I am just readingFux again (struggled last time)
In the early text he talks about the nature of intervals I am not sure and he does not make it clear if he is treating the chromatic intervals or the diatonic - he mentions the tritone and the diminished fifth as seperate intervals (pg 20 of my copy). Though I have studied this before it was some twenty years ago and I need to get out the oil can.

Here is a table I drew up - which probably has misunderstandings in it. Could someone shed some light please.

It's diatonic intervals, because he is dealing with the 6 modes, D E F G A and C. B is left out because there is not a perfect 5th.
So thirds and sixths can be major or minor.
In the F lydian mode, the B natural can become a Bb if for example you are writing a 5th below F.
Or in other instances to avoid a tri-tone that would be created not in the same measure but from two adjacent measures. See fig. 14. the fourth measure if A F (from the bottom up), and the fifth meas. is Bb D. Although B is still consonant with D, the Bb is used to avoid an F in the first meas. and a B in the second measure, creating a tri-tone.

As far as the difference between the diminished 5th and the tri-tone, (p. 20 also) I believe the reason for including both is the enharmonic difference. In F lydian, the B is not a diminished 5th, (i.e. Cb) since there is a perfect 5th in the mode. So I guess it would be called a tri-tone.

By the way, can you post a link to the table you made?

Does your edition have the second part on fugues?
It is referenced in "The Study of Fugue" - Alfred Mann, but in my edition of Fux, it ends after 139 pages with Florid counterpoint.

Hi Jw,
Thanks for the input - yes I forgot the table, buts its probably wrong. I shall include it below.
My knowledge of modes comes from Jazz the Lydian chromatic concept, and the Aebersold stuff. I suspect this is something very different based on church modes.I am still getting used to the language.


I am not sure if I understand this at all yet so I am going to paraphrase your answer to see if I am correct:

I presume all your examples are talking in the Key of C

1] OK we have all the modes and notes of these modes, except the locrian mode which has a tritone fifth. Therefore thirds and sixths can be major or minor (contrast for example the Dorian and Aoelion). How about the Phrygian, which has a flat 2nd? I presume this means we can have flat seconds?

2, You said: "As far as the difference between the diminished 5th and the tri-tone" So I suppose he is calling an Augmented (Ionian) fourth a tritone?

3] My edition is the 1965 edition with a 1965 forward by Mann - it ends on page 139 with the words Farewell and pray to God for me" So no it does not have a second part on Fugues

Here is the table:


Unison Perfect Consonant
m2 Dissonant (phrygian)
M2 Dissonant
m3 Imperfect Consonant
M3 Imperfect Consonant
Perfect fourth Dissonant
Tritone Dissonant
Perfect fifth Perfect Consonant
Minor 6th Imperfect Dissonant
Major 6th Imperfect Consonant
Minor 7th Dissonant
M7 Dissonant
Octave Perfect Consonant

Do I have the language right?

Your chart looks good except the minor 6th is an imperfect consonance. Also, be aware that Fux is referring to harmonic intervals here. For example, the harmonic major 6th can be used, as an imperfect consonance, wherever appropriate; but the major 6th cannot, at any time, be used as a melodic interval.

I posted this last week for those who want to learn to do it all by silent hearing. Of some use?

For those who want to have a go at how they learnt in the 16th century:

This is in the alto clef for ease of reading. Create a part above note-for-note using these rules:
Only consonant intervals may be used. These are:

Unison
Third major and minor
Perfect fifth
Sixth, major and minor
Octave
Tenth Twelth, etc

The spacing should rarely exceed a twelfth.

Fourths and diminished fifths are NOT consonant so are avoided

Parellel fifths and octaves are forbidden (two fifths or octaves after each other)

Exposed fifths and octaves are forbidden (moving to a fifth or octave from the same direction)

Remember, all done in your head!

I'll give it a go:

E D C G E F G ^C A G B ^C ^D B G A G B ^C

(^ means in the upper octavae)

Thanks for the replies guys. I have been doing some research and came across http://downloads.zdnet.com/download.aspx?&docid=278321&promo=100511 "The complete guide to counterpoint"

and more interestingly : http://downloads.zdnet.com/download.aspx?&docid=278321&promo=100511 "Counterpointer" a software noation package which teaches counterpoint. If anyone is interested or has used it - could they post?

