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This Question might be TwoFold, based on an uncertainty I have

I understand the construction of traditional Ionian-Locrian modes, and I have read that various notes in the modes have traditionally been slightly sharpened/flattened from what they are today, so for example instead of DEFGABCD perhaps the Dorian Mode was DE(slightly flat)FGABC

What are the original greek modes?

How do the slightly sharp/flat notes translate to playing a mode today? (how would you notate greek modes now including microtonal pitch differences)

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This may be related to just intonation vs equivalent temperament, see http://music.stackexchange.com/a/3487/28 –  Matthew Read Jul 12 '13 at 14:54
    
Maybe someone over at history.SE can provide an extensive answer? Perhaps repeat the question there? –  Ulf Åkerstedt Jul 17 '13 at 12:45
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3 Answers

up vote 10 down vote accepted
+300

I am really not knowledgable in ancient music theory so the following might be riddled with errors, but here is what I've gathered.

What are the original Greek modes?

Ancient Greek music scale theory was built upon the concept of the "tetrachord" - literally meaning four strings. A tetrachord consists of a group of four notes with three smaller intervals that together span the total interval of a perfect fourth (a 4:3 frequency proportion). The fourth was viewed as the basic unit for tuning, perhaps comparable to the octave of modern western music. In short there were three tuning genera or tonoi for a tetrachord, signified by the greatest appearing interval. (Example notes are in descending order as is apparently historically correct.)

  • Diatonic (a-g-f-e)
    {Greatest interval: approximately a whole tone}

  • Chromatic (a-f#-f-e)
    {Greatest interval: approximately a minor third}

  • Enharmonic (a-f-fd-e) [fd denotes the quarter tone below f]
    {Greatest interval: approximately a major third}

The three internal intervals of a genus could be arranged in different permutations. These tetrachord permutations or harmoniai were:

  • The dorian group (a-g-f-e)

  • The phrygian group (a-g-f#-e)

  • The lydian group (a-g#-f#-e)

Tetrachords were stacked to form greater systems of intervals. For instance, if you stack two diatonic lydian tetrachords on top of each other with a whole tone in between, you end up with what we today refer to as the diatonic major scale: e-d#-c#-b+a-g#-f#-e. The whole tone distance [9:8 frequency proportion] between the tetrachords is natural since it is the difference between a perfect fourth and a perfect fifth. Also, having two tetrachords of perfect fourths with a whole tone in between means that you end up with a perfect octave.

One important system of stacked tetrachords was the "Greater Perfect System". It can be constructed using diatonic dorian tetrachords thusly: Stack two of them (e-d-c-b and a-g-f-e) with a whole tone in between. Extend this system at each end by adding two conjunct tetrachords each sharing one note with the existing tetrachords. Finally add a whole tone to the bottom of the system in order to end up with a complete two octave span:

The Greater Perfect System

      | tetra | tetra |
 a-g-f-e-d-c-b-a-g-f-e-d-c-b-a
| tetra |           | tetra |

Starting from different places in the Greater Perfect System gives the seven different harmoniai, or ancient greek modes, as follows (with example note names to illustrate the distribution of intervals - semitone intervals in bold):

  • Dorian (e-d-c-b-a-g-f-e)

  • Phrygian (d-c-b-a-g-f-e-d)

  • Lydian (c-b-a-g-f-e-d-c)

  • Mixolydian (b-a-g-f-e-d-c-b)

  • Hypodorian (a-g-f-e-d-c-b-a)

  • Hypophrygian (g-f-e-d-c-b-a-g)

  • Hypolydian (f-e-d-c-b-a-g-f)

Although the names of the modern church modes are drawn from the nomenclature of the ancient Greek modes, their uses don't match up. A german wikipedia article speaks of a translation error regarding this. And the english version has this text

"The Greek concepts of scales (including the names) found its way into later Roman music and then the European Middle Ages to the extent that one can find references to, for example, a "Lydian church mode", although [the] name is simply a historical reference with no relationship to the original Greek sound or ethos." [Emphasis is mine]

And another site says that mediaeval European music scholars misinterpreted the Latin works of Boethius, causing the shift of what mode is refered by what name.

How do the slightly sharp/flat notes translate to playing a mode today?

The Pythagorean Philolaus defined the interval ratios of the diatonic dorian tetrachord as 9:8, 9:8, and 256:243. This is the basis for what is refered to as the Pythagorean diatonic scale. Another Pythagorean, Archytas, defined interval divisions for all three tetrachord genera as:

  • the enharmonic 5:4, 36:35, and 28:27

  • the chromatic 32:27, 243:224, and 28:27

  • the diatonic 9:8, 8:7, and 28:27

More on Pythagorean tunings of tetrachords is found here.

The following table compares the tunings of Philolaus (Pythagorean diatonic) and Archytas to the modern equal tempered tuning (12-TET).

Ancient Greek tuning comparison

How would you notate greek modes now including microtonal pitch differences?

Regarding notation of microtonal shifts from a base system, besides the use of a backwards flat sign (b) and a half sharp/hash sign (#) to denote quarter notes, I have seen various signs including plus signs and arrows to denote microtonal shifts from a base system. But how to represent the differences according to the table above is for someone more well-versed in the area to elaborate on.
My guess, though, would be that, if the whole composition is to be performed using a specific tuning, you simply state that in the liner notes and then use regular modern notation.


