is in 12-tet, there's a fair amount of beating / roughness of intervals, however we have learnt to tolerate that, and even to think it sounds nice, and I like this particular piece tuned this way. Doesn't mean however that it can only be played this way - you are welcome to try other tunings too!

1st movement, 2nd movement, 3rd movement, 4th movement.

You can hear the mp3 at http://www.mp3.com.au/RobertWalker

NWC files - don't need retuning this time.

1st movement, 2nd movement, 3rd movement, 4th movement.

Scores: I haven't done them yet.

Especial thanks to Fred Nachbaur for some very helpful criticism of it. Of course he isn't responsible for what I have done with his comments!!

twelve tone equal temperament dominant seventh

Newbie notes

Many of the scales are shown in cents or ratios notation. For an introduction to these notations: cents and ratios

If you want to convert a scale in cents or herz into ratios, see Find the closest ratios for a scale.

The reason ratios are particularly interesting in microtonal work is that they represent intervals between notes in the overtone (or harmonics) series - pure "just intonation" intervals.

For those new to ideas of the overtone series, just intonation scales, and how some of them are constructed from the overtone series, this introduces some basic ideas Harmonics and just temperament (extract from the help for Fractal Tune Smithy).

When you analyse or listen for the component notes of a timbre, you can refer to them as "partials". In many melodic instruments, the partials belong to the overtone series.

The partials (component notes of a timbre) of a string instrument are very loud. In a cello concerto, the partials will at times be louder than many instruments of the orchestra, though they are usually not heard as separate notes unless one listens out for them with a keen ear.

In this clip cello partials, try listening to hear how the note played on the pan pipes continues into the cello note that follows it (like a kind of continuing resonance after the note stops sounding).

You can also try singing the note, and in fact it a nice way to get into playing just intonation is to play a sustained drone note on, say, a cello (or on the cello voice of your synth or soundcard) and then sing the partials in turn, or sing tunes going up and down the harmonic series.

Here is a drone cello note to sing along with.

Cello drone on C2 (lowest note on the cello).

Here is a link to a page with audio clips of the Scottish Bagpipe Bass drone with two of the partials repeatedly turned on / off so that you can hear them in the drone at the end of the clip.

Here is a frequency analysis reconstruction of a 'cello voice from its partials, and as with the bagpipe clips, I've switched each partial on / off so that you hear it sound. Starts with the partial switched on, then switches it off a couple of times, and ends with it on. (You may often hear another note seem to come in when the partial gets switched off - this is one of the partials that is there all the time but becomes more prominent when the other one is removed).

Cello partials switched on/off: 1, 2, 3, 4, 5, 6, 7, 11, 13. - plays C (Midi notation C3 - octave below middle c)

The just intonation major third may sound flat to a classically trained western musician when first heard. On the other hand, to a classically trained Indian musician, the major third of classical Western music may sound extremely sharp on first hearing. Perhaps after singing along with the cello drone you may understand why - Indian music uses drones a lot.

The seventh harmonic may sound very flat to a classically trained western musician on first hearing. It too is used in many types of music (not in Indian music). For instance, it is sometimes used in Blues / Jazz. It is a seven limit note, so called because it is divisible by 7, rather than powers of 2, 3 and 5 as in the numbers used for the five limit just intonation scale. It is used in the seven limit dominant seventh. This gives a wonderfully consonant chord, for those who have the taste for it.

There are two commonly used five limit versions of this note - 9/5 or 16/9, which give the notes used for the two common just intonation five limit dominant sevenths.

Here is a clip of the three dominant sevenths to compare: (each time resolving to the just intonation major third triad).

7/4 dominant seventh

16/9 dominant seventh

9/5 dominant seventh.

The 9/5 one is perhaps less decisive than the other two. To my ear, the 16/9 dominant seventh sounds the most decisive of the three. What do you think?

Here is the twelve tone equal temperament dominant seventh. It can sound quite harsh after the other three.

Here they all are as mp3s - just the dominant sevenths, then a triad to try out the resolution. You may well hear them better this way as the midi clips depend a lot on how your sound card or soft synth plays the instrument.

7/4 dominant seventh

16/9 dominant seventh

9/5 dominant seventh.

Resolve to just triad on 2/3

Resolve to twelve equal triad on -800 cents.

Special note on the scores:

The idea for these scores is to use your notation software to compose as usual. But instead of playing directly through your soundcard, set the output of your composing software so that it plays the score via retuning software. You can notate your score with accidentals if you like, but your notation software doesn't have to be able to play them. Instead, just add a note to the score (hidden if you like) to say what scale you need to use to retune it when you play it - and set up your retuning software to retune your score to that scale whenever you play it.

You can then use all the capabilities of your notation software to their full. It is really easy and intuitive to compose this way - you hear the pitches as you enter the notes on the score. Just as you do when you compose in twelve equal :-). Of course the ideal is a purpose designed microtonal notation software but until such a thing is written, this is the next best thing.

These tunes can be taken as examples of the technique. The midi clips were made by simply playing the scores and recording them to midi in my Fractal Tune Smithy program. You can also pick up your keyboard and try out ideas before entering them on the score, as many composers do - either play your keyboard through your notation software if you normally do it that way - in which case it gets sent on to the retuning program and retuned as desired. Or, you can hook up your keyboard to FTS instead, - you can simultaneously retune your midi keyboard in FTS to the same tuning as is used for the score.

The scores are to be retuned according to the scale shown in the lyric line.

Of course, the easiest way is to just use a twelve tone score and a twelve tone scale for retuning. There are several examples on this page. I will choose a more exotic example however, to show how flexible it is.

