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Listening to the midi clips, 17-et improvisation, Commas as scale steps, Al Farabi's very septimal Dorian, Seven Equal, Strange Times, Subminor and minor, Golden pentatonic, Koto and shakuhachi, Thorvald Kornerup's golden ratio scale, Prinz temperament, 1 2 5 7 Hexany kalimba, 13 equal divisions of 32/3, Improv. in 13-tet, Spiralling hexany, Margo Schulter's two Pythagorean scales at interval of 7/6, 24-tet, David Canright's 13 limit twelve tone, Werckmeister III, 55-tet, j.i. diatonic. 7-tet, Setherised timbre for 7-tet, Improv. in Chopi Scale, 1 3 5 7 Hexany, 17-tet diatonic,

Here are some improvisations in various tunings. Notation: tet stands for "tone equal temperament". E.g. 55-tet is a scale with 55 equally spaced notes per octave. If new to the use of ratios and cents to describe a scale, have a look over the newbie notes for the Tunes page.

These pieces are inspired by the various tunings - the feel of them when improvising. I find this is quite marked, when playing, but notice it far less when listening, especially for the subtler temperaments (temperament = the exact tuning used for the notes of the scale, and the sharps and flats).

In the nineteenth century and earlier, listening out for the qualities of a temperament must have been something musical listeners were quite accustomed to, as the keys varied in their tuning depending on how near or remote one was from C major. However nowadays we aren't that used to this way of listening to Western diatonic / twelve tone music, as all the keys sound the same in the standard 12-tet tuning of modern pianos.

At any rate, it is interesting to hear them in other tunings, such as 12-tet. I think the inspiration of the original tuning carries through when one does this, or can do.

Here is a truly glorious realisation of one of my pieces in 12-tet by Mary Ackerley - Thanks Mary :-).


Mary Ackerley's realisation of the 55-tet improvisation7 for a 'cello voice, - see the Improv. in 55-tet section for more about the piece.

It's quite a large file for those using modems - several megabytes.


17-et improvisation

Just a couple of short improvisations in seventeen equal major.
This is a rather beautiful tuning, it isn't particularly noted for its pure harmonies, but is somehow just a gentle tender lovely tuning. 


Commas as scale steps

The syntonic comma is the interval between the Pythagorean (pure fifths based tuning) interval and the corresponding just intonation interval such as 5/4, 6/5 etc using pure harmonic thirds. The septimal comma is the interval between the Pythagorean interval and the corresponding septimal interval such as 7/6, 7/4 etc.

Syntonic coma as scale step

10/9 9/8 5/4 3/2 5/3 27/16 2/1

Here the syntonic comma is between 5/3 and 27/16 (the just and Pythagorean sixths). See lattice diagram below:

Syntonic coma as scale step eight tones

10/9 9/8 5/4 3/2 5/3 27/16 15/8 2/1

Added a 15/8.

Comma and septimal comma steps

10/9 9/8 5/4 21/16 4/3 3/2 5/3 27/16 7/4 16/9 15/8 2/1

This is a scale by Gene Ward Smith posted to the tuning list. (Mon, 09 Aug 2004 post to tuning at Yahoogroups, subject "Comma as unison vector")

It has many pure harmony chords of 3, 4 and 5 notes.

lattice diagram

Here is a lattice diagram of the first scale - fifths shown horizontally, thirds diagonally, and notes shown up to octave equivalence

            1 ... 3 ...  9 ... 27 
             \  /
5/9 ...5/3 ... 5

So, it only has one pure triad: 1/1 5/4 3/2. Adding the 15/8 gives us another triad to explore.

The standard just intonation seven tone scale is some variant on:

 1/3... 1 ... 3 ... 9  
  \ / \  /  \  / 
5/3. . 5.. .15

There, if you continue one step to the right you reach 27 in the top row, which is a syntonic comma away (up to octave equivalence) from the 5/3 at bottom left. It normally gets left out because normally you don't want to use these tiny scale steps. In temperaments, the third and fifth get put slighty out of tune compared with the pure ratios in order to make the 27 and the 5/3 the same and so allow unlmited modulation without any syntonic comma steps.