The following are some of my sources:

http://en.wikipedia.org/wiki/Musical_system_of_ancient_Greece
http://www.midicode.com/tunings/greek.shtml
http://en.wikipedia.org/wiki/Ancient_Greek_music
http://en.wikipedia.org/wiki/Tetrachord
http://en.wikipedia.org/wiki/Pythagorean_tuning
http://de.wikipedia.org/wiki/Musiktheorie_im_antiken_Griechenland
http://sv.wikipedia.org/wiki/Den_grekiska_antikens_musik
http://no.wikipedia.org/wiki/Tetrakord
http://sv.wikipedia.org/wiki/Tetrakord
http://mixolydian.askdefine.com

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Brilliant answer, but might I suggest you include a link to what stacking is? there's a great one introduction at youtube.com/watch?v=hV7UDp5V1AE –  Alexander Troup Jul 15 '13 at 17:56
    
I'm sooooo tempted to give you the bounty now. But I'm gonna wait a few days at least to try to attract more. You're definitely the front-runner. :D –  luser droog Jul 16 '13 at 15:59
    
@luserdroog :-) I'm hoping for an ancient music expert historian to turn up and give a complete exposé on tetrachords, early music theory, and the evolution of the concept of modes. –  Ulf Åkerstedt Jul 16 '13 at 20:57
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Good answer, supplement Harry Partch's "Genesis of a Music" where he covers the Greeks very nicely. –  filzilla Jul 17 '13 at 22:34
    
+1 for mentioning Pythagorean tuning. –  Garan Jul 23 '13 at 19:44
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What you are asking about doesn't really have to do with modes or accidentals. Essentially what you are talking about is the difference between modern equal temperament on the one hand, and just intonation on the other.

Up until the late 1800s, musical instruments could play the traditional Greek modes and scales based on pure intervals, but a given instrument could only play in tune in a few keys, certainly not all 12 keys. Since the late 1800s, a new system in tuning instruments was developed: that system is called equal temperament.

With modern equal temperament, our instruments are calibrated to play 12 notes in an octave, all of which are equally spaced apart by 100 cents. This system is only about 125 years old.

Before that, people used various kinds of compromises on just intonation, where intervals are pure. In a just-intonation C-major scale, there are half-steps and whole steps, and we call the pitches by the same names that are used in equal temperament, but the various half-step intervals all have a slightly different size, as do the whole-step intervals.

The music written by Bach or Mozart or any of those composers, and certainly all of the ones that came before, was not composed to be played in equal temperament like we usually do today.

This is a deep and complex subject that requires a lot of mathematics to explain, and I don't choose to go into that here. You can find plenty of references on just intonation online.

Here is one observation for you to think about: On a modern piano or guitar, which is tuned to equal temperament, all the major third intervals are considerably sharp compared to a pure, just-intoned major third. The equal-temperament system was developed to permit an instrument like a piano or guitar to play all the modes in all 12 keys and have them reasonably in-tune, but not perfect. Before about 125 years ago, a piano would be tuned to a mean-tone system, a modification of just intonation, that enabled certain keys and certain intervals to sound really in tune, and certain other keys would sound really out-of-tune.

Today, when singers in an acapella choir sing together, they can make pure major-third intervals (and other intervals as well), and so can a string quartet, where the instruments have no frets. But if singers or string instruments play along with a piano or guitar, they can no longer make those pure intervals because they clash very slightly with the equal-tempered pitches on the piano or guitar. So, really without even thinking about it, singers and string players adjust their intonation away from just intonation and toward equal-temperament, depending on the situation.

None of this distinction between just intonation, or meantone intonation, or equal temperament, has any impact on how the music is notated. We use the same 12 pitches and the same sharps, flats and naturals (although G-sharp and A-flat may be distinct pitches which are tuned slightly differently, for example). The difference is how it is performed and on which instruments and how they are tuned.

Again, this is a deep subject and I can't thoroughly explain it without writing a book on it, but there are plenty of books already written on the subject and you can go looking for them if you would like to.

Update: See my post from December 23, 2012 where I present a chart showing the differences between equal temperament and just intonation, measured in cents. (A cent is 1/100th of a modern equal-tempered half-step interval.)

http://music.stackexchange.com/a/8022/1044

enter image description here

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There are two things you need to be aware of when it comes to modes.

  1. the modes of the major key (Ionian, Dorian, etc)
  2. the "modal" style of jazz created by McCoy Tyner and others in the 1960s which includes, among other things, the use of quartals and side-slipping of pentatonic scales

So when you talk about modes be sure to specify which one.

Regarding the Greek modes, there aren't/weren't any "slightly sharp or flat" notes, you must have misinterpreted something you read. There are certain cases where a sharp might occur in order to produce a secondary dominant.

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As has been mentioned in other answers, modes have Existed long before jazz. I already know modern modal theory, but modern modes originated from greek modes. As far as slightly flat/sharp goes, the equal temperament system didn't come around until much later, so playing a D Dorian scale would not have the same frequencies as a modern D dorian. (IE some notes would be slightly sharp, and some flat). –  Alexander Troup Aug 21 '13 at 13:02
    
First two points: correct. Your last point: doesn't matter when it comes to making music; completely irrelevant to being a musician. –  Michael Martinez Aug 21 '13 at 17:29
    
@Michael: I find it odd that you find differences between tuning systems irrelevant to musicians. I believe a lot of people would disagree with you including Bach, Pythagoreans, middle east musicians, etc, and anyone not using or limiting themselves to the equal tempered scale or tuning. –  Ulf Åkerstedt Aug 21 '13 at 20:49
    
@MichaelMartinez I think the history of music, and the way it was originally played is very relevant to being a musician. The way our entire system ended up is the result of one set of paths from that origin. Going back and seeing what the sound is like allows not only music that hasn't been heard in centuries, but a deeper understanding of sound and the potential to open peoples ears to music that they've never heard. If you're claiming that's useless then by extension you're claiming all music theory is useless. –  Alexander Troup Aug 21 '13 at 21:41
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the -2 on your answer would suggest that you're alone in that assertion. –  Alexander Troup Aug 26 '13 at 20:49
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