Scala 2.0 tutorial

First I would like to thank Manuel Op de Coul for his tireless
engineering and curating efforts on behalf of the microtonal
community. Where would we be without you, Manuel.

Thanks to all of those who helped Ara Sarkissian get Scala up and
running on his machine. I went throught the tutorial, and had some
comments.

SHOW DATA

- The word "interval" in the first part of the output means "step",
while in the second part of the output it really means "interval".
This is potentially very confusing.

- It is easy to show that the least squares estimator of a quantity
is simply its mean. Hence it's odd that "least squares average
interval" is different from "average interval" -- and both should
really say "step" or something rather than "interval".

- The example given is a 12-tone Pythagorean scale. SHOW DATA says
it's maximally even. Maximal evenness, though, is only defined
relative to a larger chromatic universe. The next line of output says
that the scale is a subset of 665-tET. But it's _not_ a maximally
even subset of 665-tET. So something's wrong.

- Odd number limit . . . Why is the "O" limit different than the "U"
limit? The Pythagorean scale is symmetrical as to otonality or
utonality -- these should be the same.

EQUAL/DATA

- Highest harmonic represented uniquely O . . . O/U . . . the first
doesn't seem right at all. Perhaps you mean highest harmonic
represented uniquely when octave equivalence is not assumed, and then
where it is assumed?

- Pyth. third up, Pyth. third down. This doesn't make any sense.
Perhaps you mean Pyth. major third, Pyth. dim. fourth?

- Pythagorean means a chain of 3/2s. Pythagorean in your program
means a linear tuning with any generator. If you call the Pythagorean
diminished fourth a Pythagorean "third down", that's like three
meanings of Pythagorean. The popular account, unfortunately, is that
Pythagorean means all simple ratios, including 5/4. Stop the
confusion!

CHORD

- Added sixth & ninth -- it looks like you're using a just
interpretation of this chord where the sixth and ninth are a 27:20
apart -- I object strongly; that's very ugly -- this is not a JI
chord!

Hope you take these criticisms with the highest respect for the work
you have done and know that it is our common desire to make these
concepts as accessible to as many users as possible.

Thanks!

Paul,

Thanks for your comments.

>- The word "interval" in the first part of the output means "step",
>while in the second part of the output it really means "interval".

I'll change the wording.

>- It is easy to show that the least squares estimator of a quantity
>is simply its mean. Hence it's odd that "least squares average
>interval" is different from "average interval"

I should make this clearer too. The value given is the linear
regression, the straight line through the pitch values at equidistant
degree numbers.

>- The example given is a 12-tone Pythagorean scale. SHOW DATA says
>it's maximally even. Maximal evenness, though, is only defined
>relative to a larger chromatic universe. The next line of output says
>that the scale is a subset of 665-tET. But it's _not_ a maximally
>even subset of 665-tET. So something's wrong.

You are correct, I should choose a different name for it, and add a
check for the official "ET maximal evenness".

>- Odd number limit . . . Why is the "O" limit different than the "U"
>limit? The Pythagorean scale is symmetrical as to otonality or
>utonality -- these should be the same.

Ah, there's a bug.

>- Highest harmonic represented uniquely O . . . O/U . . . the first
>doesn't seem right at all. Perhaps you mean highest harmonic
>represented uniquely when octave equivalence is not assumed, and then
>where it is assumed?

No, the first is when inversional equivalence is not assumed, and then
when it is assumed.

>- Pyth. third up, Pyth. third down. This doesn't make any sense.
>Perhaps you mean Pyth. major third, Pyth. dim. fourth?

Yes, that's better.

>- Pythagorean means a chain of 3/2s. Pythagorean in your program
>means a linear tuning with any generator. If you call the Pythagorean
>diminished fourth a Pythagorean "third down", that's like three
>meanings of Pythagorean.

I'll see how it can be improved.

>The popular account, unfortunately, is that
>Pythagorean means all simple ratios, including 5/4.

This I don't understand.

>- Added sixth & ninth -- it looks like you're using a just
>interpretation of this chord where the sixth and ninth are a 27:20
>apart -- I object strongly; that's very ugly -- this is not a JI
>chord!

Ok, will remove it. Let me know if you have any chords not
in the list.

Manuel

--- In tuning@y..., <manuel.op.de.coul@e...> wrote:
>
> >- It is easy to show that the least squares estimator of a quantity
> >is simply its mean. Hence it's odd that "least squares average
> >interval" is different from "average interval"
>
> I should make this clearer too. The value given is the linear
> regression, the straight line through the pitch values at
equidistant
> degree numbers.