The idea here though is to use the syntonic comma scale steps as a feature to be explored.

Note on harmony and melody.

Sometimes when newbies to all this start doing lattice diagrams of scales, they may get the impression that this is a kind of objective measure of a scale - the more triads it has the better it is. There's no doubt, scales with many triads are harmonically interesting, and lattice diagrams are a useful tool to have in ones scale construction tool kit. However there are other factors to consider as well, so it doesn't have to be ones only criterion for making a scale.

It depends on ones aims. Ones criterion may be to design a melodically interesting scale which is something which is less easy to quantise, and even if you decide you want all your chords in your piece to be pure, maybe sometimes one wants a sparse type of harmonisation rather than a rich one with tetrads and quintads. It just depends. In middle ages they were quite happy to use just intonation 1/1 3/2 2/1 chords (or 1/1 4/3 3/2) as the only form of pure harmony - with other chords as dissonances which resolve to the pure harmonies..

If one wants to have the 5/4s, for a richer harmonic basis, adding in a single 1/1 5/4 3/2 or 1/1 6/5 3/2 triad creates a point of interest in a five limit way - and you have a great deal already in the way of harmonic resourses for a still fairly sparse five limit just intonation harmonisation of a melody with just that one five limit chord.

Sometimes of course you have both - rich in harmony and melodically interesting - this is just to say that harmony doesn't always have to be your prime criterion and one shouldn't rule out a scale just because it looks sparse in harmonic resources on paper. A scale may be harmonically sparse, and melodically superb.

Then, sometimes indeed, one wants to use the dischords as well, as in the middle age tunings - to deliberately play an "out of tune" just intonation chord which provides a contrast of course if all the other chords are pure. Or there is just intonation tempering, where you have a few low number just intervals, and other chords are intended as tempered notes just as in a normal tempered scale except that they use ratios rather than cents values for the piches.

(added 19th August 2004)


Al Farabi's very septimal Dorian

Sorry everyone, all the mp3s I had here are mp3s of something completely different. Mary Ackerley's realisation of my Improv. in 55-tet.

I don't know how it happened and seem to have lost the original recording, probably when I did a big clean up of my disk recently and removed loads of .wav files for fractal tune experimental saves in Csound format - some in the same tuning - and didn't back it up (I'm not so methodical about backing up .wavs as c-code). Or it is possible that I just never recorded it at all.

I'll upload another improvisation in this tuning at some point. and put it here.

Here is the description anyway to whet your appetite :-)

very_septimal_dorian_improvisation.mp3 (variable bit rate up to 128 kbit, 4.1 Mb), Same file streamed

[ACTUALLY short section from the start of Mary Ackerley's rendering of my Improv. in 55-tet.]

Here are some smaller versions of it:

64 Kbit, 2.3 Mb, Same file streamed

48 Kbit, 1.7 Mb, Same file streamed

This is played in a symmetrical very septimal Dorian scale. As ratios it is:

1/1 8/7 7/6 4/3 3/2 12/7 7/4 2/1

The whole tone is very wide, and the minor third small, so the step between them is a really tiny semitone (about a third of the size of a normal one) which you hear often in this improvisation.

This scale is very rich in harmonic possibilities, including chords such as 1/1 7/6 4/3 3/2 which consists of the 6th to the 9th consecutive partials of the harmonic series, so a very harmonious chord. Also its inversion is there since the scale is symmetric.

This is a pretty ancient tuning as it was found (amongst other tunings) by Al Farabi, a ninth to tenth century musician and scholar from Turkestan.

My tune Sad very septimal dorian is in this same tuning.

