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Here are some tunes I've written; many of the recent ones being in microtonal scales. Some new, some from other cultures or the past.
Newbie notes, Listening to the midi clips, Special note on the scores, Jacob Van Eycks Boffons + accompaniment as exercise for just intonation major third, Hexany phrase transformations, 7 limit adaptive puzzle, Adaptive puzzle, Graham Breed's Blues scale, Jacky Ligon's golden mean-tone non octave scale, Chopi Scale, 11 and 13 flavour 7 note scale, 7 equal trio Dan Stearn's diatonic in 20-tet, Andante in C Major for Flute and Obbligato keyboard, 17-tet hurdy gurdy player, Day's end song, Twilight Bells, Hexany recorder trio, Hexany phrase, 2001 A MOS Odyssey, 7 Limit octony lullaby, String Quartet.
For those new to the idea of ratios and cents describing a scale, this introduces the terminology:
Here is a page for those who want to convert a scale in cents or herz into ratios Find the closest ratios for a scale
If new to ideas of the overtone series, just temperament 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).
See also Why two notes of the harmonic series sound good together.
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).
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).
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.
The scores are to be retuned according to the scale shown in the lyric line.
Take the 7 Limit octony lullaby as an example.
Here is the Score.
It is played in the scale 1/1 35/32 5/4 21/16 3/2 105/64 7/4 15/8 2/1 which has 8 notes. These are played on the white notes, as if they were successive strings of a harp (say), retuned to this scale.
c = 1/1, d = 35/32,... c' = 15/8, d' = 2/1, e' = 35/16,...
(If you are completely new to all this, 15/8 in this notation means a pitch that is 15/8 times the pitch of the 1/1. So for ex. if the 1/1 was C at 261.63 Hz, 15/8 will be 490.55 Hz which is a B).
So, as an eight note scale played on the white notes, octave of the score = four lines + four spaces instead of the usual four lines plus three spaces.
So the octaves are:
G'' = 1/8, A' = 1/4, B = 1/2, c = 1/1, d' = 2/1, e'' = 4/1, f''' = 8/1.
One could think of this score as a kind of tablature for a harp - a harp which can be tuned to any pattern of pitches one likes. Then each note will correspond to a string.
A harpist could read this score exactly as it is, just as she would play any other harp piece. If playing a harp retuned in that way, it would sound with the new pitches. When the harpist plays the G'' string, it will sound the 1/4 below the c string, and so play a C' instead of a G''.
One can also retune ones midi keyboard to the scale in this same way, 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.
So basically, all these scores are playing scores - retune the midi keyboard, then play the score exactly as it is written, and because of the way the notes are retuned, it will sound as intended.
Quite useful also for following along too, and I find when composing that one soon gets used to the new layout of the pitches.
Scores 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.
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, especially, Retune the staves of a score editor.
For list of microtonal groups at Yahoo: Tuning2
The one particularly devoted to practical microtonality is: Crazy music. Because of a regular poster to Crazy music who, though an interesting writer on microtonality, often uses language and metaphors that are offensive to some, a new moderated group has recently been started called MakeMicroMusic. Crazy music continues unmoderated. However it is prob. not suitable for young children ???!! :-)
The Microtonal Activists e-zine:
http://www.geocities.com/jacky_ligon/TMA_archive_index.htm
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_with_zany_accompaniment.mid
Score for the recorder parts only
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).
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:
hexany_phrase_transformations.mid
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).
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).
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 the Fractal Tune Smithy beta preview, 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.
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. In fact, first time it was an F#, but at
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, as two wide 8/7 whole tones below 1/1 (at least, I think 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:
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:
5 limit, soft vertical springs, 7-limit tuning, soft vertical springs , 5 limit, rigid vertical springs, 7 limit, rigid vertical springs.
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
NWC file (for retuning to the scale). Uses new option for root control in the Fractal Tune Smithy beta preview, 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.
Also here it is with an extra section I've added at the end:
adaptive_puzzle_continued.mid, NWC file
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):
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 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.
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
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.
Midi format:
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.
Midi format:
NWC files (for retuning to the scale).
Score:
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.
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
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.
Score as zip, Score as Word 7 document.
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.
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.
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".
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.
Andante in Quarter comma meantone
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.
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
Here's a little tune I wrote at the end of a rather nice day.
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.
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.
You can hear the mp3 at http://www.mp3.com.au/RobertWalker/TwilightBells
NWC file (for retuning to the scale).
For Rhodes piano.
Recorder trio playing in the hexany - a musical geometry scale with six notes at vertices of an octahedron.
NWC file (for retuning to the scale).
This may be the first of several movements.
NWC file (for retuning to the scale).
This may be seed for larger piece later on.
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.
Here is a little lullaby I did in the 7-limit octony - a musical geometry scale with eight notes at vertices of a cube.
NWC file (for retuning to the scale).
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!!
