Yahoo Tuning Groups Ultimate Backup tuning Septimal tunings in traditional music cultures?

On 5/16/2012 8:26 AM, hstraub64 wrote:
> Yesterday I had a short conversation with Julien Weiss, about, among
> other things, septimal intervals, such as the harmonic seventh 7/4 or
> the septimal whole tone 8/7. Apparently intervals of similar sizes
> appear in Persian tuning systems (Dastgah). There arose the question
> which traditional music cultures of the world use septimal intervals.
> I did a short search of the archives here and found two results:
> indeed Persian music, and Indonesian (pelog/slendro). I would like to
> ask the round here: is there knowledge of more existing musical
> cultures that incorporate septimal intervals? (African maybe?) And
> could it be said that the use of septimal intervals is a distinctive
> property of Persian tuning systems, compared to Arabic and Turkish
> ones?

I wouldn't count pelog/slendro in general, but the jegog scale gives the impression of being septimal, and Bill Alves for instance has used a septimal JI tuning for slendro.

Barbershop quartet singing uses justly tuned septimal tetrads.

The harmonic series (including the 7th harmonic) is widely used, including for instance overtone flutes, or natural horns such as alphorns. Overtone singing could produce the 7th harmonic, but from what I've heard is typically pentatonic.

🔗Keenan Pepper <keenanpepper@...>

🔗Vaughan McAlley <ockegheim@...>

🔗Tim Sabin <tim@...>

🔗Andy <a_sparschuh@...>

--- In tuning@yahoogroups.com, "hstraub64" <straub@...> wrote:
>... septimal intervals, such as the harmonic seventh 7/4 or the
> septimal whole tone 8/7...

Hi Hans,

1.)
apperently the old persians and later the arabs overtook that
aboriginally from the prior earlier
Archytas's three divsions of the tetrachord
http://de.wikipedia.org/wiki/Archytas_von_Tarent
"
enharmonisches Tetrachord: (28:27)(36:35)(5:4)
chromatisches Tetrachord: (28:27)(15:14)(6:5)
diatonisches Tetrachord: (28:27)(8:7)(9:8)
"
more en detail
http://www.ex-tempore.org/ARCHYTAS/ARCHYTAS.html
even dicussed already here in that group under
/justintonation/topicId_unknown.html#362

http://en.wikipedia.org/wiki/Musical_system_of_ancient_greece
"
The three divisions of the tetrachords of Archytas were: the enharmonic 5:4, 36:35, and 28:27; the chromatic 32:27, 243:224, and 28:27; and the diatonic 9:8, 8:7, and 28:27.
"

2.)
For traditional usage of 7-limit intervals in old India see again:
/tuning/topicId_72699.html#72853
quote from there (Madras 1954):
""The shrutis bearing septimal ratios 7/6
(280 vibrations per second sa=240)....
...7/5 (336cps)...7/4...."
&ct.

3.)
/tuning/topicId_78905.html#79050
"
"Pygmy" scale

1/1 7/8 21/16 3/2 12/7 2/1
"
http://infohost.nmt.edu/~jstarret/pygmies.html
quote:
"
...According to my measurements, the basic intervals used seem to be 7/6, or 267 cents (small minor third), 8/7, or 231 cents (large second), and 9/8, or 204 cents (whole tone).

If intervals between degrees are determined by differential notes which can change according to the musical context, the polyphony can produce intervals very close from each other, but with a different function : 7/6 + 8/7 (498 cents, the perfect fourth) is only a few cents apart from 8/7 + 8/7 (462 cents) or 7/6 + 9/8 (471 cents); 7/6+7/6+8/7+8/7+9/8 (1200 cents, the octave), from 7/6+8/7+8/7+8/7+9/8 (1164 cents), etc. This could explain 8/7 (462 cents) or 7/6 + 9/8 (471 cents); 7/6+7/6+8/7+8/7+9/8 (1200 cents, the octave), from 7/6+8/7+8/7+8/7+9/8 (1164 cents), etc. This could explain the often quoted presence of "major sevenths" in Pygmy polyphonies..."