The instrument I'm using here is the FM7 soft synth's Crystal preset, and it is tuned to this tuning using my program Fractal Tune Smithy. I just improvised it and recorded it in real time as I played (as with all the improvisations on this page) - no further processing or overlaying of tracks.

Sorry I don't have a midi clip for this one, as I forgot to record it to midi, just recorded it direct to waveform audio.

You may think there are many instruments being used here, as you hear so many timbres, including maybe brass instrument like ones, and human voice and others, but all the notes you hear here are actually played on the same instrument. It is just that its tone varies tremendously depending on your touch.

16th May 2004.


Seven Equal


Improvisations in this tuning can sound surprisingly like a classical pieces with cadences and everything. Well at least, can do so to my ears - what do you think? Does this sound like a normal triadic piece or does it sound microtonal to your ears?

If you sit down to actually play in this tuning, rather than listen to it, you find all your expectations turned on their heads.

The fifth is no longer a consonance, but a dissonance to be resolved, somewhat like a dominant seventh in flavour. Generally the diads are so rich sounding they sound more like triads. The third is a neutral third half way between a minor third and a major third. That makes the third a moderately consonant interval (11/9), which as a diad stands in good service for a normal triad. Then your cadences, rather than II V I etc. (e.g. D G C) progressing downwards through the circle of fifths become V III I etc. (G E C) progressing through a circle of neutral 11/9 thirds.

Here are some waveform audio clips of them to try using a synthesized rounded triangle wave to bring out the harmonics

V - III - I (seven equal thirds diads)

This is only one take on seven equal I'm sure. Also it sounds classical probably because that is just how it comes naturally when I improvise - don't assume this is the only way seven equal can sound.

This is a good tuning for a newbie microtonalist comoposer - few intervals and notes to think about, and a delightful tuning to play in, an interesting microcosm.

Near seven equal tunings are used in Thailand, and by the few remaining Chopi musicians in Africa.


Strange Times

The idea

for this piece was to use ratios with all the numbers involved odd, and to use a clarinet as the voice because it has only odd partials  

The scale is  

21/19 11/9 9/7 7/5 3/2 5/3 11/6 2/1  

Played on the koto and clarinet voices of my SB Live!, with its metallic room 1 reverb at 50%.  

strange_times.mp3   (32 bit, 405 Kb)  

You can find a higher quality recording of it on my site

I suppose it is probably meant to evoke how terrifying it must be to be an ordinary citizen in Baghdad today (6th April 2003).

Here it is as a midi clip - though the sound probably is rather dependent on what sound card you have:



Subminor and minor


The scale is: 9/8 7/6 6/5 3/2 7/4 9/5 2/1

Subminor refers to the septimal minor third 7/6. This scale consists of two minor triads stacked on top of each other, and two subminor triads on the same roots.


Golden pentatonic


The scale is: 5/4 21/16 3/2 13/8 2/1


Koto and shakuhachi


This is a traditional combination in Japanese music.

Koto is a kind of horizontal harp or zither, and the shakuchachi is an end blown flute, and they go with each other wonderfully.

Again improvising on the voices of my S/w Synth, the default one - I particularly like its koto and shakuhachi.

This music and scale has a kind of timeless quality to it.

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


Thorvald Kornerup 's golden ratio scale

This is a scale with pure major third (or nearly so) and the fifth and major third in golden ratio. Independently invented by Danny Wier, but he tells me that Thorvald Kornerup came up with it much earlier.

improv_in_Thorvald_Kornerups_scale.ra [674 Kb, 32 kbps]

Listening to Real Audio


The complete nineteen tone scale is

1/1 167.525 51.768 219.293 386.818 271.061 438.586 606.11 490.354 657.878 542.122 709.646 877.171 761.414 928.939 1096.46 980.707 1148.23 1032.48 2/1

- the first of the two in his post (tuning post 30054).

Here I use just the diatonic mode of it:

1/1 219.293 386.818 490.354 709.646 877.171 1096.46 2/1

While improvising on this occasion and in this timbre, I felt that the major thirds were the consonances to resolve to and the fourths were comparative dissonances. The third in this scale is very pure, within half a cent of 5/4.