Again, can you clarify? I still think this should come out equal to
the average interval -- you can reply with a fuller clarification on
the tuning-math list.
>
> >- The example given is a 12-tone Pythagorean scale. SHOW DATA says
> >it's maximally even. Maximal evenness, though, is only defined
> >relative to a larger chromatic universe. The next line of output
says
> >that the scale is a subset of 665-tET. But it's _not_ a maximally
> >even subset of 665-tET. So something's wrong.
>
> You are correct, I should choose a different name for it,

So what's "it"? What property _does_ the Pythagorean 12-tone scale
have that you were testing for here?
>
> >- Highest harmonic represented uniquely O . . . O/U . . . the first
> >doesn't seem right at all. Perhaps you mean highest harmonic
> >represented uniquely when octave equivalence is not assumed, and
then
> >where it is assumed?
>
> No, the first is when inversional equivalence is not assumed, and
then
> when it is assumed.

Well, that's what I meant by octave-equivalence. Octave-equivalence
of pitches, which implies inversional equivalence of intervals. So,
why the terms O and O/U? These have nothing to do with the
distinction between inversional equivalence and lack thereof.

> >The popular account, unfortunately, is that
> >Pythagorean means all simple ratios, including 5/4.
>
> This I don't understand.

That's the "popular" account -- just like the "popular" account that
Bach invented equal temperament.
>
> Ok, will remove it. Let me know if you have any chords not
> in the list.

Can you post the list? But for a start, look at
http://artists.mp3s.com/artists/140/tuning_lab.html from about 1/3 of
the way down, to the bottom. I give JI interpretations for _most_ of
the chords (i.e., those compatible with JI). Dave Keenan may be able
to help in giving "conventional" names for these chords.

Paul wrote:
>Again, can you clarify? I still think this should come out equal to
>the average interval -- you can reply with a fuller clarification on
>the tuning-math list.

Ok, it's not so difficult. Imagine the scale in the X-Y plane.
Horizontally the degree numbers and vertically pitch in cents. Then
imagine the straight line going through the origin and coming as close
as possible to the points that represent the tones. Some points will be
above the line and some below. If it would be equal to the average
interval, the line would go through the octave point, which is not
required.

>So what's "it"? What property _does_ the Pythagorean 12-tone scale
>have that you were testing for here?

I was taking maximal evenness out of the ET-subset context. Because it
bothered me that a scale could me ME in N-tET, while the same scale
wouldn't be ME in 2*N-tET. So I replaced the one step difference
requirement with the requirement that L/S <= 2.

>Well, that's what I meant by octave-equivalence. Octave-equivalence
>of pitches, which implies inversional equivalence of intervals. So,
>why the terms O and O/U? These have nothing to do with the
>distinction between inversional equivalence and lack thereof.

With O and U I meant overtonal and undertonal. To take a simple example,
in 4-tET, the nearest step to the 3rd overtone (mod octave: 3/2) is 600
cents.
The nearest step to the 3rd undertone (mod octave: 4/3) is also 600 cents.
So the 3rd harmonic is not represented uniquely O/U in 4-tET.

>Can you post the list?

Ok, reluctantly because of the length. I haven't put a lot of work in it,
so it could grow considerably. Could somebody check "the" Sacre chord,
because I've seen it mentioned differently.