http://www.pygmies.org/aka/music-dance.asp
http://www.musicearth.name/organology/the-musical-instruments-of-the-pygmy-tribes-baaka/

4.)
Pelog & Slendro

Further infromation about the histrically usage
of septimal-intervals in other cultures one can find
in the corresponding standard reference:
http://de.wikipedia.org/wiki/Martin_Vogel
'Die Lehre von den Tonbeziehungen, Bonn 1976'

bye
Andy

🔗Andy <a_sparschuh@...>

🔗David Bowen <dmb0317@...>

🔗Graham Breed <gbreed@...>

"hstraub64" <straub@...> wrote:
> Yesterday I had a short conversation with Julien Weiss,
> about, among other things, septimal intervals, such as
> the harmonic seventh 7/4 or the septimal whole tone 8/7.
> Apparently intervals of similar sizes appear in Persian
> tuning systems (Dastgah). There arose the question which
> traditional music cultures of the world use septimal
> intervals. I did a short search of the archives here and
> found two results: indeed Persian music, and Indonesian
> (pelog/slendro). I would like to ask the round here: is
> there knowledge of more existing musical cultures that
> incorporate septimal intervals? (African maybe?) And
> could it be said that the use of septimal intervals is a
> distinctive property of Persian tuning systems, compared
> to Arabic and Turkish ones?

From the Scala archive:

! bagpipe4.scl
!
Highland Bagpipe, Ewan Macpherson in 'NZ Pipeband', Winter
1998 9
!
7/8
1/1
9/8
5/4
4/3
3/2
5/3
7/4
1190.00000

It turns out that bagpipes are traditionally tuned to blend
with the drones, and pipers recognize some of the results
as being out of tune with the normal musical scale. It was
recognized that bagpipes shouldn't play with an orchestra.
Modern instruments are more likely to veer towards equal
temperament.

Mouth organs have been tuned to 7-limit JI:

http://www.angelfire.com/music/harmonica/hohnertuningsbyepping.html

According to that article, equal temperament isn't even
ubiquitous on modern instruments.

There's been talk about 7-limit scales in traditional
Tanzanian music. Hukwe Zawosa may be an example. He's
worth listening to, anyway.

Another one from the Scala archive:

! pygmie.scl
!
Pygmie scale
5
!
8/7
21/16
3/2
7/4
2/1

I don't know how reliable that is.

Graham

🔗Carl Lumma <carl@...>

🔗kraiggrady <kraiggrady@...>

🔗hstraub64 <straub@...>

That's quite interesting! Thanks to all for your answers! The pygmies scale is quite noteworthy.

Trying to summarize: use of septimal intervals in traditional musical cultures can be found

- (most) everywhere where overtones play a prominent role (that one is quite straightforward). Hukwe Zawosa seems to fit in here.
- in some systems of african music, notably pygmies
- in some scales of indonesian music
- occasionally in arabic/turkish/persian systems (but not in persian more than in arabic/turkish, if I understand correctly - ?)
- occasionally (rarely) in indian scales

Right?