Got hold of the difference between harmonic and melodic intervals. Harmosis So the major sixth cannot be used in melodies- "Curiouser and curiouser!" said Alice.

Thanks KKlutz

I shall ponder that one. 2nd note creates an exposed fifth.

You did do it all in your head I hope!

ZeroZero, this sounds like you're not
clear on intervals themselves. This
is not uncommon, since the subject of
intervals is about the most confusing
thing around.

An interval is a musical standard that was
devised hundreds of yrs. ago by theorists
so that theory texts could be written,
based on it. It is a prerequisite to
the study of any book on theory.

Like any technical standard, it is completely
arbitrary; for example, the foot, the
standard of length for many yrs. could
have been longer or shorter than it is,
but some standard had to be agreed upon
so that everyone would be on the same
page when some measurement was quoted, so
the foot was established at the length
that it is today.

Similarly, in order for books on theory
to be written, some standard way of
measuring the distance between two pitches
had to be devised, so that everyone
would know what you were talking about
when you said: "The distance between
pitch X and pitch Y is an interval of
a ___." The "___" is the intervals
as we know them today.

What was agreed upon hundreds of years ago
is that the way to measure the distance
between two pitches would be by "intervals,"
and these "intervals" would be based
on the MAJOR scale ONLY. The standard
is that the distance between the first
tone of a major scale and itself is
a unison (middle C to middle C in the
C maj. scale); the distance between
the first an second tones in a maj.
scale is a maj. 2nd (C to D in C maj.);
the distance between the first and
third tones of a maj. scale is a maj. 3rd;
between the first and fourth tones, a
perf. 4th; between the first and fifth,
a perf. 5th, between the first and sixth,
a maj. 6th; between the first and 7th,
a maj. 7th; and between the first and
8th, a perf. octave.

Thus, C to E is always a maj. 3rd, regardless of
whether you're talking about a major
scale, a min. scale, a chromatic scale,
a blues scale, or any kind of scale or
mode or chord. This is the benefit of
a standard: everyone knows what you are
talking about in any musical situation.
If it were otherwise--if the interval
between C and E varied with the context--
it would be impractical to write books
on theory, since there are an almost
infinite number of scales that can be
devised, and then C to E would have
to be defined differently in each of
them, resulting in an almost infinite
number of terms for the interval between
C and E.

The other varieties of intervals, min.,
augmented, diminished, etc., are also based
the intervals of the maj. scale as
defined above: any maj. interval
lessened by a half tone is a min. interval,
thus, C to Eb is a min. 3rd; any perf.
interval lessened by a half tone is
a diminished interval; etc.

Thus, what is the interval between C# and
G? In a C# maj. scale the distance
between the first and fifth tones of
the scale, C# to G#, is a perf. 5th,
by the the above definition. In our example,
this 5th tone is lessened by a half tone,
to G, so C# to G is a diminished 5th.
And so forth.

Hi Gyro,
Thanks but I am clear on the intervals, and can play any of them at a moments notice in any key, plus any chord you care to name. its just the sixteenth century grammar that is throwing me a tad. Thanks for the thought though.

Intervals are not as arbitrary as the measurement of a foot, they are based on simple mathematical ratios, e.g. the octave is a doubling of the frequency. This is not a man made measurement - but a naturally occuring ratio.
Zero

Zero, what we don't know is - are consonants man made? Pythagoras thought not - that they were eternal truths. Probably this position held sway more or less until the 18th century when the Euclid's fifth postulate got into trouble. Up until then geometry perfectly mapped onto the 'real' world. When the fifth postulate fell the music of the spheres went with it.

Am I talkin rubbish?

zero - you asked:
1] OK we have all the modes and notes of these modes, except the locrian mode which has a tritone fifth. Therefore thirds and sixths can be major or minor (contrast for example the Dorian and Aoelion). How about the Phrygian, which has a flat 2nd? I presume this means we can have flat seconds?
---

The minor second is dissonant and not allowed in any mode, neither is the major second. Just because Phrygian has a minor second between the tonic and the second does not give it any special place. In Dorian you would not harmonize the E with the F either.