Manuel

! chordnam.par by Manuel Op de Coul, 2000
! Belongs to program <a href="http://www.xs4all.nl/~huygensf/scala/">Scala</a>,
! see command CHORDS/MATCH.
!
! number of names:
264
! maximum number of intervals:
9
! There are four different ways to specify a chord in this file:
! - As a list of absolute frequency ratios separated by colons, like
! 4:5:6
! - As a list of relative intervals in cents or as a ratio, like
! 50.0 50.0 400.0 or 28/27 36/35 5/4
! - As a list of relative steps of an equal temperament separated by dashes,
! like 4-3-3. Which equal temperament to be used should be specified by a
! statement like <SCALA_SCALE_DEF 2^(1/31)>
! - As a list of tones in a certain notation system (see command SET NOTATION)
! separated by spaces, like C Eb G Bb. The octave can also be specified if
! necessary with a period, like F.-1 C.0 F.0 C.1
! The notation system to be used should be specified by a statement like
! <SCALA_NOTATION E31>
! The rest of the line with the chord must contain the chord name. If it
! begins with a digit, then the name must be preceded by a '='-character.
! If the chord is a list of note names, then the name must always be
! preceded by a '='-character. Don't forget this!
! A backslash may be used to break long lines.
! Any violation of this file's syntax will result in a "Error in file format"
! message.
!
! Note that for chord spellings there is no consistent system and names can
! be different and more than one for the same chord.
!
! BP stands for Bohlen-Pierce.
! just chords:
!
1:2:3:5:8:13 Fibonacci Chord
1:3:4:7:11:18 Lucas Chord
1:3:5:15 Genus [35]
3:4:5 Major Triad 2nd inversion
3:5:7 BP Major Triad
4:5:6 Major Triad
4:5:6:7 Dominant Seventh "7"
4:5:6:7:9 Dominant Ninth "9"
4:5:6:9 Added Ninth "add9"
4:10:14:19 Hendrix Chord
5:6:7:9 Half-diminished Seventh
5:6:8 Neapolitan Sixth, Major Triad 1st inversion
5:7:9 BP Minor Triad
5:7:9:12 Tristan Chord, Half-diminished Seventh
6:7:9 Subminor Triad
7:9:15 BP Major Triad 2nd inversion
8:10:12:14:17 Dominant Ninth Minor "7b9"
8:10:12:15 Major Seventh "*" "maj7"
8:10:12:15:18 Major Ninth "maj9"
9:10:12:15 Seconds Chord "2"
9:12:16 Quartal Triad
9:12:16:20 Carol King Chord
10:12:14:17 Diminished Seventh "dim"
10:12:15 Minor Triad "m"
10:12:15:17 Minor Sixth "m6"
10:12:15:18 Minor Seventh "m7"
10:12:15:18:22 Minor Ninth "m9"
10:12:15:18:22:27 Minor Eleventh "m11"
(4:5:6:7) Harmonic Half-diminished Seventh, Subharmonic Seventh
12:14:18:21 Subminor Seventh
12:15:20 Minor Triad 1st inversion
12:15:18:20 Added Sixth "6"
14:18:21 Supermajor Triad
14:18:21:24 Supermajor Added Sixth
15:18:20:24 Third-Fourth Chord "4/3"
15:20:24 Minor Triad 2nd inversion
16:20:25 Augmented "aug"
16:20:25:28 Augmented Dominant Seventh "7#5"
16:20:25:30 Augmented Seventh "*#5"
18:22:27 Neutral Triad
20:25:30:36 Dominant Seventh
20:25:36 Augmented Sixth "aug6"
25:30:35:42 Diminished Seventh
25:30:36:45 Half-diminished Seventh
32:36:45:54 Double Diminished Seventh
49:56:64 Double Septimal Whole-Tone
60:64:75:80 Tartini neochromatic
!
! Sethares' 10 tone chords:
<SCALA_SCALE_DEF 2^(1/10)>
!
3-3-4 =10-Tone Neutral Chord
5-2 =10-Tone Tritone Chord II
5-3 =10-Tone Tritone Chord I
!
! 12 tone chords:
<SCALA_SCALE_DEF 2^(1/12)>
<SCALA_NOTATION E12>
!
1-1 Semitone Trichord
1-2 Phrygian Trichord
1-3 Major-Minor Trichord I
2-1 Minor Trichord
2-1-2 Minor Tetrachord
2-1-2-2 Minor Pentachord
2-1-2-2-2 Minor Hexachord
2-2 Whole-tone Trichord
2-2-1 Major Tetrachord
2-2-1-2 Major Pentachord
2-2-1-2-2 Major Hexachord
2-2-2 Secundal Tetrachord
2-2-3 Added Second "2" "add2"
2-2-3-2 Added Second and Sixth "6/2"
2-3-2 Second-Fourth-Fifth Chord "5/4/2"
2-3-4 Seconds Chord
2-4 Double Diminished