--- In tuning@yahoogroups.com, "Andy" <a_sparschuh@...> wrote:
>
> --- In tuning@yahoogroups.com, "hstraub64" <straub@> wrote:
> >... septimal intervals, such as the harmonic seventh 7/4 or the
> > septimal whole tone 8/7...
>
> Hi Hans,
>
> 1.)
> apperently the old persians and later the arabs overtook that
> aboriginally from the prior earlier
> Archytas's three divsions of the tetrachord
> http://de.wikipedia.org/wiki/Archytas_von_Tarent
> "
> enharmonisches Tetrachord: (28:27)(36:35)(5:4)
> chromatisches Tetrachord: (28:27)(15:14)(6:5)
> diatonisches Tetrachord: (28:27)(8:7)(9:8)
> "
> more en detail
> http://www.ex-tempore.org/ARCHYTAS/ARCHYTAS.html
> even dicussed already here in that group under
> /justintonation/topicId_unknown.html#362
>
> http://en.wikipedia.org/wiki/Musical_system_of_ancient_greece
> "
> The three divisions of the tetrachords of Archytas were: the enharmonic 5:4, 36:35, and 28:27; the chromatic 32:27, 243:224, and 28:27; and the diatonic 9:8, 8:7, and 28:27.
> "
>
>
> 2.)
> For traditional usage of 7-limit intervals in old India see again:
> /tuning/topicId_72699.html#72853
> quote from there (Madras 1954):
> ""The shrutis bearing septimal ratios 7/6
> (280 vibrations per second sa=240)....
> ...7/5 (336cps)...7/4...."
> &ct.
>
>
> 3.)
> /tuning/topicId_78905.html#79050
> "
> "Pygmy" scale
>
> 1/1 7/8 21/16 3/2 12/7 2/1
> "
> http://infohost.nmt.edu/~jstarret/pygmies.html
> quote:
> "
> ...According to my measurements, the basic intervals used seem to be 7/6, or 267 cents (small minor third), 8/7, or 231 cents (large second), and 9/8, or 204 cents (whole tone).
>
> If intervals between degrees are determined by differential notes which can change according to the musical context, the polyphony can produce intervals very close from each other, but with a different function : 7/6 + 8/7 (498 cents, the perfect fourth) is only a few cents apart from 8/7 + 8/7 (462 cents) or 7/6 + 9/8 (471 cents); 7/6+7/6+8/7+8/7+9/8 (1200 cents, the octave), from 7/6+8/7+8/7+8/7+9/8 (1164 cents), etc. This could explain 8/7 (462 cents) or 7/6 + 9/8 (471 cents); 7/6+7/6+8/7+8/7+9/8 (1200 cents, the octave), from 7/6+8/7+8/7+8/7+9/8 (1164 cents), etc. This could explain the often quoted presence of "major sevenths" in Pygmy polyphonies..."
>
> http://www.pygmies.org/aka/music-dance.asp
> http://www.musicearth.name/organology/the-musical-instruments-of-the-pygmy-tribes-baaka/
>
> 4.)
> Pelog & Slendro
>
>
> Further infromation about the histrically usage
> of septimal-intervals in other cultures one can find
> in the corresponding standard reference:
> http://de.wikipedia.org/wiki/Martin_Vogel
> 'Die Lehre von den Tonbeziehungen, Bonn 1976'
>
> bye
> Andy
>

🔗Andy <a_sparschuh@...>

--- In tuning@yahoogroups.com, "hstraub64" <straub@...> wrote:
>
>
> ...summarize:
> use of septimal intervals in traditional musical cultures can be found
>
> - in some ... african music, notably pygmies
> - in some scales of indonesian music
> - occasionally in arabic/turkish/persian systems
> - occasionally (rarely) in indian scales
>
please do not forget our own western historically view
on that topic:
http://www.teoria.com/articulos/balsach/03.htm
"
Prime 7-harmonics

Since Aristoxenos, and for the majority of later theorists including Zarlino and Rameau, harmonic theory was based on the arithmetic ratios of the numbers 1, 2, 3, 4, 5, and 6 (known as the "senario") 6. Some theorists, such as Leibnitz (see Luppi, 1989), Tartini (1754 and 1767) Euler, Kirnberger and Vogel (see Vogel, 1993) add the number 7 to the "senario".

Since Rameau (1750) it has been known that harmonically establishing the arithmetic relations of the first 6 or 7 natural numbers is equivalent to considering the relations for the human auditory system of the first 6 or 7 partials of a (harmonic) tone. Those in favor of including the number 7 consider the minor seventh as harmonic ratio 7/4, while those in favor of the "senario" consider the minor seventh as the ratio of two fifths (C:F:Bb) (3/2 x 3/2). Riemann himself, while being in favor of the "senario" for tonal relations, accepted that the interval of a minor seventh was a direct gift of nature (Elementar-Musiklehre, Hamburg, 1883.) 7.

In psycho-acoustics it is considered that there is no reason to "cut off" the harmonic series for a given position, especially for such a low number as 6 or 7. If this is done it is purely for practical reasons. "Cutting off" is carried out in higher positions, from 8 (Plomb 1964) to 15 (Leman 1995).

