Intonation factoids 2

Tags

diatonic pitch set, ein feste burg, intonalism, luther

So many schemes of intonation with a goal of getting beyond equal temperament end up hitting a giant rock: the number 12, or a diatonic 7. There is NO reason for a set of pitches used in music to be twelve in number.

With the low and high RE, the FI, and the TE, a very simple diatonic set of pitches contains 10 notes, with which much music from the Renaissance to the Rock and Roll can be put into sound. Not 7!!! See “Ein Feste Burg” below, where the RE of the first two systems should be tuned low and the RE of the final system should be tuned high. The third system could easily use a FI (F#) in end of the first bar, many transcriptions do.

Intonation factoids 1

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circle of fifths, intonalism, intonation, music theory, Pythagorean comma

The note at the top of a stack of 12 pure fifths is different from the note at the top of a stack of 7 octaves.

Pythagorean Comma, defined approximately 2,500 years ago.

W. A. Mathieu, “Harmonic Experience” 1997

“Since modern Western musicians are so used to ‘the circle of fifths’ as a theoretical concept as well as a practical learning tool, it may be surprising to some that twelve perfectly tuned fifths stacked from from C do not equal some distant octave of C, and that there is hence no circle of perfect fifths.”

DO-RE Project Post 1

Tags

chord progression, harmonization, intonalism

How to harmonize melody line DO-RE. 8 progressions over DO-DO in bass, 21 progressions over DO-SI in bass. Progression labeling in progress.

Intonalism: the way forward

Tags

intonalism, intonation, music theory, serialism

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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intonalism, intonalist, intonation, performance practice

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.

Minor Seconds, Augmented Unisons, and Unisons retuned

Tags

chromatic solfege, comma of didymus, intonalism, schisma, syntonic comma, unison

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,

and

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:

DO DOI DI RA RE REI

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

REI RI, ME, MEI, MI

(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).

Sight-tuning exercise

Tags

intonalism, intonation, just intonation

As an example of how one might combine an exercise in sight reading with an exercise in tuning, I made this harmonization of the Agnus Dei from Mass XVIII chant. Needless to say, if you listen, it is a synthesis. But it is tuned correctly.

RECORDING: Mass XVII Agnus Dei (SAB a cappella)

 

PDF: Mass XVII Agnus Dei

Symbol expanations may be found in an earlier post, but briefly the heavy down triangle is a comma-low modal scale note; the circle+ tonic; the light up and down arrows dominant and sometimes supertonic over dominant, and subdominant; the double heavy down triangle two comma low implied secondary dominant third.

I intend to make this available free, but with each use requiring an explicit license stamp allowing duplication and performance. Contact me directly for such a license. For perusal, naturally, no license at all is needed.

Tuning a diminished seventh chord

Tags

intonalism, intonation, just intonation, music theory, tuning

It has been asserted that a dimished seventh chord cannot be tuned in just intonation.   While a composer may well want the sound of the equal tempered blur of a diminished seventh chord on the piano or organ, I believe as used by Mozart and Copper, and maybe Scriabin … one day i’ll look into that music more …, it can and should be tuned as shown.  Not all composers spell the chord correctly even when used as in my example as a clear substitute for a secondary dominant.

diminished_7

There is only one untunable interval, the Pythagorean minor third, the same interval  that falls in the dominant seventh chord between fifth and seventh.   The other intervals, including the resolution major and minor seconds, have clear and unambiguous tuning.

For another example of using just intonation in contemporary music, listen to my arrangement of the beautiful folk song “O Danny Boy”. https://soundcloud.com/williamcopper/o-danny-boy

Intonation Symbols

Tags

intonalism, intonalist, intonation, tuning

For reference please find a table of symbols to indicate intonation. 

intonation_symbols

Major Thirds

Tags

harmonic series, intonalism, intonation, pythagorean third

There are two major thirds used in Intonalism, and in most great music.

One of the major thirds is tunable, the pure major third of the harmonic series.    If you play the open C string on a viola, and listen carefully, you can hear that third (two octaves and a third higher).  That is the fifth overtone or fifth harmonic in the harmonic series.   If you sing the third of a major triad in pure intonation it should be the same pitch as that harmonic.

The other major third, often called the Pythagorean third, is not directly tunable.  It is much sharper than the pure, tunable major third.  However, it is used in tonal modulations along the circle of fifths.   It is the result of a stack of pure fifths. From the same open C string on a viola, each successive string is tuned a pure fifth higher:  C – G – D – A on the viola.   A fifth above the viola’s A string is the violin E string, another pure fifth higher.  So the violin E string is two octaves and a Pythagorean major third above the viola C string.

If a portion of music were to modulate from C to G and on to D and A, and finally to E, it would be properly tuned to that violin E string, a Pythagorean third away.   If a different portion of music were to modulate from C by means of a pivot on the third scale step, it would be properly tuned to the harmonic third, and the violin open E string would sound badly out of tune.

The third on the organ or piano, in equal temperament, must fulfill both purposes, so as a reasonable compromise it is somewhere in between in tuning, rather closer to the Pythagorean third than to the Harmonic third.

The audio example gives all three over a viola open C string.

In relation to the equal tempered third, the Pythagorean third is approximately 8 cents sharper  and the Harmonic third is 14 cents flatter.   (A cent is a rather artificial measurement, meaning 1 / 100 of an equal tempered semitone).

The first, the harmonic third, is played lightly by a flute, and blends almost undetectably into the viola sound.