2-4-3 Double Diminished Seventh
2-5 Suspended Second "sus2", Second-Fifth Chord "5/2"
2-5-2 Sixth Suspended Second "6sus2"
2-5-3 Dominant Seventh Suspended Second "7sus2"
2-5-4 Major Seventh Suspended Second "*sus2"
3-1 Major-Minor Trichord II
3-1-3 Major-Minor Tetrachord
3-2-3 Third-Fourth Chord "4/3"
3-3 Diminished "mb5", Minor Trine
3-3-3 Diminished Seventh "dim"
3-3-3-3 Strawinsky's Sacre-chord
3-3-4 Half-diminished Seventh "m7b5", Eulenspiegel Chord
3-4 Minor Triad "m"
3-4-2 Minor Sixth "m6"
3-4-2-5 Minor Sixth Added Ninth "m6/9"
3-4-3 Minor Seventh "m7"
3-4-3-4 Minor Ninth "m9"
3-4-3-4-3 Minor Eleventh "m11"
3-4-3-4-3-4 Minor Thirteenth "m13"
3-4-4 Minor-Major Seventh "m*"
3-4-4-3 Minor-Major Ninth "m*9"
3-4-4-3-7 Minor-Major Thirteenth "m*13"
3-4-7 Minor Added Ninth "madd9"
3-5 Neapolitan Sixth, Major Triad 1st inversion
3-6-3 Sixths Chord
3-7 Minor Quintal Trine
4-1-2 Added Fourth "add4"
4-2 Hard-diminished "b5"
4-2-4 Hard-diminished Seventh "7b5", French Sixth
4-2-4-3 Hard-diminished Ninth "9b5"
4-2-4-3-8 Hard-diminished Ninth Added Thirteenth "13b5b9"
4-2-4-5 Hard-diminished Seventh & Augmented Ninth "#9b5"
4-3 Major Triad "maj"
4-3-2 Sixte Ajoutée "6"
4-3-2-5 Added Sixth & Ninth "6/9"
4-3-2-5-4 Added Sixth & Ninth & Augmented Eleventh "6/9#11"
4-3-3 Dominant Seventh "7", German Sixth
4-3-3-3 Dominant Ninth Minor "7b9"
4-3-3-3-4 Dominant Ninth Minor Added Eleventh "11b9"
4-3-3-3-8 Dominant Ninth Minor Added Thirteenth "13b9"
4-3-3-4 Dominant Ninth "9"
4-3-3-4-3 Dominant Eleventh "11"
4-3-3-4-3-4 Dominant Thirteenth "13"
4-3-3-4-4 Dominant Ninth Augmented Eleventh "9#11"
4-3-3-4-4-3 Augmented Thirteenth "13#11"
4-3-3-5 Dominant Augmented Ninth "7#9", Altered Dominant
4-3-3-7 Dominant Seventh Added Eleventh "7/11"
4-3-3-7-4 "7/6/11"
4-3-4 Major Seventh "*" "maj7"
4-3-4-3 Major Ninth "maj9"
4-3-4-3-3 Major Eleventh "maj11"
4-3-4-3-3-4 Major Thirteenth "maj13"
4-3-7 Added Ninth "add9"
4-4 Augmented "aug" "#5", Major Trine
4-4-2 Augmented Dominant Seventh "7#5"
4-4-2-3 Augmented Dominant Seventh Minor Ninth "7#5b9"
4-4-2-4 Augmented Dominant Ninth "9#5"
4-4-2-5 Augmented Dominant Seventh Augmented Ninth "7#5#9"
4-4-3 Augmented Seventh "*#5"
4-5 Minor Triad 1st inversion
4-5-2 Sixth-Seventh Chord "7/6"
4-5-5 Sixth-Ninth Chord "9/6"
4-6 Augmented Sixth "aug6", Italian Sixth
4-7 Major Quintal Trine
4-7-3 Seventh-Ninth Chord "9/7"
5-1 Viennese Trichord
5-2 Suspended Fourth "sus4"
5-2-2 Sixth Suspended Fourth "6sus4"
5-2-3 Dominant Seventh Suspended Fourth "7sus4"
5-2-3-4 Dominant Ninth Suspended Fourth "9sus4"
5-2-3-4-7 Thirteenth Suspended Fourth "13sus4"
5-2-4 Major Seventh Suspended Fourth "*sus4"
5-2-7 Fourth-Ninth Chord "9/4" "sus4add9"
5-3 Minor Triad 2nd inversion
5-4 Fourth-Sixth Chord, Major Triad 2nd inversion, "6/4"
5-5 Quartal Trine
5-6 Fourth-Seventh Chord "7/4"
5-6-5 Quartal Tetrad
F B D#.1 G#.1 =Tristan Chord
C F# Bb E.1 A.1 D.2 =Skriabin's Mystic Chord, Promethean
7-3 Minor Quintal Trine
7-4 Major Quintal Trine
7-7 Quintal Trine
7-7-7-7 Fifths Chord
7-7-10 Quintal Tetrad
12-7-5-4-3 Overtone
3-4-5-7-12 Undertone
C E F# G A# C#.1 =Petrushka Chord
C Bb E.1 D.2 G#.3 F#.3 =Whole-Tone Chord
!
! Messiaen's chords:
!
B.-1 E Gb G Bb Eb.1 F.1 A.1 =orangé
A.-1 D F G Ab Eb.1 Gb.1 Bb.1 =gris et or
Bb.-1 Eb F G B C.1 Gb.1 Ab.1 =rouge
D.-1 E C#.1 A.1 =bleu
C#.-1 C G# D.1 =vert pâle et argent
G A C.1 D.1 F.1 B.1 E.2 =Accord sur dominante
C E G Bb D.1 F#.1 G#.1 B.1 =Accord de la résonance
Db G C.1 F#.1 B.1 F.2 =Accord en quartes
!
! 15 tone chords:
<SCALA_SCALE_DEF 2^(1/15)>
!
4-5-4 =15-Tone Minor Seventh
5-4-4 =15-Tone Dominant Seventh
5-4-5 =15-Tone Major Seventh
!
! 16 tone chords:
<SCALA_SCALE_DEF 2^(1/16)>
!
4-5-4 =16-Tone Diminished Seventh
!