For more detail see the examples in:
http://www.lamadeguido.com/artangles.htm

Literature reference:
http://www.syntropia.de/die-naturseptime-p-11601.html
An referee wrote about that an rescension:
"
Die Natur-Septime

Ihre Geschichte und ihre Anwendung

Von Vogel, Martin

Orpheus, 510 S.

ISBN: 978-3-922626-61-9

Hier geht es um das Schlüsselthema der Neuen Musik: die Einbeziehung der nächsten Primzahl, der Sieben, in die musikalische Theorie und Praxis. Schönberg wusste" es, sagte es 1911. Seinen Vorgängern hielt er vor, dass sie beim 5. Oberton stehengeblieben seien. Das hätte nicht sein dürfen. Bis knapp vorher war man auf dem rechten Weg, als man dem Gebot des Materials folgend die Obertöne nachahmte. Aber nun temperierte man das System, und das System temperierte den heißen Drang zum Suchen. Man hätte nie vergessen dürfen, dass das temperierte System nur ein Waffenstillstand war, der nicht länger währen darf, als die Unvollkommenheit unserer Instrumente ihn nötig macht. Ich meine: wir stehen erst im Anfang. Wir müssen weiter!
Schon über der Einführung der Intervalle der Zahl 5 vergingen Jahrhunderte. Es kann deshalb nicht verwundern, dass sich auch die Auseinandersetzung um die Siebenzahl über Jahrhunderte erstreckt. Bereits 1636 sprach sich Mersenne für die Siebenerintervalle aus. Große Männer der Wissenschaft und der Philosophie: Huygens und Euler, Descartes, Leibniz und Rousseau, beschäftigten sich mit der Frage ihrer Einführung. Tüchtige Musiker wie Tartini, Sorge, Kirnberger und Fasch traten für die Naturseptime ein.

Als Schönberg davon hörte, dass auch der große Carl Stumpf für die Sieben eintrat und der Meinung war, nach und nach würden auch die Verhältnisse 4:7, 7:8, 5:7 u. dgl. zu Konsonanzen erhoben werden, bekannte er von sich: In solchen Momenten bedaure ich, dass ich so wenig weiß. Ich muss das alles erraten. Wenn ich geahnt hätte, dass ein Gelehrter vom Ruf eines Stumpf die gleiche Ansicht vertritt wie ich! Mir fehlen alle diese Quellen; ich bin auf eine einzige angewiesen: aufs Denken. Da kommt man langsamer vorwärts! Wer schneller vorwärts kommen will, sollte dieses Buch zur Hand nehmen. Sein erster Hauptteil handelt von dem bisherigen Verlauf der Diskussion um die Sieben und ihre Intervalle, der zweite Hauptteil bespricht ihre Anwendung.
"

Here comes my attempt to translate that:
'
The natural seventh

Its history and usage

Author: Vogel, Martin

ISBN: 978-3-922626-61-9

This concerns about the key issue of new music:
How to include of the next prime number (after 5), the seven,
into musical theory and practice of performing.
Already Arnold Schoenberg "knew" about that overdue step.
He complained in the year 1911, that his predecessors,
had stopped at the 5th-partial in the overtone-series.
He judged about that limitation,
to restrict it to the 5th partial,
that there is no reason to stop there.
His forerunneres were right on track,
when they followed the natural material
of imitating the overtone-series.
But now the system got temperature controlled,
and just that system tempers down the search to escape from that.
the hot desire to search what comes next after 5...

One should never forget, that the tempered system is only a
provisional truce, which should not last longer than the imperfection of our musical instruments enforces tempering as necessary.
I mean, this is just the beginning to exploit the 7th harmonics.
We must go on!
Even if the introduction of the 5-limit intervals last centuries
it is therefore no surprise, that the debate about the number seven continues over so many centuries. Mersenne as early as 1636
wanted to introduce the 7-limit intervalls.
Also did so famous men of science and philosophy: Huygens and Euler, Descartes, Leibniz, and Rousseau.
They dealt with the question of their introduction.
Experimentalistic musicians such as Tartini, anxiety, Kirnberger and Fasch advocated the natural seventh.