When you get to the ligatures (ch. 4) then you have dissonances that get resolved. So you do have seconds, fourths and sevenths.

Also, as far as the modes go, as I understand them, they are the same as the modes used in jazz, with the exception of not using the Locrian mode.

Harmosis -
what are harmonic and melodic intervals?

keyboardklutz - "are consonants man-made?"
Considering the perfect consonances can be found at the first and second harmonic, it seems unlikely that it would be "man-made". Now the question is - "where does the harmonic series come from?"

JW, I mean the fact they are consonant while others are dissonant. Is that socially constructed? I'm not being quite so naive here. I've never found a 10 year old who could natively distinguish dissonant from consonant.

KK there is a lot of stuff on how much of music -is natural (mathamatical) or cultural constructs but maybe off topic for this thread?

[/b]

Right you are, Sir! When going through these exercises, you'll find yourself wishing you could (use a major 6th melodically)!


[/b]

By this, I mean the interval created by two notes sounding simultaneously vs. the interval created by two notes sounding consecutively. So, the melodic skip of a major 6th is not allowed here.

Although dissonances are prohibited in the first species, they are allowed in every other species.

When you're going through the two-part exercises, the rules are very strict. As you add more voices, some of the rules are relaxed to account for the fact that more melodic lines are involved (don't worry about this until you get there).

OK so I have been reading Fux again today and gettting used to the language - I think I have the basics right. Here is my summary (which you shouldn't take as gospel (if you forgive the terrible joke)

These are the intervals Fux refers to and their properties in his terminology:

Unison is a Perfect Consonant
m2 Dissonant
M2 Dissonant
m3 Imperfect Consonant
M3 Imperfect Consonant
Perfect fourth Dissonant (when compared to a base note - but not when compared in say dominant/tonic (GC) relationship.
Tritone Dissonant
Perfect fifth Perfect Consonant
Minor 6th Imperfect Consonant
Major 6th Imperfect Consonant
Minor 7th Dissonant
M7 Dissonant
Octave Perfect Consonant

Three motions:
Direct: Parts ascend or descend together
Contrary: Parts go in different directions
Oblique: One part remains stationary

All four motion rules of Fux can be perculated down to one rule "The only motion forbidden is direct motion to a perfect consonance" (Mann p22)

Cantus firmus:
A melody to which other parts are compared.
On this point (creating cantus) there is a very strange (to me) note by Mann p27 "The possibility of vocal performance should always be taken into consideration, therefore no augmented, diminished, or chromatic intervals or intervals larger than a fifth". I do not understand this as surely a vocalist can sing chromatics and intervals larger than a fifth???

Fux postpones explanation of the rules for creating a cantus - I presume it can be anything diatonic?

Basically in the first part/species? we are using
perfect consonances (unisons fifths and octaves) to start and end the cantus then imperfects (thirds and sixths) to fill the inbetweens (allowed a few perfect consonants if we must for melodious integrity).
In the penultimate bar we go for a sixth above or a minor third below - for some reason that I dont yet understand. Maybe some form of obligatory cadence?

Lastly I not that Fux is writing his examples in D dorian and I know that some early chruch music was written in this style - I probably need to know more here.

I KNOW I MUST be wrong here, but so far, at least, this text seems like an arbitrary and outdated set of incomplete rules, expressed in a slightly dodgy fashion, using outdated concepts of harmony - apologies to Fux!

I shall be persevering!

Zero

Up to the 16th century you were not allowed to make the singer uncomfortable.
Fux was extrapolating rules from centuries of practice. So artificial they definitely are. According to CPE Bach JS didn't teach Fux. He started with alto and tenor lines for chorales with soprano and bass supplied.

OK I have read on again: Concerning the rules for motions
The only motion that is not allowed is the direct motion from imperfect consonance to perfect consconance.
Why?
On pg 32 Fux's pupil asks this:

This is his (answer (condensed):


"One may not do it because two fifths follow each other immediately of which one is apparent and the other concealed or hidden and would stand for the diminution of the interval"

Personally, though two consecutive fifths may be a little bland, - I can also see when this might be useful.