! 17 tone chords:
<SCALA_SCALE_DEF 2^(1/17)>
!
5-5 =17-Tone Neutral Triad
!
! 19 tone chords:
<SCALA_NOTATION E19>
!
Ab.-1 C D F# =19-Tone French Sixth
Ab.-1 C D# F# =19-Tone English Sixth
Ab.-1 C Eb F# =19-Tone German Sixth
Ab.-1 C F# =19-Tone Italian Sixth
!
! 24 tone chords:
<SCALA_SCALE_DEF 2^(1/24)>
!
2-4-4 Kurdi
2-6-2 Higazi, Nagriz (Hisar)
3-3-2-6 Saba
3-3-3-5 Rakb
3-3-4 Bayati
3-4 Sikah
3-4-3 Mojahira
3-5-2 Athanasopoulos Byzantine Chromatic
3-6-1 'awg 'ara
3-6-2 'awg 'ara
4-2-4 Busalik, Nahawand
4-2-6 'ajam
4-3-3 Rast, Nagdi, Neutral Diatonic, Islamic Diatonic
4-3-5-2 Zawil
4-4-2 Cahargah, Nakriz
5-2 Musta'ar
6-1-3 Sazkar
6-2-2-4 Sipahr
8-2 Baharsurak
!
! 31 tone chords:
<SCALA_NOTATION E31>
!
C E G =31-Tone Major Triad
C Eb G =31-Tone Minor Triad
!etc.
!
! 48 tone chords:
<SCALA_SCALE_DEF 2^(1/48)>
!
3-3-6-8 Hemiolic Chromatic and Diatonic Mixed
5-5-5-5 Equal Pentachord
!
!
! 53 tone chords:
<SCALA_SCALE_DEF 2^(1/53)>
!
4-9-9 Kurdi
5-8-13 Araban
5-9-5 Huzzam
5-9-8 Iraq, Segah
5-9-8-5-9 Iraq hexachord
5-12-5 Sedaraban, Hicaz
5-13-4 Evicara
8-5-5 Saba
8-5-9 Acem, Tahir, Beyati, Huseyniasiran, Nisabur
8-5-9-8-5 Huseyniasiran hexachord
9-4-9 Buselik, Nihavend
9-5-8 Mustear
9-5-12 Neveser
9-8-5 Turkish Rast
9-8-5-9 Turkish Rast pentachord
!
! Miscellaneous chords:
!
14/13 8/7 13/12 Buzurg
59/54 66/59 12/11 Dudon Tetrachord A
13/12 59/52 64/59 Dudon Tetrachord B
17/16 19/17 64/57 Finnamore Tetrachord
16/15 75/64 16/15 Helmholtz Chromatic
2187/2048 16777216/14348907 2187/2048 Palmer Tetrachord
25/24 128/125 5/4 Salinas Enharmonic
96/95 19/18 5/4 Wilson Enharmonic
23/22 22/21 28/23 Wilson Chromatic
9/8 12/11 88/81 Modern Rast
9/8 36/35 35/34 34/33 33/32 256/243 Al-Hwarizmi's tetrachord division
256/243 531441/524288 256/243 256/243 2187/2048 256/243 Al-Kindi's tetrachord division
!
! From John Chalmers: Divisions of the Tetrachord, 1993
!
28/27 36/35 5/4 Archytas' Enharmonic
28/27 243/224 32/27 Archytas' Chromatic
28/27 8/7 9/8 Archytas' Diatonic
40/39 39/38 19/15 Eratostenes' Enharmonic
20/19 19/18 6/5 Eratostenes' Chromatic
256/243 9/8 9/8 Eratostenes' Diatonic, Pythagorean Diatonic, \
Ptolemy's Diatonon Ditoniaion
32/31 31/30 5/4 Didymos' Enharmonic
16/15 25/24 6/5 Didymos' Chromatic
16/15 10/9 9/8 Didymos' Diatonic
256/243 2187/2048 32/27 Pythagorean Chromatic
46/45 24/23 5/4 Ptolemy's Enharmonic
28/27 15/14 6/5 Ptolemy's Soft Chromatic
22/21 12/11 7/6 Ptolemy's Intense Chromatic
21/20 10/9 8/7 Ptolemy's Diatonon Malakon, Soft Diatonic
28/27 8/7 9/8 Ptolemy's Diatonon Toniaion
16/15 9/8 10/9 Ptolemy's Diatonon Syntonon, Intense Diatonic
12/11 11/10 10/9 Ptolemy's Diatonon Homalon, Equable Diatonic
40/39 26/25 5/4 Avicenna's Enharmonic
36/35 10/9 7/6 Avicenna's Chromatic
14/13 13/12 8/7 Avicenna's Soft Diatonic
16/15 15/14 7/6 Al-Farabi's Intense Chromatic
8/7 8/7 49/48 Al-Farabi's Diatonic
27/25 10/9 10/9 Al-Farabi's 10/9 Diatonic
512/499 499/486 81/64 Boethius Enharmonic
256/243 81/76 19/16 Boethius Chromatic
21/20 64/63 5/4 Pachymeres/Tartini Enharmonic
21/20 15/14 32/27 Perrett's Chromatic
18/17 17/16 32/27 Quintilianus Chromatic
83.333 83.334 333.333 Parachromatic
! Many more can be typed in from John Chalmers' book...
!
! Aristoxenos' tetrachords:
!
50.0 50.0 400.0 Enharmonic
50.0 150.0 300.0 Chromatic/Enharmonic
50.0 250.0 200.0 Diatonic with Enharmonic Diesis
66.667 66.667 366.666 Soft Chromatic
66.667 133.333 300.0 Unnamed 1:2 Chromatic
66.667 233.333 200.0 Diatonic with Soft Chromatic Diesis
75.0 75.0 350.0 Hemiolic Chromatic
75.0 225.0 200.0 Diatonic with Hemiolic Chromatic Diesis
100.0 100.0 300.0 Intense Chromatic
100.0 150.0 250.0 Soft Diatonic
100.0 200.0 200.0 Intense Diatonic
166.667 166.667 166.666 Equal Diatonic

