[tuning] Diminished chord

I'm searching for the smoother diminished chord between the most plausible versions.
What is your ranking order (not considering any voice leading necessity)?

15:18:21:25
25:30:35:42
20:24:28:33
10:12:14:17

The question could be: admitting that the best diminished triad is 5:6:7 what is the best diminished seventh between 5/3 17/10 42/20 and 33/20?

Lorenzo

Hi Lorenzo,

> I'm searching for the smoother diminished chord between the
> most plausible versions.
> What is your ranking order (not considering any voice leading
> necessity)?
>
> 15:18:21:25
> 25:30:35:42
> 20:24:28:33
> 10:12:14:17
>
> The question could be: admitting that the best diminished
> triad is 5:6:7 what is the best diminished seventh between
> 5/3 17/10 42/20 and 33/20?

I think you meant 42/25 instead of 42/20 in your last sentence.
A scale with 10 tones supports all of these chords:

!
Lorenzo Frizzera's four dim7 chords.
10
!
10/9 !20
7/6 !21
4/3 !24
25/18 !25
14/9 !28
5/3 !30
33/18 !33
17/9 !34
35/18 !35
2/1 !18
!

I synthesized them with a fundamental of 400 Hz. in Cool Edit...
http://lumma.org/stuff/dim7s.zip
(about 450K)

To me the smoothest chord is the one with 17:10. It's
modulations are the most orderly.

The chord which reminds me most of the 12-ET dim7 chord is
the one with 5:3. The 42:25 chord sounds very similar to
that, except its difference tone (or virtual fundamental??)
is less in tune.

33:20 is the most piquent/unusual sounding, but actually
its modulations sound more orderly and its difference tone
better in tune than the 5:3 or 42:25 versions. To a
listener familiar with extended JI, it probably sounds
better than those versions, while to a listener used to
Western music, it probably sounds more 'out of tune'.
Nevertheless, I'm sure it could be slipped into many pieces
of music without causing a fuss.

Interestingly, the pitch of the lowest prominent difference
tones of these chords goes from highest in the 33:20 version,
down about a whole tone to the 5:3 version, down to being
very out of tune in the 42:25 version, and finally to the
lowest clear one in the 17:10 version. This of course is
only something you'd notice when listening to these perfectly-
synthesized versions sharing an identical fundamental pitch.

-Carl

I think they all work fine depending on the context.

----- Original Message -----
From: "Carl Lumma" <clumma@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 28 Aral�k 2006 Per�embe 6:06
Subject: [tuning] Re: Diminished chord

> Hi Lorenzo,
>
> > I'm searching for the smoother diminished chord between the
> > most plausible versions.
> > What is your ranking order (not considering any voice leading
> > necessity)?
> >
> > 15:18:21:25
> > 25:30:35:42
> > 20:24:28:33
> > 10:12:14:17
> >
> > The question could be: admitting that the best diminished
> > triad is 5:6:7 what is the best diminished seventh between
> > 5/3 17/10 42/20 and 33/20?
>
> I think you meant 42/25 instead of 42/20 in your last sentence.
> A scale with 10 tones supports all of these chords:
>
> !
> Lorenzo Frizzera's four dim7 chords.
> 10
> !
> 10/9 !20
> 7/6 !21
> 4/3 !24
> 25/18 !25
> 14/9 !28
> 5/3 !30
> 33/18 !33
> 17/9 !34
> 35/18 !35
> 2/1 !18
> !
>
> I synthesized them with a fundamental of 400 Hz. in Cool Edit...
> http://lumma.org/stuff/dim7s.zip
> (about 450K)
>
> To me the smoothest chord is the one with 17:10. It's
> modulations are the most orderly.
>
> The chord which reminds me most of the 12-ET dim7 chord is
> the one with 5:3. The 42:25 chord sounds very similar to
> that, except its difference tone (or virtual fundamental??)
> is less in tune.
>
> 33:20 is the most piquent/unusual sounding, but actually
> its modulations sound more orderly and its difference tone
> better in tune than the 5:3 or 42:25 versions. To a
> listener familiar with extended JI, it probably sounds
> better than those versions, while to a listener used to
> Western music, it probably sounds more 'out of tune'.
> Nevertheless, I'm sure it could be slipped into many pieces
> of music without causing a fuss.
>
> Interestingly, the pitch of the lowest prominent difference
> tones of these chords goes from highest in the 33:20 version,
> down about a whole tone to the 5:3 version, down to being
> very out of tune in the 42:25 version, and finally to the
> lowest clear one in the 17:10 version. This of course is
> only something you'd notice when listening to these perfectly-
> synthesized versions sharing an identical fundamental pitch.
>
> -Carl
>
>

I wrote...
> I synthesized them with a fundamental of 400 Hz. in Cool Edit...
> http://lumma.org/stuff/dim7s.zip
> (about 450K)
>
> To me the smoothest chord is the one with 17:10. It's
> modulations are the most orderly.