I understand the term "diminution" in this context refers to sung improvisation, where the singer would fill in passing diatonic tones between intervals, but why we are talking of a hidden fifth? Is this a fifth that MIGHT be there if a singer chose to improvise a line?
This does not seem to my novice understnding to be a very good reason for forbidding direct motion to the fifth - especially in modern context where diminution is not practiced.

If we can disregard this last remaining rule " direct motion to the perfect consonances is not allowed" (the other rules being condensed to this one) then we are left with no rules (??)

More and more puzzling...

I've not done any of this for nearly 20 years. I do remember why parallel fifths are not allowed - it takes what is a multi-texture and because of the similar motion has an aural effect of losing a part.

If it was a complete set of rules, it would by necessity be even more arbitrary.
But I don't know what is arbitrary in this book.

I feel the information is presented clearly and systematically, but I think you are responding to the temporary gaps where he tells the student about something but doesn't explain why because it is too soon to understand.

Well, yes the harmonic concepts are not exactly what we have today, but it is the basis for it.

As far as Bach not teaching from Fux; he didn't have to.

The point is CPE said it was an advantage NOT to start with Fux - get straight into the meat, so to speak.

Fux's cantus firmi are written for the specific purpose of exercise. They are composed to illuminate the rules and, at the same time, present difficulties for the student. I would hold off on composing your own cantus firmi until you've gone through all the exercises in the book.

Yes, this is cadential. Notice that the required intervals render a leading tone and a major 2nd to the tonic (except for the Phrygian mode).


Keep in mind that Fux's use of the modes here are for pedagogical purposes and do not necessarily represent actual practice (notice the requirement of the leading tone in most of the exercises), especially in regards to ecclesiastical modes. Fux's book is more about counterpoint and treatment of dissonances than modes.

That's part of the rationale behind the rule. But, you can think of it the same way as parallel 5ths - it's a matter of texture and line independence.

I don't know why anyone here is referring to any of these rules as arbitrary - none of these rules are arbitrary. When you are learning a style of music, you must learn the "rules" of that music. As Fux himself writes, "...the lawgivers of an art have set down nothing pointlessly and without deliberation." That is a true statement.

"What was agreed upon hundreds of years ago
is that the way to measure the distance
between two pitches would be by "intervals,"
and these "intervals" would be based
on the MAJOR scale ONLY."

Gyro- just for clarification. Rarely have any two theorists ever agreed on anything, much less a community of theorists agreeing on some universal language for measuring pitch distance. Also, intervals were discussed waaaay before the "Major scale" was even used.

BTW- keyboardklutz, you shouldn't have any unisons except on the first or last note, especially one approached in similar motion (m.4). Also, this is not a cadence. A cadence in 16C terms is defined by Zarlino as a Major 6th moving to an octave (or its inversion, a m3 to unison.) Always do the cadence first and then go back. This M6-P8 formula is a result of a compositional preference for leading tones moving to octaves in the upper ("cantizans") voice while the lower ("tenorizans") voice moved down from scale degree 2 to 1. That's why the Phrygian mode requires no leading tone, the Major 6th interval is "built in."

So, jumping in and responding to your reply to KBKlutz - trying to decipher your reply...
FUx sees a cadence in terms of a move from a major sixth to the 1, (only?) but where does the leading tone stuff come in?
A leading tone in my language is a major 7th tone - which is not in the phrygian mode - which sports a minor second???
If I sm right Fux is thinking of church modes or somthing similar, and using these so that the tonic or root is the first note of the mode?

Well, thanks so much for that. My examples from Swindale (you may know it?) has unisons but as you say they're approached contrary. I thought my cadence came too easy!

The cadence is M6 to 8. In phrygian (in E as an ex.) it is D and F to E and octave E. Instead of a leading tone you have a minor second (F - E).

Zero, for the most part, the church modes are the same as the Jazz modes (well, not really, but alot of it is the same.) Take a D Dorian scale for example. If you were working with two voices only, the bottom voice would go from scale degree 2-1 at the cadence (E-D), while the top voice would move from scale degree 7-8 (C-D). If those notes were not altered, the penultimate vertical interval would be a minor 6th and the final interval would be an octave. In order to get a Major 6th, one must raise the C to C#, resulting in the leading tone. In the Phrygian mode, the penultimate interval (using the same scale degrees as in the Dorian example) is already a Major 6th.