Schoenberg knew that even the great Carl Stumpf
for the seven-limit entered and the opinion
would be 4:7, 7:8, 5:7 considered as consonances, he confessed of themselves: in such moments, I am sorry that I know so little. I have to guess everything. If I had known that a reputation as a scholar from the stump takes the same view as me! I miss all of these sources, I am dependent on a single: Think again. As one progresses more slowly! Who wants to move forward more quickly, should take up this book. His first main part deals with the progress so far the debate on the screens and their intervals, the second main section discusses their application.
"

🔗kraiggrady <kraiggrady@...>

🔗dkeenanuqnetau <d.keenan@...>

🔗Carl Lumma <carl@...>

"hstraub64" <straub@...> wrote:

> Trying to summarize: use of septimal intervals in traditional
> musical cultures can be found
>
> - (most) everywhere where overtones play a prominent role
> (that one is quite straightforward). Hukwe Zawosa seems to
> fit in here.

No pre-Zawose Wagogo recordings of extended JI exist, to my
knowledge. I'd be more than happy to discover some.

> - in some systems of african music, notably pygmies

I have several recordings of pygmie music, again without
7-limit JI present. I'd love to discover some.

> - in some scales of indonesian music

I've never heard 7-limit intervals in Indonesian music,
and again, I'd love to discover such a thing.

> - occasionally in arabic/turkish/persian systems (but not in
> persian more than in arabic/turkish, if I understand
> correctly - ?)

Again, no evidence this was ever practiced.

> - occasionally (rarely) in indian scales

Again, no evidence.

> Right?

Unfortunately not.

In my experience, people claiming indiginous use of
extended JI never have the one thing that could
establish their theory beyond doubt: a recording.

In point of fact, if extended JI were to come from
somewhere, we'd expect it from the only indigenous
peoples to use 5-limit JI systematically: Europeans.
And, it did:

http://en.wikipedia.org/wiki/SPEBSQSA

-Carl

🔗hstraub64 <straub@...>

🔗Carl Lumma <carl@...>

"hstraub64" <straub@...> wrote:

> OK - so short answer to my question is: no - at least not
> melodically...

Well... :)

I guess we should say, the earliest evidence of a musical
form using septimal harmony seems to be for barbershop.
Zawose was born in 1938, so barbershop predates him, though
not by much (actually it is the same year SPEBSQSA was
founded... 7-limit intonation in barbershop has improved
considerably since that time, but early recordings show it
did exist already when O.C. Cash founded the society).

It would be interesting to read a biography of Zawose to
see how he learned his trade. It may be that extended JI
was used in his tribe before him - I'm just not able to
find any solid evidence of this.

Aside: Hey! Get that thing out of there!
http://www.youtube.com/watch?v=a71aACAWri4&#t=0m20s

The highland pipes were mentioned. My recording circa 1991
is definitely 7-limit. Is it harmonic use? Not the way we
usually think of "harmony", though when pieces are moving
slowly, some obvious 7:4 dyads can be heard. Dave Keenan
suggested this tuning practice may be fairly recent for the
highland pipes. I don't know the history.

Maqam (Arabic, Persian, etc) intonation has been vigorously
debated here. Ancient Greek theorists were fascinated with
numbers and obtained complex ratios with procedures like the
"freshman sum" (mediant). This may have been necessary for
want of logarithms. There's little evidence musicians took
these writings as anything more than bearing plans -- that
is, not to the point of accurately performing these ratios.
Maqam music is an improvised form that uses intonation
expressively. The practice clearly draws on a richer
variety of basic scales than does Western music, and some
interval(s) smaller than a whole tone is necessary to express
them all. But it doesn't seem there has ever been consensus
on exactly which interval(s) those should be (present day
included). The maqamat were passed down aurally, from
teacher to student, and recordings I've heard and analyzed
don't present evidence of a consistent use of > 5-limit
intervals.