I should say that by "modulations" here, I mean the
undulating amplitude modulations of different partials
in the sound, not anything to do with the usual musical
meaning of 'a change between two chords or keys'.

-Carl

>I think they all work fine depending on the context.
>

I'm agree Ozan. But is it possible to have an 'absolute' value of consonance for a diminished chord as we have for a major chord (which I think is 4:5:6 for anybody even if other solutions are maybe useful in some other contests)?

lorenzo

>
> ----- Original Message -----
> From: "Carl Lumma" <clumma@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: 28 Aral�k 2006 Per�embe 6:06
> Subject: [tuning] Re: Diminished chord
>
>
>> Hi Lorenzo,
>>
>> > I'm searching for the smoother diminished chord between the
>> > most plausible versions.
>> > What is your ranking order (not considering any voice leading
>> > necessity)?
>> >
>> > 15:18:21:25
>> > 25:30:35:42
>> > 20:24:28:33
>> > 10:12:14:17
>> >
>> > The question could be: admitting that the best diminished
>> > triad is 5:6:7 what is the best diminished seventh between
>> > 5/3 17/10 42/20 and 33/20?
>>
>> I think you meant 42/25 instead of 42/20 in your last sentence.
>> A scale with 10 tones supports all of these chords:
>>
>> !
>> Lorenzo Frizzera's four dim7 chords.
>> 10
>> !
>> 10/9 !20
>> 7/6 !21
>> 4/3 !24
>> 25/18 !25
>> 14/9 !28
>> 5/3 !30
>> 33/18 !33
>> 17/9 !34
>> 35/18 !35
>> 2/1 !18
>> !
>>
>> I synthesized them with a fundamental of 400 Hz. in Cool Edit...
>> http://lumma.org/stuff/dim7s.zip
>> (about 450K)
>>
>> To me the smoothest chord is the one with 17:10. It's
>> modulations are the most orderly.
>>
>> The chord which reminds me most of the 12-ET dim7 chord is
>> the one with 5:3. The 42:25 chord sounds very similar to
>> that, except its difference tone (or virtual fundamental??)
>> is less in tune.
>>
>> 33:20 is the most piquent/unusual sounding, but actually
>> its modulations sound more orderly and its difference tone
>> better in tune than the 5:3 or 42:25 versions. To a
>> listener familiar with extended JI, it probably sounds
>> better than those versions, while to a listener used to
>> Western music, it probably sounds more 'out of tune'.
>> Nevertheless, I'm sure it could be slipped into many pieces
>> of music without causing a fuss.
>>
>> Interestingly, the pitch of the lowest prominent difference
>> tones of these chords goes from highest in the 33:20 version,
>> down about a whole tone to the 5:3 version, down to being
>> very out of tune in the 42:25 version, and finally to the
>> lowest clear one in the 17:10 version. This of course is
>> only something you'd notice when listening to these perfectly-
>> synthesized versions sharing an identical fundamental pitch.
>>
>> -Carl
>>
>>
>
>
>
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>

You mean when the chord gives the feeling of being at absolute rest? I
hardly think it is possible for the corde diabolique.

----- Original Message -----
From: "Lorenzo Frizzera" <lorenzo.frizzera@cdmrovereto.it>
To: <tuning@yahoogroups.com>
Sent: 28 Aral�k 2006 Per�embe 13:33
Subject: Re: [tuning] Re: Diminished chord

> >I think they all work fine depending on the context.
> >
>
> I'm agree Ozan. But is it possible to have an 'absolute' value of
consonance
> for a diminished chord as we have for a major chord (which I think is
4:5:6
> for anybody even if other solutions are maybe useful in some other
> contests)?
>
> lorenzo
>
>
>

--- In tuning@yahoogroups.com, "Lorenzo Frizzera" wrote:
>
> I'm searching for the smoother diminished chord between the most
plausible versions. What is your ranking order (not considering any
voice leading necessity)?
>
> 15:18:21:25
> 25:30:35:42
> 20:24:28:33
> 10:12:14:17
>
> The question could be: admitting that the best diminished triad is
5:6:7 what is the best diminished seventh between 5/3 17/10 42/20 and
33/20?
>
> Lorenzo

Hi Lorenzo,

While I haven't yet listened to Carl's syntheses,
I believe his interpretations of them in light of
his extensive experience will give some valuable
insights into when to use each version most aptly.

