Intonalism: the way forward


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I would like to name two paths forward:  “intonalistic serialism” and “intonalistic pitch class sets”.  


For the first, I can imagine a new serialism where a) there are more than 12 notes, b) certain intervals are allowed (those that can be tuned) and others disallowed (those that can’t be tuned).  The question of the fifth to minor seventh (untunable as used in traditional dominant-tonic harmony) is left for later.


For the second, it would be great to develop some of the tools used in manipulation 12-member pitch class sets (and sets with fewer members) with regard to the tuning of the individual elements, and perhaps extend to sets of greater numbers.   As a first step, perhaps, the 8 note diatonic set of tonic, low supertonic, high supertonic, mediant, subdominant, dominant, submediant, leading tone.  Then the 10 note set of tonic-dominant harmony.  usw.

Description of intonalism


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I attempt to describe better what it is that I do:

Each and every note for orchestra and chorus is assigned a specific intonation, such that through a major work like my Moses at the Jordan River, approximately 50-60 distinct pitches are used.   As a rapid summary example, for the note C# on a piano, the score might use a high Db, a low Db, a high C# quite distinct from the Db, a low C#, and a two-comma flat double low C#.  For the most part, however, these pitches are tuned in pure relation to fairly traditional harmonic structure and come naturally to the naive singer and to a good musician.   The special rules of Intonalism guide the composition of every melodic line such that every interval may be tuned perfectly.  This is not to imply that a normal variation in pitch and even outright mistakes during performance will spoil the work in any way, more than they would spoil any other composition.

Josquin “Ave verum corpus” a5

Yet another Ave Verum Corpus:

Free pdf edition of Parts I-II of Josquin’s five part “Ave verum corpus” now available from my Hartenshield Music website. In my opinion, this setting is better than the very good Mozart and Byrd settings, and completely within the capabilities of a good amateur choir.

(Another version, with old notation, is available from, by C. H. Giffen, and I consulted his edition a great deal.)

I tried to explain as clearly as possible the intonation markings used, in an appendix, and footnoted places where the intonation implied musica ficta.

Pdf score of the three parts at

Part 1:
Part 2:
Part 3:

Anyone that tries this with their choir, I’d love to hear about it!

Mozart K618 Ave Verum Corpus, final

Here is the complete pdf file for the Mozart Ave Verum Corpus, k618.   Earlier I posted an analysis phrase by phrase, but never got around to putting it all together into one score and one recordings.  


A full recording on soundcloud here

Pdf here .    Let me know if an orchestra-only recording, or individual tuned voice parts with orchestra, would be of interest, they’d be relatively easy to create now that the whole piece is done.  

Tuning Byrd “Ave Verum Corpus”

Update: remainder of first section below as well; text underlay copying error corrected; two intonation markings corrected. I have received comments like “I can’t hear the difference”. First, it’s likely to be your equipment. Music can be enjoyed even if half-heard, in the car, say. But you need to have a reasonably good sound output and a quiet environment. Ask yourself, if this is you, if you can hear the four voices clearly. If not, that’s a pre-requisite to hearing the tuning.

Update 2: complete score and recording now.

Phrase one of the Byrd gave me some trouble.    But when I finally figured it out, the solution was elegant: a kind of raising of the Host in intonational form.  Did Byrd intend this?  

Intonation for Byrd Ave verum Corpus

Please see the pdf and listen from a soundcloud link.

It is a synthesis. I have in mind trying to match the expression in the Sixteen recording, while keeping the tuning perfect.

In Phrase 2, Byrd does the intonational elevation again, this time not a full comma, but a few twists along the chain-of-fifths: at “on the cross”.

Another update: the remainder of the first section. Will Byrd lower the overall intonation in a parallel for “esto nobis”? He will! New recording, same link. New score showing intonation through the first section, link above.<

Earlier posts had the symbol definition reference.

Adam W kind enough to provide a link to a recording that sounds to me as if they agree: the sixteen

Yet another update: on “unda fluxit” I felt it necessary to correct the Es in the alto and bass to E natural. On the Sixteen recording, the alto sings a very high Eb (tuned to the C above) then has to slide around to find the low D (tuned to the Bb above). I believe it more natural to use a low E natural, which is, in fact, not too much sharper than the high Eb. I put an asterisk in the revised score. Footnote to come. If you care to compare by ear, it is at about 01:14 in my current recording, and about 01:23 in the Sixteen video. Another approach is by the Tallis Scholars . I don’t like their tuning at all at any point in the piece: here, (01:15) they sing the Eb low as if it were a modern seventh of a dominant seventh.

