Modified Dual Harmonic Scale Tuning

Whereas a tuning is something that is best experienced to be appreciated, what follows is a rather technial approach to describing this particular tuning. It is an experiment or exploration on my part to structure the description in thsi particular way, and what follows is probably best thought of as my working outline of a work in process, which I have put into gramatically correct form as a first step toward a description that will be useful for broader consumption.

For experiencing this particular tuning, the recordings above may be helpful, and no doubt the convey some of its unique qualities. It is another job to connect up that experience with a description such as the one the follows, and for that purpose it would be good to have some short sound samples to go along with the description. If you are interested in this, please ask me for it. See the contact information below.

Meanwhile, broadly stated, among tunings for instruments that produce a fixed number of well-defined pitches, this tuning falls toward one extreme of attempting to make "extended just" harmony available with the "extending" being done at the expense of the freedom to modulate to a lot of different keys. The "extending" is also done with a focus on what Harry Partch called the "otonal" harmonies, those that are based on a series of overtones (harmonics), as opposed to "utonal" harmonics which are based on a series of undertones (subharmonics). The overtone (harmonic) approach creates the possibility for a particular sonic synthesis that results when the pitches of the notes involved fall in a harmonic relationship to some fundamental pitch. If the pitches involved are tuned with sufficient accuracy, the nervous system experiences the sound to some degree as if it were a single note with overtones rather than several separate notes.

I will have to gloss over this point for now, but I should state briefly that typically, regardless of "tuning" a single note will tend to have overtone pitches that can sometimes be discerned as separate pitches embedded within the single "note" being played. Yet the fact that the overtones fall in a harmonic relationship to the original pitch tend to result in the sound being heard as a single note rather than several separate pitches. Some or all of the additional overtone pitches present in a note can be heard distinctly if the listener is aware of that possibility of hearing, but usually the additional overtone pitches fuse into a single experience of the note, and are experienced as the quality of the note rather than as distinct pitches. This is convenient since it permits qualitative judgement of various naturally-occuring sounds rather than a barrage of experiences of independent pure pitches.

In the case of playing more than one finely-tuned note together, when the notes themselves fall in a harmonic scale, this "illusion" of fusion can be extended to some degree so that multiple notes with their respective overtones fuse into a single "virtual" note, a single note that contains the overtones of all the separate notes being played. This illusion is usually somewhat disrupted by an awareness that more than one note is actually involved. This is true because typically the listener is hearing music, in which various notes come and go, typically played on a limited number of instruments, perhaps one single keyboard instrument. Thus notes are heard individually at times and this experience no doubt creates a familiarity with the possible sounds an instrument can make with its various available notes. With such experience the listener is probably able to identify the notes being played in spite of some sense of "fusion" that may also occur. Also tuning is imperfect on a real physical instrument and is likely to be effectively perfect only on in the case of synthesized sounds. Slight deviations from perfect harmonic tuning will result in some "interference" between the overtones present in the various notes being played, and this interference will have some kind of wavering or pulsing quality that results from the various notes with slightly inconsistent tunings "rubbing" against each other, so to speak. Thus the listener may simultaneously experience individual notes, qualities of indivual notes, a synthesis of multiple notes into a unity, and artifacts of deviation from that unity resulting from "imperfect" tuning. In spite of the word "imperfect" I do not mean to imply that this is anything to be regretted. I personally find it to be one of the most interesting possible sonic experiences.

So to create a tuning in which these particular kinds of sonic synthesis are emphasized, it is necessary to create a scale in which the pitches of the notes present fall on harmonics of other pitches. Usually some single useful pattern of such pitches is chosen, and then repeated in every octave. The octave itself is a harmonic (the second harmonic) so this works out well. A scale in which such possibilities for synthesis or fusion of tones are taken toward the furthest practically realizable and perceptible extreme is called a "extended just" scale, and since the relational structure is based on harmonics rather than subharmonics, it is a harmonic scale. Extended just scales tend to permit more notes to be included in chords without sonic conflict. In harmonic scales the synthesis has a particular unity and clarity that sometimes permits the "virtual" note that underlies the fusion to be explicitly perceived. The Dual Harmonic scale is constructed to compromise slightly the notion of "extension" to allow a tiny bit of modulation, maingly between two keys, although the "Modified" notion adds the possibility limited use of 3 additional keys in the case of this particular tuning.