However, I've done a little primitive number
crunching on your ratios, as follows:

Each of your diminished seventh chords has the
ratios: 5n : 6n : 7n : 8n+a(n), where the
addendum a depends on n. The table below shows
these chords, along with values for a(n) and the
ratio a(n)/n; I've also extended the table
hypothetically to the cases when n is 0 and when
n is 6 or 7:

n__5n__6n__7n_8n+a__a__a/n
1___5___6___7____8__0__0 <---| (These are moot!)
1___5___6___7____9__1__1 <---|
2__10__12__14___17__1__1/2
3__15__18__21___25__1__1/3
4__20__24__28___33__1__1/4
5__25__30__35___42__2__2/5
6__30__36__42___50__2__1/3
7__35__42__49___58__2__2/7

I interpret the trend for the ratios implied for
a/n by your choices as follows, where "too big" means
"greater than ideal" and "too small" means "less than
ideal":

(n=3) 1/2 is too big, so you next chose 1/3, rather
than 2/3. 0.3333
(n=4) 1/3 is too big, so you next chose 1/4, rather
than 2/4=1/2. 0.2500
(n=5) 1/4 is too small, so you next chose 2/5, rather
than 1/5. 0.4000
This last suggests your ideal is closer to 0.40 than
to 0.25, ie greater than 0.3250; but less than 0.3333.

Extrapolating, speculatively:
(n=6) 2/6=1/3. 0.3333
(n=7) 2/7. 0.2857
(n=8) 3/8. 0.3750
(n=9) 3/9. 0.3333
(n=10) 3/10. 0.3000

Of course, by the time n=10, we already have very large
ratios, probably impractically so. So my conclusion
from the numbers you supplied, up to 25:30:35:42, is
that your ideal is very close to 0.3300. That value
is best approximated, for n<6, by the case when n=3:
or the chord 15:18:21:25.

Anyone with a preference for smaller-numbered ratios
would also choose the 15:18:21:25 chord over all other
candidates.

So it looks like the 5/3 seventh is the go.

Regards,
Yahya

Hi Yahya.

> (n=3) 1/2 is too big, so you next chose 1/3, rather
>than 2/3. 0.3333
>(n=4) 1/3 is too big, so you next chose 1/4, rather
>than 2/4=1/2. 0.2500
>(n=5) 1/4 is too small, so you next chose 2/5, rather
>than 1/5. 0.4000
>This last suggests your ideal is closer to 0.40 than
>to 0.25, ie greater than 0.3250; but less than 0.3333.

>your ideal is very close to 0.3300. That value
>is best approximated, for n<6, by the case when n=3:
>or the chord 15:18:21:25.

It seems that my ideal should be included between 0.3 and 0.375 since

(n=3) 0.16<x<0.5
(n=4) 0.125<x<0.375
(n=5) 0.3<x<0.5

Do you think that my ideal is just a cultural habit or there could be something different?

>Anyone with a preference for smaller-numbered ratios
>would also choose the 15:18:21:25 chord over all other
>candidates.

It seems to me that 25:30:35:42 has simpler ratios than 15:18:21:25 (considering all internal ratios).

lorenzo

> You mean when the chord gives the feeling of being at absolute
> rest? I hardly think it is possible for the corde diabolique.

I think Lorenzo is just asking for a ranking. Wouldn't you
say it's possible to rank chords by consonance even if none
of them are absolutely consonant?

-Carl

--- In tuning@yahoogroups.com, "Lorenzo Frizzera"
<lorenzo.frizzera@...> wrote:
>
> I'm searching for the smoother diminished chord between the most
plausible versions.

You are assuming the most plausible versions are JI versions, but
really you should include tempered versions if you are going to put it
up to the ear. For a specific example, what about
0-64-119-183 in 247-et?