--- In tuning@y..., <manuel.op.de.coul@e...> wrote:
>
> Paul wrote:
> >Again, can you clarify? I still think this should come out equal to
> >the average interval -- you can reply with a fuller clarification
on
> >the tuning-math list.
>
> Ok, it's not so difficult. Imagine the scale in the X-Y plane.
> Horizontally the degree numbers and vertically pitch in cents. Then
> imagine the straight line going through the origin and coming as
close
> as possible to the points that represent the tones. Some points
will be
> above the line and some below. If it would be equal to the average
> interval, the line would go through the octave point, which is not
> required.

Can you write me off-list, or on the still-extant tuning-math list,
with a simple, perhaps pentatonic, example? I need to see the details
of what you're doing.
>
> >So what's "it"? What property _does_ the Pythagorean 12-tone scale
> >have that you were testing for here?
>
> I was taking maximal evenness out of the ET-subset context. Because
it
> bothered me that a scale could me ME in N-tET, while the same scale
> wouldn't be ME in 2*N-tET. So I replaced the one step difference
> requirement with the requirement that L/S <= 2.

Ah . . . there may be a name for the property that you thought was
ME. Ask John Chalmers. It may be equivalent to a logical AND of
propreity and distributional evenness.
>
> >Well, that's what I meant by octave-equivalence. Octave-equivalence
> >of pitches, which implies inversional equivalence of intervals. So,
> >why the terms O and O/U? These have nothing to do with the
> >distinction between inversional equivalence and lack thereof.
>
> With O and U I meant overtonal and undertonal. To take a simple
example,
> in 4-tET, the nearest step to the 3rd overtone (mod octave: 3/2) is
600
> cents.
> The nearest step to the 3rd undertone (mod octave: 4/3) is also 600
cents.
> So the 3rd harmonic is not represented uniquely O/U in 4-tET.

Manuel, I think you're falling into the trap of thinking of dyads as
otonal or utonal, that I keep talking about on the list. Any dyad is
equally otonal _and_ utonal . . . only triads can lean toward one or
the other characterization. But even so, this explanation would seem
to fall short. For example, can you explain the 19-tET case in this
way? It's really inversional equivalence of intervals, and hence
octave equivalence of pitches, that differentiates the two cases, not
otonal/utonal considerations . . . right?
>
> >Can you post the list?

I'll take a look at this . . . and of course I'll have some 22-tET,
31-tET, and 72-tET chords to add.

Paul wrote:
>Can you write me off-list, or on the still-extant tuning-math list,
>with a simple, perhaps pentatonic, example?

Sure.

>Ah . . . there may be a name for the property that you thought was
>ME. Ask John Chalmers.

I doubt it. Would you know John?

>It may be equivalent to a logical AND of
>propriety and distributional evenness.

No, for example 3 1 3 1 is DE and strictly proper, but not
"L/S<=2 maximal even".

>Manuel, I think you're falling into the trap of thinking of dyads as
>otonal or utonal, that I keep talking about on the list.

No, wait, I haven't used the words otonal and utonal here. That's
not the right connotation. Indeed I was only speaking of inversional
equivalence of intervals. It's about the step numbers, hence ET
intervals, that are being represented uniquely or not.

>I'll take a look at this . . . and of course I'll have some 22-tET,
>31-tET, and 72-tET chords to add.

Splendid.

Manuel

--- In tuning@y..., <manuel.op.de.coul@e...> wrote:

> >Manuel, I think you're falling into the trap of thinking of dyads
as
> >otonal or utonal, that I keep talking about on the list.
>
> No, wait, I haven't used the words otonal and utonal here.

But you used O and U, which presumably stand for overtone and
undertone?

> That's
> not the right connotation. Indeed I was only speaking of inversional
> equivalence of intervals.

Yes -- this has nothing to do with O and U.

> It's about the step numbers, hence ET
> intervals, that are being represented uniquely or not.

Yes.

--- In tuning@y..., <manuel.op.de.coul@e...> wrote:
>
> 4:10:14:19 Hendrix Chord

That's Monzo's Hendrix Chord. My Hendrix chord is 1/1:5/4:7/4:7/3, or
12:15:21:28.

> 5:6:7:9 Half-diminished Seventh
> 5:7:9:12 Tristan Chord, Half-diminished Seventh

Hmm . . . these should be called "Harmonic Half-Diminished
Seventh" . . .