Prinz temperament

This is a temperament with large contrasts between the keys

1/1 256/243 193.157 32/27 5/4 4/3 1024/729 696.578 128/81 889.735 16/9 15/8 2/1

improv_in_prinz.mid, improv_in_prinz2.mid, improv_in_prinz3.mid, improv_in_prinz4.mid

Played on harp MIDI voice.

I found it encouraged chromaticism. The C major scale has a pure major third at 5/4, but the fifth is tempered at 696.6 cents.


1 3 5 7 Hexany, Kalimba voice

This is a scale with no tonal centre, which gives a sense of weightlessness.

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

Played on the African kalimba, or thumb piano.

hexany_kalimba.mid, hexany_kalimba2.mid

See also 1 3 5 7 Hexany


3 equal divisions of 32/3

giants_synth.ra (40 kbps, 325 K, 1 min 6 seconds)

Listening to Real Audio

giants_synth.mp3 (128 kbps, 1 Mb)

This is on the tremulo strings voice of the default S/W synth of my SbLive! sound card. I rather like the way this voice distorts in the extremes of its range.

Note on mp3s


This is in the mode 0 2 3 6 7 9 12 13 of the scale with 13 equal divisions of 32/3. It's the same mode as the next improvisation, but all the steps are three and a half times larger (approx).

So, really huge steps - the "semi-tone" in terms of fingering is a minor third, and it's a non octave scale repeating at 32/3. The thirteenth root of 32/3 happens to be within half a cent of a just intonation minor third at 6/5.



1st minute or so in real audio: 13-tet_dim7th_scale_1st_min.ra [300 Kb]

Complete piece in real audio format 13-tet_dim7th_scale.ra [3 Mb = 13 minutes 28 secs]

You can hear the mp3 at

For the mp3:

This is a very exotic tuning as it has no pure fifths. This minor mode has the fifth more than a quarter tone flat and the fourth more than a quarter tone sharp.

In terms of degrees of 13-tet it is:

0 2 3 6 7 9 12 13

As twelve tone equal note names:

C D- Eb-" F+++' G---" Ab++ B' C

where - = 2/13th of a tone flat, + = 2/13th of a tone sharp, and ' = 1/13th of a tone sharp " = 1/13th of a tone flat. I've written the middle two notes as F+++' and G---" instead of F#--' and F#++" because in the mode they are playing the role of F and G, if one thinks of it as a kind of minor scale.

In cents, its

0.0 184.615 276.923 553.846 646.154 830.769 1107.69 1200.0

Such tunings only sound sweet if you choose the timbre carefully. This one uses the Sitar voice of my SB live! S/w Synth,. which is the one used for the original improvisation, and happens to work well.

The mode includes the diminished 7th 0 3 6 9 13, which is interesting in 13-tet as it is made from stacking three minor thirds at 276.92 cents and a major third at 369.23 cents, instead of the four minor thirds that one is used to in 12-tet.

If you stack seven 13-tet "minor thirds" on top of each other,

0 3 6 9 12 15 18

then reduce into the octave, you get 0 2 3 5 6 9 12 13, which differs from this scale by a single note. It has the three note diminished seventh in five places.

0 2 3 6 7 9 12 13 itself consists of a six note 13-tet diminished seventh 0 3 6 9 12 15 and one other note, the flattened "fifth" at 7, and has the three note diminished seventh in four places.

I recorded it using the Sitar voice of the SB LIve! soundcard Creative S/W synth.

13-tet has no pure fifths at all, nearest notes is at 738.46 cents, a third tone out, and this improvisation uses instead a "fifth" of 646.154 cents, more than a quarter tone flat. However, if you choose the right timbre, 13-tet is a great scale. It is very timbre specific, what sounds sweet on one timbre sounds raucous on another.