So in short, this tuning is designed to provide the possibility for extended harmonic fusion, but with some limited possibilities for modulation. An extremely irregular pattern of intervals between the pitches of consecutive notes also results as a side-effect of the extended harmonic pitch structure. This irregularity can be used to create a lot of melodic variety.

Now on to the specifics.

Non-technical readers occasionally find things that are interesting to them in the process of reading material that is a bit "over their head". If you find that you are having such an experience and there is something you are beginning to glean that you would like to see developed further or clarified, please let me know.

The upper octaves (Middle C and above) are tuned to a purely constructed dual Harmonic Scale, which is to say two harmonic scales interleaved into the same 12-tone octave. The two scales are rooted in the keys of C and G. The lower octaves are modified because some of the harmonic notes are not useful that low on the keyboard, and thus some notes can be freed up for another purpose.

I have over time used several different variations on the lower-octave modifications, but the tuning described below is the one I have settled on for now as a useful compromise, and a is the one used in the 2nd recording. This tuning gives 13-limit just chords in C and G, which includes the pure major and the rather rare pure dominant seventh chords in these two keys. Altered tuning of 4 black keys in the bottom two octaves offers the additional resource of pure major chords in D and E as long as the 3rd is only played in the lower 2 octaves of the instrument. A furrther tweak of the F in the low octave adds the F dominant 7th with missing 5th giving an additional possibiity for a chord in the bass, but it is only useful as a transitional chord because there are not many possibilities for what can be played along with it melocially, since the rest of the tuning really has nothing in the key of F except for C. The F elsewhere is tuned to the just dominant-7th pitch for the key of C.

In addition the pure subminor is available in the keys of C, G, and D, and the subminor seventh in C and G.

There is also a good blues chord (not strictly just) built from the D major triad plus the unsharpened 7-limit F an octave above, and another interesting chord built from the E major triad with the same F above it.

I will add a full technical description of this tuning and the chords named above when I get the chance. For questions see the contact information below.

There are some tuning terms that I use that need some explanation that I haven't yet provided. For the time being you can find technical tuning terms in the Tonalsoft™ encyclopedia of microtonal music theory.

Background: the Harmonic Scale.

Before getting to the "dual" harmonic scale I'll describe the harmonic scale, with some reference to typical 12-tone nomenclature familiar to western music theory. If you have no music theory background you can probably skip most of this, or skim it quickly.

A Harmonic scale (not to be confused with the term "Harmonic Minor" scale from modern music theory) is a scale whose notes are taken entirely from the harmonic series, which is to say that the frequencies of the notes in the scale are all multiples of some root frequency. A fairly common interpretation of the harmonic scale concept within a single octave is to create a sequence of notes assigned to successive harmonic numbers, usually with the first note of the scale having a harmonic number that is a power of two. The simplest useful harmonic scale is proably the scale in which an octave is built from harmonics 8 through 16, i.e. the scale consists of the following harmonic numbers:

8 9 10 11 12 13 14 15 16

The first note is harmonic number 8, and the last is harmonic number 16. 16 is twice 8, reflecting the fact that an octave equals a factor of 2 in frequency, and therefore in harmonic number, since a harmonic number represents a multiplier applied to a root (fundamental) frequency.

Typical examples of Tuvan throat singing (as I recall) use harmonic numbers 8, 9, 10, 12, 14, 16 to form an octave. The 11 and 13 are probably omitted for aesthetic reasons. Maybe the 15 is present sometimes--I'll have to check, or if you have any information please let me know. The complete sequence of harmonics 8 through 16 offers a lot of interesting variety in terms of melodic potential, because every scale step is a different size--the very opposite of the approach used in modern equal temperament, in which the entire scale can be decomposed into semitones which are exactly the same size. Harmonically, any series of notes from the scale can be combined freely to create a chord that tends to be heard as fusing into a unified sonic experience, with no beating, as is typical for chords in "Just Intonation". The Harmonic Scale is a special case of Just Intonation, and a case for which the quality of fusing comes more readily in a larger variety of chords. A single harmonic scale however, can only be used to create chords in a single key. For example if the fundamental pitch in which the scale is rooted is a C, then all chords constructed will be in the key of C, except for the fact that if the harmonic numbers can be extended indefinitely, it is possible to create chords in an indefinite number of keys, but this is a subtlety that I can't go further into at this point, but let's just say for now that would be "cheating". Without cheating much however, it would be fairly easy to construct the G major triad in a typical harmonic scale in the key of C, just because G will be the 3rd harmmonic of a C note. Beyond that harmonics would have to go far beyond 16 to create common chords in additional keys.