Take 7lim_octony_lullaby (midi clip) as an example.

Here is the Score., or open in new window.

Notice that the lyric line shows a scale 1/1 35/32 5/4 21/16 3/2 105/64 7/4 15/8 2/1.

Newbie note: here 15/8 means a pitch that is 15/8 times the pitch of the first note of the scale, the 1/1. See cents and ratios. So for ex. if the 1/1 was concert pitch middle c at 261.63 Hz, 15/8 will be 490.55 Hz, which is the b above middle c.

This scale has eight notes to an octave. It is known as the hexany because it has six interlocked triads in an eight note scale.

It is a bit awkward to fit with the seven white and five black notes of a twelve tone scale, which is why I give it as an example.

One could choose to use one of the black keys for one of the eight notes, and just extend it to a twelve note scale, or repeat notes for the other black keys. But there is no particularly natural way to choose which notes to use as the black or white keys - and also as you have eight notes, then no matter what mapping you use, you can't show it using a traditional key signature so will need to keep inserting accidentals.

Another solution is to simply play the notes from successive white keys, ignoring the black ones, as if the white keys were successive strings of a harp tuned to this scale.

The score here is then a kind of scordatura score for use with a retuned keyboard. A keyboard player can read it and play the notes using the standard eye to fingers coordination - though the actual pitches played are not as expected.

Basically you have a choice - to make the score corresponds as closely as you can get it to the expected pitches - but with awkward accidentals and some notes still far away from the actual notated pitches.

Or you can make it easy to play but make it into a scale which doesn't repeat at the keyboard octave,which is how I have done it here.

I most often use this method, so the convention I use for the scores is that if nothing is said otherwise, you play the scale from the white keys in this fashion.

The octaves correspond to these notes on the keyboard.:

G'' = 1/8, A' = 1/4, B = 1/2, c = 1/1, d' = 2/1, e'' = 4/1, f''' = 8/1.

Let's suppose it was played on a zither which only has strings for the white keys - or possibly a small diatonic folk harp if it can be retuned to this extent. Then the player could play this score without any special training on how to read it, just as she would play any other harp piece.

When the harpist plays what she would usually think of as the A' string for instance, it will sound the 1/4 below the c string, and so play a C' instead of an A'. So the pitch heard may be a surprise :-). However you have the same hand / eye coordination as before, and can read the score with no more training, so it should be very easy to learn to play on any fixed pitch instrument such as harp, marimba, keyboard instruments etc.

So this type of score may also be of interest if you have made a custom built fixed pitch musical instrument in some scale which you want players to use. This may be the easiest kind of score for your players of your instrument to learn to read if they are classically trained, and have limited time in which to learn a new notation system.

One can also retune ones midi keyboard to the scale, using a midi relay program such as my Fractal Tune Smithy . If one does that, one can play from the score just as one would from any other score. Again, all the keys will be retuned accordingly, so that the key which usually plays c' will instead play the 15/8 above c , and the key which usually plays the G'' will instead play a C'', etc. And again a classically trained keyboard player will be able to play from your score instantly with no special training.

So basically, all these scores are playing scores - you retune the midi keyboard in FTS (say), then play the score exactly as it is written, and because of the way the notes are retuned, it will sound as intended.

You can also follow along reading the scores as the piece plays. If you want to comment on a particular chord or note in a score, it seems to be no great handicap to use a scordatura score like this rather than an ordinary one, you can still follow the notes, and say - I like that chord at the start of bar 23 or whatever. The only drawback really is that you can't use your previous experience of twelve tone scores to anticipate what the chord will sound like - but you can follow the notes easily enough and can get to learn what to expect to hear too.

The one big drawback of this approach is - if you want to make a score with microtonal accidentals as are needed for wind players, string players etc etc, then you will need to do that as a separate score - unless your piece can be notated in twelve tone and retuned like that.

There is no way to automatically convert these playing scores to ones with accidentals. If your piece requires both type of player, then you will need both types of score. But you can compose using this method and then convert your composition by hand to the other type of score. A truly microtonal sequencer would enable one to do that conversion automatically with a press of a button. Well wait for that whenever it comes - will be great of course!

Ths scores on this page are for printing out (or viewing on very high resolution screen, e.g. 1600 pixels wide).

The dynamics / tempi are more precise than they need to be - every detail has to be spelt out for the midi clip. So, if you want to play them, or make mp3 realisations of them, feel free to be flexible with timing and dynamics.

I used NoteWorthy Composer, which is a great program, very popular with amateur composers, and I think deserving the attention of the professionally trained ones too. But this method can be used with absolutely any notation software or sequencer program. Don't feel you have to use NWC to use it - it is ideal for some but depends on your needs, some composers use it, and many amateur musicians because it is easy to enter the notes, well easier than with some of these programs. Many composers use Sibelius, or Finale, and you can use the same method with them.

Here is a little eulogy about NWC and how nice I find it to use.

I used it for both the scores and the midi clips, using the output of the NWC playback relayed through my Fractal Tune Smithy program for the midi clips.

You can also open the NoteWorthy Composer files using NoteWorthy Player or Noteworthy Composer and print the scores out from there.

The NoteWorthy Composer files can also be played through midi relaying software retuned according to the scale shown in the lyric line. I use Fractal Tune Smithy; any midi retuning program can be used.

For more details about how this works, see the Midi Relaying help for FTS.

For list of microtonal groups at Yahoo: Tuning2

The one particularly devoted to practical microtonality is: MakeMicroMusic.