> 9:10:12:15 Seconds Chord "2"

I don't understand this terminology. Where does it come from? A "2"
chord is usually 8:9:10:12.

> 9:12:16:20 Carol King Chord

It's spelled Carole.

> 10:12:14:17 Diminished Seventh "dim"

Should say "17-limit" or something.

> 10:12:15:17 Minor Sixth "m6"

Ditto.

> 10:12:15:18:22 Minor Ninth "m9"

Goodness, no. The 9th must be consonant with the 5th and 7th. So
20:24:30:36:45.

> 10:12:15:18:22:27 Minor Eleventh "m11"

The m11 is not a JI chord but a meantone chord and should be omitted
from this portion of the list, I feel.

> (4:5:6:7) Harmonic Half-diminished Seventh, Subharmonic Seventh

You mean 1/(4:5:6:7)? That should be Subharmonic Half-diminished
Seventh, not Harmonic Half-diminished Seventh . . . and "Subharmonic
Seventh) should be 1/(5:6:7:9).

> 15:18:20:24 Third-Fourth Chord "4/3"

Odd . . . this is just the 2nd inversion of the minor seventh
chord . . . a 4/3 is the classical notation for the 2nd inversion of
_any_ seventh chord. Anyway, shouldn't your program check for
inversions automatically?

> 15:20:24 Minor Triad 2nd inversion
> 16:20:25 Augmented "aug"
> 16:20:25:28 Augmented Dominant Seventh "7#5"
> 16:20:25:30 Augmented Seventh "*#5"

"Seventh" normally means "Dominant Seventh" So you mean "Augmented
Major Seventh"? Is "*" some sort of shorthand for "Major Seventh"?
Where does it come from?

> 18:22:27 Neutral Triad

1/(18:22:27) is a virtually indistinguishable neutral triad. Perhaps
a non-JI specification would be more appropriate.

> 20:25:30:36 Dominant Seventh
> 20:25:36 Augmented Sixth "aug6"

This is rather contradictory. A dominant seventh with no fifth is not
the same as an augmented sixth . . .

> 25:30:35:42 Diminished Seventh

You have it above as 10:12:14:17 . . . it's OK to use the same name
for different chords?

> 25:30:36:45 Half-diminished Seventh

Ditto.

> 32:36:45:54 Double Diminished Seventh

Where did this come from?

> 49:56:64 Double Septimal Whole-Tone
> 60:64:75:80 Tartini neochromatic

Tell me more about that one.

> ! 24 tone chords:
> <SCALA_SCALE_DEF 2^(1/24)>
> !
> 2-4-4 Kurdi
> 2-6-2 Higazi, Nagriz (Hisar)
> 3-3-2-6 Saba
> 3-3-3-5 Rakb
> 3-3-4 Bayati
> 3-4 Sikah
> 3-4-3 Mojahira
> 3-5-2 Athanasopoulos Byzantine Chromatic
> 3-6-1 'awg 'ara
> 3-6-2 'awg 'ara
> 4-2-4 Busalik, Nahawand
> 4-2-6 'ajam
> 4-3-3 Rast, Nagdi, Neutral Diatonic, Islamic Diatonic
> 4-3-5-2 Zawil
> 4-4-2 Cahargah, Nakriz
> 5-2 Musta'ar
> 6-1-3 Sazkar
> 6-2-2-4 Sipahr
> 8-2 Baharsurak

Very confused . . . these names refer to tetrachords or something as
well as the usual larger scales that they refer to?

> !etc.

Does that mean you omitted something here?

--- In tuning@y..., "Daniel James Wolf" <djwolf1@m...> wrote:
> Just one remark: It is far from clear that classical "dominant
> seventh" functions are best represented in JI by chord in the form
> 4:5:6:7. In non-classical repertoire (blues, barbershop, etc.),
one
> may well make the case for such labels, but the classical treatment
> of the minor seventh, as a dissonance to be resolved, does not
> necessarily map best onto a 7:4 interval.

I agree (though John deLaubenfels may not). You may not have noticed,
though, that Manuel had at least two different JI tetrads listed
as "dominant seventh", not just 4:5:6:7, but also a 5-prime-limit
version. Thus I asked him about his having the same name for
different chords. Perhaps "7-limit" needs to be included in the
description for 4:5:6:7, just as I suggested "17-limit" should be
included in the description for 10:12:15:17 and 10:12:14:17.

> ----- Original Message -----
> From: Paul Erlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Wednesday, June 06, 2001 12:47 PM
> Subject: [tuning] Re: Scala 2.0 tutorial
>

> > 9:12:16:20 Carol King Chord
>
> It's spelled Carole.

This chord is commonly known among R&B and pop
musicians as a "dominant 11th" chord, or just
plain "11th". That terminology should definitely
be in Scala.

>
> > 10:12:14:17 Diminished Seventh "dim"
>
> Should say "17-limit" or something.