I had noted that every scale step is a different size. The first step in the scale shown above is the step from harmonic 8 to harmonic 9, involving a frequency ratio of 8 to 9. In modern tuning theory the "interval" of this step is commonly notated as 8:9. It turns out that the 8:9 interval is a Just whole step. So if the scale is rooted in C, then the 8th harmonic will also be a C, and the 9th harmonic will be a D. The frequences of C and D will have the exact ratio 9/8, and thus this whole step is a "Just" interval. A Just interval is defined as a interval involving two frequences whose ratio can be expressed as a pair of small integers. The definition of "small" may vary depending on context but the numbers involved will not often exceed 16.

The second scale step is from harmonic 9 to harmonic 10. The interval is thus 9:10, and the ratio of the frequencies of the two consecutive tones forming the "step" is exactly 10/9. 10/9 is just slightly smaller than 9/8, but still qualifies (just barely) as a whole step. Continuing to assume the key of C, the 10th harmonic would fall on the note E,. The sequence, C, D, E thus formed from harmonics 8, 9, and 10 would be a typical interpretation of C, D, and E in a Just Intonation scale rooted in C. Beyond the E however, the harmonic scale deviates from the most typical just intonation scales by the continuing narrowing of the steps as the top of the octave is approached, and by the lack of possibilities for good-sounding triad chords in different keys. Continuing up the scale...

The next step is from the 10th harmonic (on E) to the 11th. The interval is thus 10:11 and the frequency ratio, 11/10 is again narrower. In fact 11/10 is enough narrower than 9/8 that the interval can not be considered a whole step. Yet it is not nrarrow enough to be a near a half step. It is perhaps a wide 3/4 step if you had to name it that way. If you had to place it on a note on the familiar 12-tone keyboard, it would proably go on the F#. It is a bit flat of what would usually be expected for F#, almost a half-semitone flat. The pitch of this F# would be rather unfamiliar, falling just about halfway between what would ordinarily be the F and F# in familiar western tunings.

The next step is from the 11th harmonic to the 12th. The interval is 11:12, and has frequency ratio 12/11, narrower again, almost smack on to what would "measure" as a 3/4 step (or 1.5 semitones). But the 12th harmonic would without argument be assigned to the note G.

The next step is from the 12th to the 13th harmonic. The interval is 12:13 (frequency ratio 13/12), a narrow 3/4 step. This step takes us to the 13th harmonic, another unfamiliar tonality, to my ear even stranger than the 11th harmonic. Convenience would probably put this note on the G#, although it is almost a semitone sharp of a typical G#. Since the 12:13 interval is a little narrower than the 11:12 interval, an the 12 falls on G, this 13 is just a little less sharp of the normal G# than the 11 is flat of a normal F#. Remember the steps keep getting smaller as we go up the harmonic series, since the frequency ratios 9/8, 10/9, 11/10, 12/11, 13/12 etc. are progressively smaller numbers.

The next step is from the 13th harmonic to the 14th, interval 13:14, frequency ratio 14/13, just a little narrower than 13/12 but still distinctly wider than a semitone. The best place to put the 14th harmonic on the familiar 12-tone keyboard is probably on the Bb. It is about a third of a semitone flat of a typical Bb, but probably corresponds to the Bb on common types of bagpipes. Its flatness make the interval from G to Bb not technically a minor third. Thus G-Bb-D would not create a minor triad. However, the G-Bb interval has a name in Just Intonation theory: it is called the subminor third. And the chord G-Bb-D would be called a subminor triad, which sounds probably more sad than a minor triad, but may also be a slightly more consonant chord because it involves lower harmonic numbers than the ones required to construct a normal minor triad, but that is getting ahead of the current topic. This flatter than usual Bb has its other utility in that it can be used to create a pure just Dominant 7th chord in the key of C. In other words, in the scale described, the notes C-E-G-Bb form a just dominant-7th chord, something we rarely hear except in Barbershop, Gospel, and occasional other choral interpretations of a dominant-7th chord. It is natural actually to make the dominant-7th Bb a little flatter than "usual" because this makes that note fall into place in a chord constructed on a harmonic series, which the ear will hear as very consonant (harmonious), a result that the voice will seek to produce since it has the flexibility to do so. However, I think only very well-trained choirs know to do this, and the information is not widely available that this pitch in a dominant-7th context should be about a third of a semitone flat of its "normal" pitch. Apparenty this information is at least informally (or unconsciously) available in the culture of Gospel and Barbershop singing, although I have not had the opportunity to talk to a choir director or singer about this.