I'm not sure why it sounds so nice on the SB LIve! sitar. Even on the SB live, works on the Creative S/W synth and not on the SB Live! Midi Synth 8 Mb soundbank.

One can experiment to see which voice it sounds best on. It sounds pretty strange on (say) the piano timbre.

Here is the midi clip.



Spiralling hexany


Midi instrument: kalimba, or African thumb piano.

This is in the hexany, which has special characteristic of having no tonal centre, so the music never goes anywhere, and has a sense of weightlessness to it.

However, rather than use the six note hexany, I've used a spiralling hexany:

1/1 8/7 12/7 96/35 12/5 32/15 4/3

It's a non octave scale repeating at 4/3, and the pitches goes up and down in waves across the keyboard.

It's 5*7 5 3*5 3 3*7 7/3 5*7/3   where the 3*7 7/3 5*7/3 = 1 1/9 5/9

so next chord is 7/3 5*7/3 5/3  and it continues across the break as consecutive triads.

Also four of the six triads you get from playing groups of three alternate notes are also consonant, and the two remaining ones are quite interesting crunchy ones too.

Though a non octave scale, it has some perfect octaves in it with span in degrees of seven notes (between the 8/7 and the 16/7 of the next turn of the spiral).


Margo Schulter's two Pythagorean scales at interval of 7/6


You can hear the mp3 at

This little plainsong like piece uses two copies of the usual pythagorean scale

1/1 256/243 9/8 32/27 81/64 4/3 729/512 3/2 128/81 27/16 16/9 243/128 2/1

at an interval of 7/6 with each other.

The piece is based on

C C+ D+ E+ G F D C   where + = + 7/6,

which is repeated three times, each time with more elaborations, ending with a little closing phrase to round it off.




This explores use of the "inconsistency" of 24-tet - not really the scale itself that is inconsistent, but if you try approximating the intervals 5/4 and 7/4 in 24-tet, you find that the approximations for 5/4 and 7/5 don't add up to the one for 7/4. So, assumption that one can add approximations for intervals and get the approximation for the multiple of the two is inconsistent for this scale.

This can be used to modulate up and down by a quarter tone, which I'm doing here in this short melody.

Go up by closest approximation to 7/4, then down by closest approx to 5/7, and you end up at 5/4 plus a quarter tone. Do this twice and you can modulate from C to C#.

Exactly that happens in this short example melody - and at the end it drops down to the original C so you can hear how remote it has become by then.

In the opening dom7th chord, notice how the 7/5 interval from the major third to the seventh is a bit "sour". A little later the same interval is played with the third a quarter tone sharp, making it sweeter.

Notes played:

C, E, G, A+, G, A+, C, A+, E+, A+, B+, A+, E+, A+ ,... (where + = a quarter tone sharp).

In this next one I'm just playing around with the dom7th and the E and E+ (where + = quarter tone sharp)



David Canright's 13 limit twelve tone


You can hear the mp3 at

1/1 13/12 9/8 7/6 5/4 4/3 11/8 3/2 13/8 5/3 7/4 11/6 2/1

Uses chord 3/2 + 13/8 + 7/4 or 1/1 13/8 7/4 as the centre and resting point.

This has an unbelievable sadness, from the 13/8, with warmth in it as well.

I think, something of a lament for all those killed in the recent world events.

Score (very free rhythm).

Nwc file (for retuning to the scale).

This is what David Canright says about this scale:

" The final, intriguing tuning goes all the way to a 13-limit (see Table). This includes a complete eight-tone harmonic scale on the 4/3 and a seven-tone one on 1/1. Such scales sound fascinating in their variety, coherence, and newness. However, the other tonalities (except 3/2 minor) are pretty strange."