How about "harmonic diminished 7th"?

That's an important point, Paul.

In "common-practice" harmonic theory a diminished 7th
chord would have been based on 5-limit JI, meantone,
well-temperaments, and finally 12-EDO tunings
(in roughly correct chronological order).

I'd even venture to say that more frequent use
of the diminished 7th chord was one of the things
that led to the wider acceptance of 12-EDO
during the 1800s.

The only musical genre where this 10:12:14:17
"harmonic diminished 7th" is common is barbershop.

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
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Get your free @yahoo.com address at http://mail.yahoo.com

> ----- Original Message -----
> From: Paul Erlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Wednesday, June 06, 2001 1:30 PM
> Subject: [tuning] Re: Scala 2.0 tutorial
>

> You may not have noticed,
> though, that Manuel had at least two different JI tetrads listed
> as "dominant seventh", not just 4:5:6:7, but also a 5-prime-limit
> version. Thus I asked him about his having the same name for
> different chords. Perhaps "7-limit" needs to be included in the
> description for 4:5:6:7, just as I suggested "17-limit" should be
> included in the description for 10:12:15:17 and 10:12:14:17.

Similar to what I said about "harmonic diminished 7th",
I think a good name for 4:5:6:7 is "harmonic dominant 7th"
or just plain "harmonic 7th" chord.

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

Paul wrote:
>But you used O and U, which presumably stand for overtone and
>undertone?

Right. And from the overtone 3 it's only a small leap to the
interval 3/2 and from the undertone 3 to the interval 4/3.

>Yes -- this has nothing to do with O and U.

We have a different opinion then.

Manuel

--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> The only musical genre where this 10:12:14:17
> "harmonic diminished 7th" is common is barbershop.
>
Perhaps brass sections gravitate toward this tuning, in middle-to-
high-register close voicings?

--- In tuning@y..., <manuel.op.de.coul@e...> wrote:
>
> Paul wrote:
> >But you used O and U, which presumably stand for overtone and
> >undertone?
>
> Right. And from the overtone 3 it's only a small leap to the
> interval 3/2 and from the undertone 3 to the interval 4/3.

If you mean the _pitch_ 3/2 and the _pitch_ 4/3, then maybe. But the
intervals? Nah. And how about the 19-tET case?
>
> >Yes -- this has nothing to do with O and U.
>
> We have a different opinion then.
>
Well, you did agree with me that inversional equivalence was really
the operative distinction here. So how about "IE" (meaning
inversional equivalence" instead of O/U? I'd never understand what
you meant by O/U had I not been able to able to speak with you
directly.

Sorry if I seem anal about this, but there's so much confusion in
this field anyway that I think more ambiguity is just going to cause
more people to disregard all this stuff.

Paul wrote:
>If you mean the _pitch_ 3/2 and the _pitch_ 4/3, then maybe. But the
>intervals? Nah.

3/2 and 4/3 are frequency ratios so they are intervals too.
Calling 3/2 a pitch is rather meaningless without specifying
the frequency to multiply it with. In Scala, a scale pitch is
synonymous with the interval with respect to 1/1.

>And how about the 19-tET case?

You were right they were the wrong values. I will fix this bug.

>I'd never understand what
>you meant by O/U had I not been able to able to speak with you
>directly.

Maybe you were impeded by too much knowledge? :-)
Ok, if you think it will make it clearer for everyone.

>Sorry if I seem anal about this, but there's so much confusion in
>this field anyway that I think more ambiguity is just going to cause
>more people to disregard all this stuff.

I'm not so pessimistic.

Manuel

--- In tuning@y..., "monz" <joemonz@y...> wrote:

/tuning/topicId_24103.html#24444
>
> I'd even venture to say that more frequent use
> of the diminished 7th chord was one of the things
> that led to the wider acceptance of 12-EDO
> during the 1800s.
>

Gee... I don't know about that, but I *do* believe it is generally
viewed that it resulted in a greater acceptance of "unrooted"
tonality, and the unrestrained use in the 19-th century led to
Schoenberg and all the other "pan-tonals.." Just the ambiguous
resolution of the dimininished seventh could contribute to that...

________ _______ ______
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> /tuning/topicId_24103.html#24444
> >
> > I'd even venture to say that more frequent use
> > of the diminished 7th chord was one of the things
> > that led to the wider acceptance of 12-EDO
> > during the 1800s.
> >
>
> Gee... I don't know about that, but I *do* believe it is generally
> viewed that it resulted in a greater acceptance of "unrooted"
> tonality, and the unrestrained use in the 19-th century led to
> Schoenberg and all the other "pan-tonals.." Just the ambiguous
> resolution of the dimininished seventh could contribute to that...

Well, I think that's Monz's point . . . as more Romantic composers
relied on the ability of the diminished seventh chord to resolve
_equally convicingly_ to four different tonal centers, tunings that
differed from 12-tET became less and less aesthetically appropriate.