Keenan Pepper is our resident expert on Indonesian intonation.
Daniel Wolf and others have commented in the past. I have
never heard extended JI in Indonesian music. The "American
gamelan school" (Lou Harrison, Other Music, etc.) *does* use
extended JI, and it doesn't sound like its traditional
counterpart.

-Carl

🔗lobawad <lobawad@...>

The chromatic tetrachord of Al Farabi can be demonstrated on a fretless instrument in this way:

Play a pure fourth.
Place a finger halfway between this fourth and the nut, play.
Place a finger halfway between that point and the nut, play.
Play the open string.

That is it- strings tuned to open fourths will give you conjuct
tetrachords, tuned to fifths, disjunct.

Now, if we look at the extreme simplicity of this, and consider that
Al Farabi was noted as practising musician, and that his book on
music includes illustrations and descriptions of musical instruments
as well as theory, surely Occam's razor favors the idea that this chromatic tetrachord represents an actual musical practice.

The tetrachord is 1/1, 16/15, 8/7, 4/3

and is supposedly still in use, see John Chalmer's Divisions of the Tetrachord for references.

As to accurarcy of intonation in practice (placing ideally thin fingers perfectly on points measured to these ratios would make for slightly flat intonation on a real instrument, for example), I would suggest playing, say, an oud and judging for yourself whether the instrument sings more when playing the ratios purely as you can.

Regardless of the perfection of the intonation, it seems clear to me that the burden of proof lies on the claim that this tetrachord was not in actual use. Why would something so simple, teachable, allegedly still in use, lovely sounding, documented by a practising musician and organologist, NOT have been in use?

🔗Andy <a_sparschuh@...>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

> ... I guess we should say, the earliest evidence of a musical
> form using septimal harmony seems to be for barbershop...

> ... Ancient Greek theorists were fascinated with
> numbers and obtained complex ratios with procedures like the
> "freshman sum" (mediant). This may have been necessary for
> want of logarithms. There's little evidence musicians took
> these writings as anything more than bearing plans -- that
> is, not to the point of accurately performing these ratios.

In deed Carl,
that agrees with the conservative point of view
in that still ongoing controversy:

http://plato.stanford.edu/entries/archytas/
See last clause in chapter 2.2
Quote:
"
Archytas' final contribution to music theory has to do with the structure of the scale. The Greeks used a number of different scales, which were distinguished by the way in which the fourth, or tetrachord, was constructed. These scales were grouped into three main types or genera. One genus was called the diatonic; one example of this is the Pythagorean diatonic described above, which is built on the tetrachord with the intervals 9 : 8, 9 : 8 and 256 : 243 and was used by Philolaus and Plato. There is no doubt that Archytas knew of this diatonic scale, but his own diatonic tetrachord was somewhat different, being composed of the intervals 9 : 8, 8 : 7 and 28 : 27. Archytas also defined scales in the two other major genera, the enharmonic and chromatic. Archytas' enharmonic tetrachord is composed of the intervals 5 : 4, 36 : 35 and 28 : 27 and his chromatic tetrachord of the intervals 32 : 27, 243 : 224, and 28 : 27. There are several puzzles about the tetrachords which Archytas adopts in each of the genera. First, why does Archytas reject the Pythagorean diatonic used by Philolaus and Plato? Second, Ptolemy, who is our major source for Archytas' tetrachords (A16), argues that Archytas adopted as a principle that all concordant intervals should correspond to superparticular ratios. The ratios in Archytas' diatonic and enharmonic tetrachords are indeed superparticular, but two of the ratios in his chromatic tetrachord are not superparticular (32 : 27 and 243 : 224). Why are these ratios not superparticular as well? Finally, Plato criticizes Pythagorean harmonics in the Republic for seeking numbers in heard harmonies rather than ascending to generalized problems (531c). Can any sense be made of this criticism in light of Archytas' tetrachords? The basis for an answer to all of these questions is contained in the work of Winnington-Ingram (1932) and Barker (1989, 46-52). The crucial point is that Archytas' account of the tetrachords in each of the three genera can be shown to correspond to the musical practice of his day; Ptolemy's criticisms miss the mark because of his ignorance of musical practice in Archytas' day, some 500 years before Ptolemy (Winnington-Ingram 1932, 207). Archytas is giving mathematical descriptions of scales actually in use; he arrived at his numbers in part by observation of the way in which musicians tuned their instruments (Barker 1989, 50â"51). He did not follow the Pythagorean diatonic scale because it did not correspond to any scale actually in use, although it does correspond to a method of tuning. The unusual numbers in Archytas' chromatic tetrachord do correspond to a chromatic scale in use in Archytas' day. Barker tries to save Archytas' adherence to the principle that all concordant intervals should have superparticular ratios, but there is no direct evidence that he was using such a principle, and Ptolemy may be mistaken to apply it to him. Archytas thus provides a brilliant analysis of the music of his day, but it is precisely his focus on actual musical practice that draws Plato's ire. Plato does not want him to focus on the music he hears about him (âheard harmoniesâ) but rather to ascend to consider quite abstract questions about which numbers are harmonious with which. Plato might well have welcomed a principle of concordance based solely on mathematical considerations, such as the principle that only superparticular ratios are concordant, but Archytas wanted to explain the numbers of the music he actually heard played. There is an important metaphysical issue at stake here. Plato is calling for the study of number in itself, apart from the sensible world, while Archytas, like Pythagoreans before him, envisages no split between a sensible and an intelligible world and is looking for the numbers which govern sensible things.
"