However, a fixed-pitch keyboard instrument has no possibility of getting this pitch right without compromising other requirements for the user of that note in the common literature, and thus the dominant-7th chord we are used to hearing is not really a just dominant-7th chord, but rather a concatenation of a minor 3rd on top of a major chord. It takes a subminor 3rd on the top to put the entire chord in the harmonic series so that the chord achieves what might be called its natural fruition. Curiously as a child I never liked the sound of the dominant-7th chord, but over the years I somehow got used to it. Yet Barbershop always did something specially right with this chord, something I also recognized early on, although I had no idea of what was behind this difference.

Note that the dominant-7th chord as described, built as C-E-G-Bb with the special flatter Bb, corresponds to harmonic numbers 8, 10, 12, 14 in our scale. All these harmonic numbers are divisible by 2, so the chord can be "reduced" to lowest terms and expressed in typical modern-just-intonation form as the 4:5:6:7 chord. Note this chord is built from 3 stacked intervals: the 4:5 interval (C to E in our scale), which is a pure major third, the 5:6 interval, somewhat narrower, the pure minor 3rd, and finally the 6:7 interval, narrower yet, the pure just "subminor 3rd". This 4-note chord is buillt from alternate notes in our harmonic scale, thus it makes sense that the chord intervals also get narrower toward the top. But the chord could equally be thought of as being taken from the simpler harmonic scale built from harmonics 4 through 8 to form an octave, and so however you look at it the observation that intervals get narrower as you go up the harmonic scale still applies.

Returning from the Dominant-7th diversion, the next step is from the 14th to the 15th harmonic. This interval is still too wide to be a semitone. However the 15th harmonic would unarguably be assigned to the note B, a pure major 3rd above G.

The remaining step, from the 15th to th 16th harmonic is therefore from B to C and qualifies as a half step, or semitone. So note that our 8 through 16 harmonic scale started with a whole step at the bottom of the octave and ended with a half step at the top.

The "Dual" Harmonic Scale

The dual Harmonic Scale I used is just a way of weaving two 8 through 16 harmonic scales into an octave, one in the key of C, and one in the key of G. You will note that the numbers 8 through 15 (leaving out 16 because it is just the octave of 8) makes a total of 8 notes. To put 2 of these in an octave, you might thing off-hand this would requrie 16 notes, but as was observed above, even the C harmonic scale starts to have some of the resources needed for the key of G, and the result of this numerical synchronicity is that two 8 through 15 harmonic scales can be woven into just 12 notes.

(dual harmonic scale charts to be added here)

"Modified" Dual Harmonic Scale

This is the modification of the lower 2 octaves (below Middle C) to eliminate the 11th and 13th harmonics in the keys of C and G (so 4 notes total), which frees up C#, D#, F#, an G# for other use. The obvious use for F# and G# are to use them to complete the D-major and E-major triads, since the D-A interval and the E-B interval are already pure fifths in the Dual Harmonic Scale.

It seemed most useful to make the lower-octave D#(Eb) be a pure 5th flat of Bb. This gets 3 separate subminor thirds into the octave in 3 different keys C, G, and D, permitting the construction of the subminor 7th chords in C and G. The subminor 7th is simply two subminor 3rds combined at a pure 5th apart. This tuning of Eb also fits in with the special tuning of the low F described below.

The lower-octave C# could have been used for various functions, but I chose to make it a pure 5th flat of the lower-octave G#, permitting the pure just C# minor triad.

In the very bottom octave only, I found the F pitch used in the other octaves not to be useful since I rarely play the inversion of the G7 chord that would utilize it. So I tune that F to be a pure fifth flat of the C above it. Given that Eb is a pure 5th flat of Bb, this permits construction of an incomplete F-Dominant-7th (F7) chord. In numeric notation this chord is the 4:6:7 chord, the dominant-7th chord with the "5" missing.

(scale charts to be added here)