Simon W pointed out that my markings imply Bb as tonal center. No, it’s overall g minor. I don’t yet have the right vocabulary to describe how my markings should be used: but if I were to mark G as tonic, then the comma-high markings (up-pointing open arrow) would need to be two-comma high markings. I’ll keep working on how to indicate intonation in a minor key context.

I do these analyses from time to time. On this series of posts, Mozart’s and now Byrd’s Ave Verum Corpus. Earlier, Josquin’s Ave Verum Corpus, imo the best of the three, with the most spine-chilling effects. It’s in three parts, they are somewhere down in my soundcloud account, with scores referenced in the notes. Direct link to full score

How to play the clavichord in one lesson

When I was a student I spent some time at the school’s clavichord … but eventually gave it up, I just couldn’t make musical sounds. 


Now, after these studies in intonation, I think I could play it better.   Follows the one secret key to playing a clavichord:

Background 1: like a guitar, a lute, a theorbo, the clavichord can only tune in one direction: sharp.  

Background 2: most instrumental music follows vocal music: we tune the notes that are and should be low, and just follow ‘tonal center’ for those notes that are and should be tuned high.


HENCE(!) on a clavichord you must reverse your ideas:  the tonic, dominant, subdominant notes in a tonal area are the ones that need the extra pressure on the keys; play the thirds of a major triad very lightly.  For a minor triad, of course, play the third with pressure and the fifth and root lightly. 


Romance for ‘Cello and Orchestra

Seeking commission or consortium for completion and score preparation of a new work incorporating the beauty of tuning according to these researches. 

I don’t believe anyone has ever written with this science, so I consider it radical avant garde masquerading as beautiful tonal music. Each note in the score is marked for pitch, resulting in a tonal universe of around 30 different notes.


Soundcloud brief demo recording at

See symbol table for the markings. See earlier discussions, for example: major seconds and major thirds for discussions about tuning specific intervals.

Please visit for many other recordings including some full orchestral works. Example full scores (Holy Day Overture) at and (Symphony #4 I) at

Minor Seconds, Augmented Unisons, and Unisons retuned


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I’ve been aware that my series of posts on just intonation intervals was incomplete.  It was necessary to get the Dos, Dis, Deis, and Ducks all in a row before summarizing. 

Imagine my surprise, after carefully identifying all the small intervals, to find that there is only one minor second: it is approximately 112 cents, somewhat larger than the equal tempered minor second, and found between MI and FA (and many other places). 

In addition there are the two chromatic or augmented unisons, approximately 70 and 92 cents, as between a note and its most normal chromatic alteration (C to C#, say, with the latter having a low-tuned and a high-tuned variant). The latter, as you will see in the list below, is the sum of the small chromatic augmented unison and the 22 cent comma. In my writing, I find both necessary.

And, as we found in the Mozart Ave Verum Corpus, retuning a unison over a changing harmonic background means an intonational change in the unison of approximately 22 cents, the syntonic comma.   This comma is most commonly found in diatonic music between the low-tuned RE of the ii chord and the high-tuned RE of the V chord. 

And finally, there is the enharmonic unison, a syntonic comma less a schisma, or approximately 20 cents, between, say, a C# and a Db, with the former tuned lower than the latter.

The complete solfege scale using my modification of the chromatic solfege syllables for a given tonal center, then, might be sung in order thus:


DO DOI DI RA RE REI ||  REI RI ME MEI MI || MI MII FE FA || FA FI SE SOL from root to fifth,


SOL SI LE LA LI TE TEI TI DE DO from fifth up to octave. 

 Looking in detail at the first two steps, DO to REI (high RE). a large whole step in five subintervals, and REI (high RE) to MI:, a small whole step in four subintervals:


(in order, in cents) 70, 22, 20, 70, 22 (and in notes) C C#(low) C#(high) Db D(low) D(high)


(in order, in cents) 70, 20, 22, 70


Is all this realistic?  Testimony of one:  In the privacy of my studio, I hum and groan as Beethoven is supposed to have done, and it might sound out of tune to others, but I use the above solfege syllables to keep track of where I am intonationally.

[edit] Update: recalling the Pythagorean major third, one might find another minor second between the perfect fourth and the large, Pythagorean major third. I don’t believe the third should be used melodically, or that it can be tuned, but if it were, the remaining minor second, say from high (Phyth.) MI to FA, would be a small minor second, an interval of 90 cents (the larger chromatic second less the enharmonic schisma: 92 – 2 = 90).


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