In order to tune all the tree Achytas genera at once
in a octave on ancient coeval instruments
one needs in practice at least six different:

http://homoecumenicus.com/ancient_instruments.htm

then select among them for instance the well fitting :
"
Phorminx probably the oldest of the Cithara type instruments. From references in ancient sources (Homer, Hesiod, Aristophanes) we know that Phorminx was richly decorated with gold and ivory, and accompanied the singing of the epic singers called rhapsodes.
"
http://en.wikipedia.org/wiki/Phorminx

but how to distribute the given ratios that
Archytas heared in coeval music of hist time,
as documented in the reference-article:

http://eamusic.dartmouth.edu/~larry/published_articles
/divisions_of_the_tetrachord/index.html

there especially
http://eamusic.dartmouth.edu/~larry/published_articles/divisions_of_the_tetrachord/chapter6.pdf

[ p.101 in the original facsimile] = [p.29 in the *.pdf ennumeration]
quote
"
E. 1/1 unison
F. 28/27 appears in all three tetrachords as the same
Gbb 16/15 enh.
Gb 9/8 chr.
G. 32/27 dia.
a. 4/3 quarte
b. 3/2 quinte
c. 14/9 same within all three genera as already F.
dbb 8/5 enh.
db 27/16 chr.
d. 16/9 dia.
e. 2/1 octave

or when compiled into modern 'scala' terminology

! Archytas3genera.scl
!
All three Archytas's gerenra at once: diatonic+chromatic+enharmonic
!
11
!
28/27 ! F. enh.+chr.+dia. introducing semitone
16/15 ! Gbb enh.
9/8 ! Gb chr.
32/27 ! G. dia.
4/3 ! a. quarte: enh.+chr.+dia.
3/2 ! b. quinte: enh.+chr.+dia.
14/9 ! c. enh.+chr.+dia.
8/5 ! dbb enh.
27/16 ! db chr.
16/9 ! d. dia.
2/1 ! e. octave: enh.+chr.+dia.
!
! [eof]

But sadly today nobody knows any more
how an Homerian rhapsode might once had sounded
in such an coeval ancient 7-limit tuning,
that Archytas actually heared in his days?

bye
Andy

🔗kraiggrady <kraiggrady@...>

As far as the Greeks actually using ratio based scales~
Freshman sums do not explain many of Ptolemy's scales which have high artistic value and are real developments on the scales that preceded them. He also comments on deviations from his scales (which musically are easily explained). There is also nothing in freshman sums or medients that would lead to the scales over the vast possibilities unless they actually heard and used them. For example Archytas fixation on the 28/27 is something i have seen time and again with those who work in Just intonation. It is a very compelling interval.
It seems the present day practice where the scales in scala outnumber actual composition is being projected upon the Greeks.