See his On Piano retuning

I suppose the eight tone harmonic series would be:

1/3 (2/3) 1/1 (4/3) 5/3 (2/1) 7/3  (8/3)  3/2 (10/3) 11/3 (4)  13/3 (14/3) 5/1
1    (2)   3   (4)   5   (6)   7    (8)    9   (10)   11   (12)  13   (14)  15

and the seven tone one on the 1/1:

1/1 (2/1) 3/1 (4/1)  5/1  (6/1) 7/1 (8/1) 9/1 (10/1) 11/1 (12/1) 13/1
1    (2)   3   (4)    5    (6)   7   (8)   9   (5)    11   (12)   13


Improvisation in Werckmeister III

improv_wiii_4.mid Another little improv. in WIII.

In 7/4 most of the time which is a favourite time sig, in division into 4/4 + 3/4.

wtc_bach_improv.mid. Classic well tempering from the time of J.S. Bach, and one possible candidate for the tuning of the Well Tempered Claver.

This scale has a kind of "old well-worn" feel to it when improvising and falls naturally into gentle counterpoints - may just be imagination and association of course, who knows.

1/1 256/243 192.18 cents 32/27 390.225 cents 4/3 1024/729 696.09 cents 128/81 888.27 cents 16/9 1092.18 cents 2/1


Improvisations in 55-tet

55-tet_improv.mid. (rather quiet as I recorded it at half volume).

55-tet_improv7.mid. Here is another one. Here is a written out score for the first page (no more transcribed than this so far - done by editing the midi file. Rhythm is very free).

Here is Mary Ackerley's realisation of it for a 'cello voice, in 12-tet.

The range is perfect for a 'cello (also a favourite instrument of mine). I've no idea how practical it would be to play or how many 'cellos would actually be needed if it was. Two anyway as it has a couple of six note chords in it; perhaps three at times would really be needed.

This uses the 55-tet major scale, plus accidentals.

This is a most delightful major scale (I find). The fifth is just a bit unstable, at 698.182 cents, with a nice major third at 392.727 cents - just a little sharp on 5/4.

The mode is 0 9 18 23 32 41 50 55 in 55-tet, or as steps 9 9 5 9 9 9 5.

As the accidentals I'm using C#, F#, and G# at + 3 steps, and Eb, and Bb = -4 steps.

So as a twelve tone scale it's: 3 6 5 4 5 3 6 3 6 5 4 5.

This scale has extra-sharp major thirds are at 436.364 cents (e.g. steps 6 3 6 5 = 20), and a couple of narrow major thirds at 370.9 cents (steps 5 4 5 3 = 17), and most of the major thirds are at 18 steps = 392.727 cents.

Here is the scale in cents: 0.0 65.455 196.36 305.45 392.73 501.82 567.27 698.18 763.64 894.55 1003.6 1090.9 1200.0

If you want to hear the other five improvisations I did:

55-tet_improv2.mid, 55-tet_improv3.mid, 55-tet_improv4.mid, 55-tet_improv5.mid, 55-tet_improv6.mid

Here is another one I did with equal values for the sharps and flats: C#, F#, and G# at + 4 steps, and Eb, and Bb = -4 steps, giving as the twelve tone scale:

4 5 5 4 5 4 5 4 5 5 4 5.

Or as cents:

0.0 87.273 196.36 305.45 392.73 501.82 589.09 698.18 785.45 894.55 1003.6 1090.9 1200.0


With these new sharps, it has quite a different feeling when improvising. Reminds me of a starry sky.

This 55-tet twelve tone scale is actually a close approximation to sixth comma meantone. It originates from Tosi in 1723.

Each step of 55-tet is very close to a syntonic comma in size (closest approximation of all is one step of 56-tet).

The syntonic comma is the interval you get if you go up four 3/2 major fifths, and compare this with the result of going up a 5/4 major third plus two octaves. It is the same as the interval between the E string of a violin and the similarly pitched E high harmonic of the C string of a cello in a string quartet if the players tune using pure instead of tempered fifths, and is a little over 1/5th of a 12-tet semitone.