Consonance is best classified according to limit. For example, 81/64 is a
3-limit consonant major third, 5/4 is a 5-limit consonant, 9/7 is 7-limit
consonant, etc...

If it is a matter of gradation according to the prime limit, the lowest
prime simple integer ratio would be the most consonant.

Oz.

----- Original Message -----
From: "Carl Lumma" <clumma@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 28 Aral�k 2006 Per�embe 22:13
Subject: [tuning] Re: Diminished chord

> > You mean when the chord gives the feeling of being at absolute
> > rest? I hardly think it is possible for the corde diabolique.
>
> I think Lorenzo is just asking for a ranking. Wouldn't you
> say it's possible to rank chords by consonance even if none
> of them are absolutely consonant?
>
> -Carl
>
>
>

> Consonance is best classified according to limit. For example,
> 81/64 is a 3-limit consonant major third, 5/4 is a 5-limit
> consonant

Since 3 < 5, you consider 81/64 more consonant than 5/4?

-Carl

I personally do not prioritize one before the other. It is simply a matter
of context if you ask me. But because the 5-limit interval simpler, it has a
certain charm in homophony.

Oz.

----- Original Message -----
From: "Carl Lumma" <clumma@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 29 Aral�k 2006 Cuma 9:15
Subject: [tuning] Re: Diminished chord

> > Consonance is best classified according to limit. For example,
> > 81/64 is a 3-limit consonant major third, 5/4 is a 5-limit
> > consonant
>
> Since 3 < 5, you consider 81/64 more consonant than 5/4?
>
> -Carl
>
>
>

> > I'm searching for the smoother diminished chord between the most
> > plausible versions.
>
> You are assuming the most plausible versions are JI versions, but
> really you should include tempered versions if you are going to put
> it up to the ear. For a specific example, what about
> 0-64-119-183 in 247-et?

I rendered a comparison involving this chord back in '03,
using MIDI...

http://lumma.org/stuff/dim7-midi.zip

To those of you who haven't listened to the wavs yet, shame
on you! :)

My experience is: 10:12:14:17 is definitely the most
consonant dim7. But it isn't as *tonally ambiguous* as the
12-tET version, and seems to sound better in some inversions
than others. Gene's chord sounds noticeably better than the
12-tET version while retaining the tonal ambiguity.

-Carl

Hi Carl.

Would be nice to have the dim7 cycle with inversion you did in that old file with the other possibilities (15:18:21:25, 25:30:35:42, 20:24:28:33). I would do by myself but I'm not able with Cubase...

I've tried to rank my four versions in different way. I've considered just the ratios concerning the diminished seventh because the rest of the chord is the same for all versions (5:6:7).After the harmonic number you have: limit, n+d, n*d and a way I measure the harmonic complexity (maybe not new but to me it works):

15:18:21:25__7___97___990___39
25:30:35:42__7___90___1115__42
20:24:28:33__11__133__1672__64
10:12:14:17__17__87___612___74

The last thing is just the quantity of information necessary to describe a ratio; this means the sum of all the numbers without paying attention to the operations (/, *, ^, +, -) which are a way to use information, not his quantity.

Ranking the most common just ratios we have this:

3/2 (5)
4/3 (7)
5/3 (8)
5/4 (9)
6/5 (10)
8/5 (10)
9/5 (10)
9/8 (10)
16/9 (11)
7/4 (11)
7/5 (12)
15/8 (13)
16/15 (14)

I feel my ears agree with it. It seems that also in ranking diminished chords I've followed this way.

lorenzo

----- Original Message ----- From: Carl Lumma
To: tuning@yahoogroups.com
Sent: Friday, December 29, 2006 9:49 AM
Subject: [tuning] Re: Diminished chord

> > I'm searching for the smoother diminished chord between the most
> > plausible versions.
>
> You are assuming the most plausible versions are JI versions, but
> really you should include tempered versions if you are going to put
> it up to the ear. For a specific example, what about
> 0-64-119-183 in 247-et?

I rendered a comparison involving this chord back in '03,
using MIDI...

http://lumma.org/stuff/dim7-midi.zip

To those of you who haven't listened to the wavs yet, shame
on you! :)

My experience is: 10:12:14:17 is definitely the most
consonant dim7. But it isn't as *tonally ambiguous* as the
12-tET version, and seems to sound better in some inversions
than others. Gene's chord sounds noticeably better than the
12-tET version while retaining the tonal ambiguity.