Doug Leedy has done extensive research on the performance of Homer

/^_,',',',_ //^/Kraig Grady_^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

a momentary antenna as i turn to water
this evaporates - an island once again

🔗Andy <a_sparschuh@...>

--- In tuning@yahoogroups.com, kraiggrady <kraiggrady@...> wrote:

> ... Archytas fixation
> on the 28/27 is something
> i have seen time and again with those who work
> in Just intonation.
> It is a very compelling interval.

Yep Kraig,
just that Archytas third-tone 28/27 (~63.0 cents)
also sounds in my ears 'compelling' well in tune.
Hence i have choosed it in my recent rational 53-tone too:

/tuning-math/message/20649

! Sp53in13lim.scl
which occurs there at the third position as the third-tone:

"28/27 ! C +3 EB"

instead of the here also possible 27/26 (65.3 cents)
or the even more out of place 'chroma' 25/24 (70.7 cents).

An other guess about Archytas preference of 28/27 can be found in:
http://tonalsoft.com/enc/d/diatonic-genus.aspx
Quote:
"
Archytas used the same tuning for the note above the bottom (parhypate) in all three of his genera, making it a 28:27 above the bottom note (hypate) and a 9:7 below the top note (mese); all other ancient Greek theorists allowed parhypate to be moveable along with lichanos, its placement depending on the genus. Winnington-Ingram speculated that perhaps the reason for this strange interval was that it was important for the note above the 28:27 to be in a 7:6 "septimal minor-3rd" ratio to the note which lies a 9:8 "tone" below its lower note.
Thus, whereas the highest interval in his diatonic tetrachord is the same Pythagorean 9:8 tone indicated by Philolaus, the middle interval is the larger 8:7 "septimal tone".
"

But attend, that maquam theorist usually dislike 7-limit intervals
alike for instance group-member Ozan in his statement:
/tuning/topicId_85218.html#85251
Quote:
"
I don't see why (9/8 x 28/27) should be superior to (9/8 x 25/24). I
don't think 76/64 is a crude interval or an approximation. It is a
legitimate, nicely resonating augmented second/minor third. It's just
the right flavour I need in my formulations for maqam scales. 7/6 is
rather inappropriate for my purposes in fact.
"

While Marcel considers 28/27 versus 25/24 as indistinguishable
in his contribution
/tuning/topicId_81042.html#81750
Quote:
"
However simply because archytas wrote down 7-limit intervals doesn't mean he did so correctly.
28/27 is for instance close enough to 25/24 to be confused with it by the ear.
Or even 256/243 or 135/128. I've caught myself making the error of 25/24 vs 256/243 vs 135/128 a lot.

This could lead to something like this
making diatonic stepsizes 25/24 256/225 9/8 making
1/1 25/24 32/27 4/3 3/2 25/16 16/9 2/1

chromatic 25/24 32/27 25/27 making
1/1 25/24 100/81 4/3 3/2 25/16 50/27 2/1

enharmonic 25/24 128/125 5/4 making
1/1 25/24 16/15 4/3 3/2 25/16 8/5 2/1

Or enharmonic 256/243 81/80 5/4 making
1/1 256/243 16/15 4/3 3/2 128/81 8/5 2/1

.... &ct....
"

Conclusion
All one can say about the anew speculations about the actual
7-limit issue is the moment:
The controversy about 28/27 (and maybe 27/26) versus 25/24 persists
and remains contentious,
due to the depending personal preference
in the individual habituation of perception.

Probably alike in the classical literature about that topic:
http://www.archive.org/stream/psychologicalmo505ameruoft/psychologicalmo505ameruoft_djvu.txt
"
Authors: 6 LYLE H. LANIER

Habituation Effects of the Equally Tempered Musical Scale.
(With W. F. Smith) Proc. Ninth Int. Cong. Psychol, 1929,
p 338-339.

The Range and Modifiability of Consonance in Certain Musical
Intervals. (With W. F. Smith) Amer. Jour. Psychol.,
1930, 42, 561-572.
"
bye
Andy

Septimal tunings in traditional music cultures?

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