This makes a twelve tone scale consisiting of two sizes of semitone, one of five, and one of four approximate syntonic commas. Leopold Mozart (Wolfgang Mozart's father) was in favour of this system and wrote a couple of scales to be used as excerices for it. Teleman advocated it for this same reason.

Ross Duffin here strongly advocates use of sixth comma meantone for baroque music:

On page 4 he mentions that the tritone sounds particularly good in sixth comma meantone, being at the most optimal position in a certain sense, and this links in with the improvisations - I remember noticing that the one from B to F sounded particularly nice and enjoyed using it.

Interestingly, if one follows the logic of the meantone scale itself, it has a syntonic comma of 0 (zero) as Paul Erlich has just pointed out to me on MakeMicroMusic.

The fifth in 55-tet is 32 steps.

So going up four fifths as in c to e'', means 128 steps. Going up two octaves is 110 steps. So the difference between the two is 18 steps, which is the same as the major third of 55-tet.

So what happens to the other major third in the circle of fifths - the one that crosses the break in the circle? (down 8 fifths)

Say, F# to Bb. One gets a sharper third at 414.55 cents.

While in PYthagorean, the one going up four fifths is the one that's sharp at 81/64 = 407.82, and the one going down 8 fifths is flatter and near to the just intonation 5/4 at 8192/6561 = 384.36 cents.


j.i. diatonic

ji.mid (5 mins 15 secs).

Revelling in the wonderful melodic resources of j.i. diatonic.


7-tet (seven tone equal temperament)

7_equal_improv.mid (3 mins 4 secs).

This is exact seven equal. Much use of the 7-tet "neutral diminished seventh".


This uses the complete 7-tet "neutral diminished seventh" consisiting of two of the 7-tet "neutral diminished sevenths" stacked on top of each other to include all of the notes of 7-tet in a seven note chord. You can use it to modulate where you like just as with the 12-tet diminished seventh, but more so.

If you want to hear more:

7_equal_improv2.mid, 7_equal_improv3.mid (5 mins 54 secs),

Exploring some of the wilder chords in 7-tet (oboe voice)


More about the scale:

7-tet can be thought of as result of stacking seven 11/9s on top of each other, and then tempering to remove the Pythagorean comma. (11/9)^7 = two octaves + 31.86 cents - so you need to temper each 11/9 by -4.55 cents. It is exactly analogous to stacking twelve 3/2s on top of each other to get seven octaves plus 23.4 cents, then tempering each 3/2 by 2 cents to get the 12-tet equal temperament.

Tempering the 11/9s to 7-tet keeps them reasonably in tune, but the (11/9)^2, which is fairly close to 3/2 at 694.8 cents instead of 702 cents, when tempered to 7-tet becomes a 685.7 cents fifth - decidedly dissonant in most timbres, with fast and prominent beating. It is especially so when used as a triad with contrast of the relatively pure 11/9 neutral third (neutral = mid way between major and minor). I find the 7-tet triad best thougth of as an incomplete "neutral diminished seventh".

See also the 7-tet trio for violin, viola and glockenspiel (with extra cello part in movement 2) on the tunes page.


Setherised timbre for 7-tet

The dissonance of the fifth in 7-tet arises because of the partials present in the timbre, and one can try "Setherising" the timbre by changing the partials.

Here is an 7-tet with a timbre with the 7-tet fifth instead of the pure 3/2 fifth (actually implemented using the ocarina voice in Midi for the partials).

My sense of rhythm isn't quite good enough to bring it off with a "swing" to it, but maybe you'll hear what I'm getting at:



Chopi Scale

chopi_improv.mid (eleven minutes 18 secs)

See also Chopi Scale on the tunes page.

This scale has seven almost equally spaced intervals rather smaller than our usual whole tones, and has purer fifths than exact seven equal, an uneven temperament, and slightly stretched octaves.

It's most enjoyable and fun to improvise in.

To extend range beyond two octaves, I've left out the last five of the recorded notes, and repeated the resulting two octave scale at 2412 cents.