-Carl

Hi Lorenzo,

> Would be nice to have the dim7 cycle with inversion you did in
> that old file with the other possibilities (15:18:21:25,
> 25:30:35:42, 20:24:28:33). I would do by myself but I'm not
> able with Cubase...

Try Scala. Here's the .seq file I used for the midi demos

!
0 text "dim7 inversions"
0 exclude 10
0 frequency 220.0
0 tempo 40 pm
!
0 track 1
0 program 21
0 timesig 4/4
0 division 1
0 velocity 100
!
0 note 0 1
0 note 1 1
0 note 2 1
0 note 3 1
0 note 4 1
!
1 note 1 1
1 note 2 1
1 note 3 1
1 note 4 1
1 note 5 1
!
2 note 2 1
2 note 3 1
2 note 4 1
2 note 5 1
2 note 6 1
!
3 note 3 1
3 note 4 1
3 note 5 1
3 note 6 1
3 note 7 1

Copy and paste this into a text file with a .seq extension.
Make the current scale in Scala be the four notes of one
of your diminished chords, then render this seqence to MIDI.
Put the next chord in Scala, and repeat.

> I've tried to rank my four versions in different way. I've
> considered just the ratios concerning the diminished seventh
> because the rest of the chord is the same for all versions

Not necessarily the best approach, because adding a note
creates *four* new relationships, three of which depend on
the existing notes. This is reflected in the size of the
numbers in the simplest harmonic-series representation of
the chord, which you gave. If you rank the chords by the
size of the lowest number (or the 2nd-lowest, etc.) in each
chords, I think that gives a good ranking. Another way is
to take the product of the numbers (eg 15*18*21*25).

> a way I measure the harmonic complexity (maybe not new
> but to me it works):
>
> 15:18:21:25__7___97___990___39
> 25:30:35:42__7___90___1115__42
> 20:24:28:33__11__133__1672__64
> 10:12:14:17__17__87___612___74
>
> The last thing is just the quantity of information necessary to
> describe a ratio; this means the sum of all the numbers without
> paying attention to the operations (/, *, ^, +, -) which are a
> way to use information, not his quantity.

I'm afraid I don't follow you...

> Ranking the most common just ratios we have this:
>
> 3/2 (5)
> 4/3 (7)
> 5/3 (8)
> 5/4 (9)
> 6/5 (10)
> 8/5 (10)
> 9/5 (10)
> 9/8 (10)
> 16/9 (11)
> 7/4 (11)
> 7/5 (12)
> 15/8 (13)
> 16/15 (14)
>
> I feel my ears agree with it.

My ears closely agree with this ranking. What does
it have to do with the diminished chords?

-Carl

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> Consonance is best classified according to limit. For example,
81/64 is a
> 3-limit consonant major third, 5/4 is a 5-limit consonant, 9/7 is 7-
limit
> consonant, etc...
>
> If it is a matter of gradation according to the prime limit, the
lowest
> prime simple integer ratio would be the most consonant.
>
> Oz.

Inasmuch as no one has responded to this correctly, I couldn't keep
myself from chiming in here.

Consonant interval-ratios are classified according to *odd* rather
than *prime* limit, according to the largest *odd number* in the
ratio. 81/64 would therefore not be considered a consonance below
the 81-limit, so it would be a 3-limit dissonance. Likewise, 9/7 is
a 9-limit (rather than a 7-limit) consonance.

This relationship between consonance and harmonic limit, first
proposed by Harry Partch, is backward-compatible with common-practice
(5-limit) usage and is widely accepted in this group.

--George

I have much to learn then...

----- Original Message -----
From: "George D. Secor" <gdsecor@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 29 Aral�k 2006 Cuma 22:24
Subject: [tuning] Re: Diminished chord

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> >
> > Consonance is best classified according to limit. For example,
> 81/64 is a
> > 3-limit consonant major third, 5/4 is a 5-limit consonant, 9/7 is 7-
> limit
> > consonant, etc...
> >
> > If it is a matter of gradation according to the prime limit, the
> lowest
> > prime simple integer ratio would be the most consonant.
> >
> > Oz.
>
> Inasmuch as no one has responded to this correctly, I couldn't keep
> myself from chiming in here.
>
> Consonant interval-ratios are classified according to *odd* rather
> than *prime* limit, according to the largest *odd number* in the
> ratio. 81/64 would therefore not be considered a consonance below
> the 81-limit, so it would be a 3-limit dissonance. Likewise, 9/7 is
> a 9-limit (rather than a 7-limit) consonance.
>
> This relationship between consonance and harmonic limit, first
> proposed by Harry Partch, is backward-compatible with common-practice
> (5-limit) usage and is widely accepted in this group.
>
> --George
>
>

Hi Carl.