Now it becomes a chicken/egg argument. Zarlino was the Theorist that established this "rule," but he was really simply articulating what was already being done by the composers of his time. This is the way Theories usually are introduced. Generally theory follows practice and the "rules" clarify what the acknowledged masters of a generation were already doing.

It seems as if I am at a cadence myself in this explanation, so at the risk of eliding into another thought, I will end here.

got that Super

I'm jumping in from the Adult Beginner forum where Keyboardklutz's example here was posted so I tried to do the exercise. I probably shouldn't be doing this: I started rudimentary theory this year starting with note names but moved rather fast, and am at the very beginning of harmony - just piecing my chords together in combination in root position: smidgin of an understanding of cadences and that there is such a thing as movement.

So this is my version, strictly following the rules Keyboardklutz posted. (I've fixed the tritone, KbK). Now I am reading here that there is such a thing as a cadence, but it's a movement from 6 to 1? Wasn't there a time when there was no 7th note, so no leading note? So that would explain the 6 to 1.

So this is my first attempt, before:

Hi keystring,
It's great that you're learning this stuff!

Now, for the fun part:

1) You've got parallel octaves at the end - that penultimate interval is an octave, not a major 6th (See below for more on this).

2) You've got two unison's in the body of the exercise (mm 2, 4). As this is First Species counterpoint in two parts, the unison may only be used at the beginning or end of the piece.

3) You have a whole note in the counterpoint for two half notes of cantus firmus (m 4). Again, as this is a First Species exercise, you should have 1:1 notes in counterpoint to the cantus firmus.

4) You have a P5 to a unison, a perfect consonance to a perfect consonance, in similar motion (m 4, beats 2-3). A perfect consonance must be approached by contrary or oblique motion.

5) You started the piece with an imperfect consonance - a major 3rd. A perfect consonance must be used at the beginning and the end.

As for the leading tone, these exercises do make use of it. The intervals of the cadence should be:

B-C
D-C

This is a major 6th interval to an octave. Notice that the major 6th contains the leading tone, resolving to the tonic.

Keep up the studies!

Thank you, Harmosis, for all that feedback. I feel like I'm groping in the dark since I barely have my feet wet for the most introductory level of even ordinary harmony theory, where you learn that there is such a thing as a cadence, or such a thing as V - I movement, or there is such a thing as a V.

So I have the rules you have set out, and the ones listed by Keyboardklutz. Is the basic idea that I am moving from a unison at the beginning to a cadence which is VI-octave, where VI is the leading note and those last two intervals create a cadence, meaning that it gives a finality?

Slightly off-topic: last night we were talking about plagal cadences, which in a certain quarter is being argued is not a true cadence because it contains neither the dominant nor the leading note. But it came to me that it is called "church cadence", and the notes include the VIth degree, which is actually the leading note in the present exercise. Historically might the plagal cadence herald back to earlier times?

You're welcome!

The beginning must be a perfect consonance, but not necessarily a unison (it can be an octave or perfect 5th as well).

The cadence is a major 6th interval resolving to an octave, but that is NOT to say it is VI resolving to I. When working through species counterpoint exercises, you don't often get functional harmony, so it is best not to think in terms of it. But if you wanted to look at it that way, then the cadences in Fux's species counterpoint would be V-I (as we are discussing counterpoint in two parts, I have placed a hypothetical third part in parenthesis make the point more clear):

-B - C
-D - C
(G)-(C)

The D to B clearly forms a major 6th interval, but as you can see, it outlines a V, not a VI.

Of course this is not the case in the phrygian mode, where we would use a phrygian cadence:

-D - E
(A)-(G#)
-F - E

I you haven't already done so, you should pick up a copy of The Study of Counterpoint from Fux's Gradus ad Parnassum, translated into English by Alfred Mann. It is a good step-by-step study of the species counterpoint that we are discussing now.