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

You can hear him playing here: TIMBILA TE VENANCIO

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



1 3 5 7 Hexany

hexany_koto_shak.mid (eight minutes 14 secs)

This is in a musical geometry scale called the hexany. It has a special kind of weightless quality to it, and I'm trying to link into that in this improvisation.

Instruments: Koto and Shakuhachi.

Scale degrees - i.e. numbering the notes starting with 0 for the 1/1

 0     1     2     3      4      5    6


1/1   8/7   6/5  48/35   8/5   12/7   2/1
   8/7  21/20  8/7    7/6   15/14  7/6

Notes go on the vertices of an octahedron. That gives eight triads corresponding to the eight triangular faces. See the beginning of Exploring chords in the Wilson CPS sets.

An octahedron also has four square cross-sections, if you cut it through the middle. These are known as geodesic squares (by analogy with the great circles on a globe). It is well worth while to find these to help with hexany improvisations.

Each pair of identical intervals between scale notes gives opposite sides of a square on the octahedron. So one can see two of them straight away - the two 8/7 intervals will make one of them, and in fact do so as 8/7 6/5 7/8 5/6, and the two 7/6s make another one, as 7/6 5/4 6/7 4/5.

The last one is 7/5 3/2 5/7 2/3, which takes more finding. In terms of scale degrees, the 7/5 is the step from degrees 1 to 4. The 3/2 would then take one to degree 8 in the next octave, or if one looks for the same note in the first octave, one goes down 4/3 from 4 to 2, instead of up 3/2 from 4 to 8.

So, this last square is 7/5 4/3 10/7 2/3, and it joins the scale degrees 1 4 2 5 1

These squares are worth finding, as the remaining two notes for each square then make all eight consonant triads with the consecutive notes of the square.

The special thing about this scale is that there is no special resting point or centre, as all the triads and diads are points of rest. It is one of many scales of this type invented by Erv Wilson, collectively known as Combination Product Sets.

Though the hexany is a small scale with only six notes, it is one that takes a lot of time to get used to - if you have a try at composing in it, or improvising in it, one needn't be disheartened if ones first attempts don't work at all. It takes a fair while to get to know it. But it is well worth the effort!

This is my first hexany improvisation that has worked reasonably well, after many attempts. For my hexany compositions, see Hexany recorder trio, Hexany phrase, but they don't really exploit this wonderful weightless quality of the scale.

Kraig Grady is the master of performance and composition in CPS sets, with many years of experience of it.

See his beautiful A Farewell Ring at the Tuning Punks web site, which is in the 1 3 5 7 hexany too.

Other hexanies include the 1 3 5 11, 1 3 7 11, ...

E.g. the 1 3 7 11 hexany is the same basic idea as the 1 3 5 7, but it has 11s instead of 5s. The diads of the 1 3 5 7 hexany are 3/2 4/3 5/4 6/5 7/6 8/7, and the diads of the 1 3 7 11 hexany are 3/2 4/3 7/6 8/7 11/8 16/11, involving the numbers 3 7 and 11 instead of 3 5 and 7.

More elaborate CPS sets such as the twenty note Eikosany have the same quality that every triad is a point of rest, and have tetrads and pentands as well, and have many component hexanies. So learning the various hexanies is a first step towards exploring these more complex CPS sets.



17-tet diatonic

17_tet_diat_improv.mid (five minutes)

In Seventeen-tone Major scale = steps 3 3 1 3 3 3 1 in the system with seventeen equally spaced notes per octave.

See 17-tet hurdy gurdy player for more about it. This scale is particularly festive and exciting, however can also have a kind of subtle beauty and calm as well, and maybe some of that may come across in this improvisation.

Here is another one if you want to hear more (twelve minutes 50 secs): 17_tet_diat_improv2.mid

For those susceptible to trance, it is poss. one might go into a kind of a trance while listening to my improvisations and relaxing.