>> I've tried to rank my four versions in different way. I've
>> considered just the ratios concerning the diminished seventh
>> because the rest of the chord is the same for all versions
>
>Not necessarily the best approach, because adding a note
>creates *four* new relationships, three of which depend on
>the existing notes.

I did my calculations considering all new four relations for each version of diminished seventh.

>This is reflected in the size of the numbers in the simplest >harmonic-series representation of
>the chord, which you gave. If you rank the chords by the size of the lowest >number (or the 2nd-lowest, etc.) in >each chords, I think that gives a good >ranking.

I'm not sure that the lower harmonic number gives the best version of any collection of intervals. For example 16/15 is not the first 'semitone' in the harmonic series but it is most common than the previous.

>Another way is to take the product of the numbers (eg 15*18*21*25).

I did it reducing the ratios in lower terms (it is the fourth 'column', n*d).

> 15:18:21:25__7___97___990___39
> 25:30:35:42__7___90___1115__42
> 20:24:28:33__11__133__1672__64
> 10:12:14:17__17__87___612___74
>
> The last thing is just the quantity of information necessary to
> describe a ratio; this means the sum of all the numbers without
> paying attention to the operations (/, *, ^, +, -) which are a
> way to use information, not his quantity.

>I'm afraid I don't follow you...

I apologize, I had to explain it better. Let say that the brain does operations with units to represent musical ratios. Mathematical operations (/, *, ^, +, -) are the tools needed to work with units and this is the work of the brain. When we get a new ratio the brain compress it as much possible. For example 16/15 is compressed in these terms: (2^4)/(3*5). So to represent this ratio the brain needs 2+4+3+5=14 units. This is the quantity of information required for 16/15.

> Ranking the most common just ratios we have this:
>
> 3/2 (5)
> 4/3 (7)
> 5/3 (8)
> 5/4 (9)
> 6/5 (10)
> 8/5 (10)
> 9/5 (10)
> 9/8 (10)
> 16/9 (11)
> 7/4 (11)
> 7/5 (12)
> 15/8 (13)
> 16/15 (14)
>
> I feel my ears agree with it.

>My ears closely agree with this ranking. What does
>it have to do with the diminished chords?

It was just an example of this way to compute harmonic complexity with single ratios instead of a diminished tetrads as I did in my last 'column'.

lorenzo

> Let say that the brain does
> operations with units to represent musical ratios. Mathematical
> operations (/, *, ^, +, -) are the tools needed to work with
> units and this is the work of the brain. When we get a new ratio
> the brain compress it as much possible. For example 16/15 is
> compressed in these terms: (2^4)/(3*5). So to represent this
> ratio the brain needs 2+4+3+5=14 units. This is the quantity of
> information required for 16/15.

I think you're adding apples and oranges here, but I'll
reserve judgement.

> > Ranking the most common just ratios we have this:
> > 3/2 (5)
> > 4/3 (7)
> > 5/3 (8)
> > 5/4 (9)
> > 6/5 (10)
> > 8/5 (10)
> > 9/5 (10)
> > 9/8 (10)
> > 16/9 (11)
> > 7/4 (11)
> > 7/5 (12)
> > 15/8 (13)
> > 16/15 (14)
> > I feel my ears agree with it.
>
> >My ears closely agree with this ranking. What does
> >it have to do with the diminished chords?
>
> It was just an example of this way to compute harmonic
> complexity with single ratios instead of a diminished
> tetrads as I did in my last 'column'.

In this case it did a pretty good job, except I think
we should all agree that 7:4 is more consonant than 16:9
or 9:5. In fact, let me propose a different ranking:

3/2 (6)
4/3 (12)
5/3 (15)
5/4 (20)
7/4 (28)
6/5 (30)
7/5 (35)
8/5 (40)
9/5 (45)
9/8 (72)
15/8 (120)
16/9 (144)
16/15 (240)

One might argue whether 7:5 is really more consonant than
8:5 or 9:5, but then again no linear ranking can capture
everything we feel about consonance. It can only do a
'pretty good job', especially considering the cultural biases
we have (8:5 is more common in Western music than 7:5). But
I think this does a pretty good job,

-Carl

I cannot resist suggesting a possibility from my Handel experiments
that does *not* have 5-6-7 above the root.