The plagal cadence is absolutely a true cadence. As you pointed out, it is often associated with church music where it was used extensively (and why it is often called the "Amen" cadence). The word, "cadence," comes from the Latin word cadere - "to fall." This stems from the fact that musical passages in general tend to fall in pitch as the phrase closes. It does not mean that the major 7th (leading tone) or the 5th (dominant) must be present.

Even so, there is descending leading-tone resolution in the plagal cadence between the root of the IV chord and the 3rd of the I chord:

D-D
G-F# *
B-A
G-D

Even more so with a minor iv chord (between the 3rd of the iv chord and the 5th of the I chord):

D-D
G-F# *
Bb-A *
G-D

Plagal cadences can be very effective, even though the IV/iv chord does not have the tension of a V7 or viiº.

Now that I have stopped Christmas-hops around the country (I feel like a flea) ... thanks again for the additional explanations. I think I'll try that exercise again.

I've also just looked it up and found this write-up Counterpoint (& Species) Definition
My first thought was that I have no business going any further until I have studied basic harmony, but now I'm reading that harmony theory wasn't even invented yet. I don't think I'll mix myself up through this diversion, what do you think?
I see that Fux's book can be purchased on-line. Was the original in Latin?

Yes, it was originally written in Latin.

Harmony theory was certainly known at the time, as evidenced by Rameau's Traite de l'harmonie of 1722.

In my opinion, it would probably be better not to start Fux's counterpoint and common practice harmony at the same time. Although there is much in common between the two, there are enough differences to cause confusion (for example, crossing voices is allowed in species counterpoint, but your harmony professor will not allow it!).

I'm not that far yet: just moving chords around in root position, but I skipped ahead in the harmony book and soon there will be those kinds of rules. I would like to get this exercise right though since I started it. How about this?

That's much better, except that you have the interval of a 4th (m4, beat 3), which is a dissonance; and dissonances are not allowed in two-part First Species counterpoint. There are a couple places where contrary motion would have been better (as you have a lot of parallel motion). It is also considered bad form to approach or depart from a unison by skip, but, as the cantus firmus is skipping (m1, beats 1-2), it's less of an issue. Overall, this is good work. The only real problem is the 4th in measure four.

Thanks again. I did wonder about the 4th. I was conscious that I instinctively wanted to hop down to the dominant which sounds correct to modern ears.

Are you saying that the top voice can actually cross over and go underneath the cantus? If so, that should open up some more possibilities.

nonsensical message deleted

Yes, but be aware of the range of the voice in which you place the counterpoint in. The counterpoint must be a "singable" melody (which is one reason for all the melodic rules), and therefore cannot go past the limits of the vocal range you place it in. The cantus firmus is in the tenor range (fits alto as well), and your current counterpoint fits nicely in the soprano range. Voice crossing can confuse the distinction of the individual voices, so use it only if it makes melodic sense to do so. Keep in mind that you're doing something musical and not simply solving a puzzle.

My harmony book lists the following ranges for theory work. Bass: the F at the bottom of the bass clef to middle C, Tenor: C an octave below middle C to G above middle C, Alto: the G where tenor ends to C an octave above middle C, Soprano: middle C to G 12 notes above that. Are these the same ranges? There were no female singers at the time as I recall.

Yes, those ranges are good. Some books may differ slightly (usually on the bass range) but those will work.

This is my last try. I was reluctant to let go of the C in brackets because it sounded so nice. I would have also liked to make the first note in m. 2 a G, but I understand that would have been a parallel movement to a consonant.

The C in brackets in acceptable (preferable, actually). The voice leading is satisfactory here. The only thing that I would call attention to is the five consecutive 3rds (m 2, beat 2 through m 3, beat 2). With this much parallel motion of the same interval class in a row, the counterpoint loses its independence for a bit. Otherwise, this is good.

Last one, I promise - 2nd measure fixed.


I thought the C was not allowed because it is an octave, which can only go at the beginning or the end (I thought). It sounded like it should go there - why is it ok?
Thanks again for all the feedback, Hermosis. I feel as though I learned a lot these last few days.

What does "voice leading" mean?

Much better. Octaves are allowed in the body of the music; it is the unison that may only be used at the beginning or end.

Voice leading refers to how the individual voices (parts) that make up harmony (chords) move in relation to each other when the harmony changes.