Namely

1 : 6/5 : 10/7 : 12/7

where the only 'complicated' interval is 25/21 in the middle.

This can be put in whole numbers as

35:42:50:60 (... but this form is not particularly helpful).

See what you think!!
~~~T~~~

I love this chord. I also adored the harpsichord piece you uploaded. Did you
perform it yourself?

Oz.

----- Original Message -----
From: "Tom Dent" <stringph@gmail.com>
To: <tuning@yahoogroups.com>
Sent: 30 Aral�k 2006 Cumartesi 17:38
Subject: [tuning] Diminished chord

>
> I cannot resist suggesting a possibility from my Handel experiments
> that does *not* have 5-6-7 above the root.
>
> Namely
>
> 1 : 6/5 : 10/7 : 12/7
>
> where the only 'complicated' interval is 25/21 in the middle.
>
> This can be put in whole numbers as
>
> 35:42:50:60 (... but this form is not particularly helpful).
>
> See what you think!!
> ~~~T~~~
>
>
>

> I cannot resist suggesting a possibility from my Handel experiments
> that does *not* have 5-6-7 above the root.
>
> Namely
>
> 1 : 6/5 : 10/7 : 12/7
>
> where the only 'complicated' interval is 25/21 in the middle.
>
> This can be put in whole numbers as
>
> 35:42:50:60 (... but this form is not particularly helpful).
>
> See what you think!!
> ~~~T~~~

This is remarkably close to Gene's version!

!
126/125 planar-tempered dim7 chord.
4
!
310.7
621.4
932.1
2/1
!

Here's are the wavs

http://lumma.org/stuff/dim7-MagicVsJI.zip

To me, they sound almost identical, except there's some
very slow, gnarled beating in Gene's version that isn't
present in the JI version (which may be within error of
my synthesis method anyway).

It seems yet another "magic" chord has lost its magic.
(For those that don't know, a "magic" chord is one which
is said to be more consonant in temperament than in JI.)

-Carl

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

> It seems yet another "magic" chord has lost its magic.
> (For those that don't know, a "magic" chord is one which
> is said to be more consonant in temperament than in JI.)

Um...you can *always* temper using rational intervals, of course, if
you don't insist on absolute regularity or else don't insist on
absolutely theoretically pure octaves.

In this case, however, you've lowered 6/5 by 126/125 to get 25/21 for
one of the thirds. This is a familiar third, being 302 cents. It's not
regular tempering, since you should adjust all 6/5 by the same amount
to do that, but then it's 25-limit JI as a JI interval, so calling it
JI is a bit of a stretch, depending on where you draw the line. Anyway,
I like this chord but I've always considered it a different chord.

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > Let say that the brain does
> > operations with units to represent musical ratios. Mathematical
> > operations (/, *, ^, +, -) are the tools needed to work with
> > units and this is the work of the brain. When we get a new ratio
> > the brain compress it as much possible. For example 16/15 is
> > compressed in these terms: (2^4)/(3*5). So to represent this
> > ratio the brain needs 2+4+3+5=14 units. This is the quantity of
> > information required for 16/15.
>
> I think you're adding apples and oranges here, but I'll
> reserve judgement.

More logical would be 2*4+3+5=16 units for 16/15, since 16/15
is 2*2*2*2/3*5, and 2+2+2+2+3+5 is 16.

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:

> Consonant interval-ratios are classified according to *odd* rather
> than *prime* limit, according to the largest *odd number* in the
> ratio. 81/64 would therefore not be considered a consonance below
> the 81-limit, so it would be a 3-limit dissonance.
> Likewise, 9/7 is a 9-limit (rather than a 7-limit) consonance.
>
> This relationship between consonance and harmonic limit, first
> proposed by Harry Partch, is backward-compatible with common-
practice
> (5-limit) usage and is widely accepted in this group.
>
> --George

While I agree that this is vastly superior than judging by primes
alone (as Carl suggested above, you can make
national-debt/centimeters-from-Pluto kinds of ratios with low primes),
I don't think, or hear, that it is telling the whole story.

For one thing, the n/2^x family must have some special status, at
least out to some distance, for obvious reasons, seems to me. I
promised Carl a recording, there's curious demonstration of this
in the recording.

Another thing, taking that idea further, is... outside of what
I can articulate with words. But you can try it yourself
and see if you agree that there's something in it, or not.

Simply take the denominator of a complex ratio and octave-reduce it,
check out what you get in relation to the 1/1. These are also
"families", I believe, and consonance/dissonance, especially
in terms of the tuning as a whole, depends on family affiliation as
well as odd number. Obviously this ties the idea of consonance with
function, but I think it's a real-life and audible thing.

-Cameron Bobro

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:

> Um...you can *always* temper using rational intervals, of course, if
> you don't insist on absolute regularity or else don't insist on
> absolutely theoretically pure octaves.

And wise decisions those would be. :-)

> > It seems yet another "magic" chord has lost its magic.
> > (For those that don't know, a "magic" chord is one which
> > is said to be more consonant in temperament than in JI.)
>
> Um...you can *always* temper using rational intervals, of
> course, if you don't insist on absolute regularity or else
> don't insist on absolutely theoretically pure octaves.
>
> In this case, however, you've lowered 6/5 by 126/125 to get
> 25/21 for one of the thirds. This is a familiar third, being
> 302 cents. It's not regular tempering, since you should
> adjust all 6/5 by the same amount to do that, but then it's
> 25-limit JI as a JI interval, so calling it JI is a bit of a
> stretch, depending on where you draw the line. Anyway, I like
> this chord but I've always considered it a different chord.

I shuold listen to the inversions, but it sounds pretty damn
similar to the regular one. Don't you think?

-Carl

Thanks! The Handel on harpsichord was all my own work... which is why
you get the noises of walking across the room and sitting down.

I belatedly realised the dim chord I favoured is just the 2nd
inversion of one Lorenzo first put forward:

25-30-35-42

vs.

35-42-50-60

... if it does sound better, it may be because all the notes are
audibly just above the root, instead of the clearly dissonant 42/25,
and the dissonant 25/21 is 'hidden' in the middle.

Also because it is more 'open' than the other chords, in the sense
that the intervals are simply wider. I'm surprised 12/7 was not
considered a viable dim7 to start off with.

Now how about 35-42-49-60 ... that will have the dissonant 3rd in a
different place and the oddity of a wide minor 3rd. I'm not overly
confident about it.

Happy New Year!
~~~T~~~

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> I love this chord. I also adored the harpsichord piece you
uploaded. Did you
> perform it yourself?
>
> Oz.
>
>
> ----- Original Message -----
> From: "Tom Dent" <stringph@...>
> To: <tuning@yahoogroups.com>
> >
> > I cannot resist suggesting a possibility from my Handel
experiments
> > that does *not* have 5-6-7 above the root.
> >
> > Namely
> >
> > 1 : 6/5 : 10/7 : 12/7
> >
> > where the only 'complicated' interval is 25/21 in the middle.
> >
> > This can be put in whole numbers as
> >
> > 35:42:50:60
> >

I thought i would remind Helmholtz's proposal for the diminished of the 10-12-14-17.
which i put forth no opinion on.

On consonant harmonies his method of looking at difference tones, which when one investigates inversions, appears to say more than any idea of limits of any kinds.
It is quite easy to make a smaller limit sound more dissonant than a higher by the use of spacing and inversions.
but we have been over this before.

Consonance is determined by 'coincidences' within all the acoustical phenomenon going on. let us also not forget 'context" in which a lower limit thrown in for instance using pitches we have not been using can sound very strident.example play pandiatonic harmonies in C and thrown in a F# major chord. Ernst loch mentions such examples.

One can look at the inversions Bach used in context of Helmholtz's work and one will notice that in his fugues especially he will use some of the formers most "dissonant' inversions.
This makes sense when one wants to have greater independence of individual lines.
--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/index.html>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main/index.asp> 88.9 FM Wed 8-9 pm Los Angeles

> I belatedly realised the dim chord I favoured is just the 2nd
> inversion of one Lorenzo first put forward:
>
> 25-30-35-42
>
> vs.
>
> 35-42-50-60
>
> ... if it does sound better, it may be because all the notes are
> audibly just above the root, instead of the clearly dissonant
> 42/25, and the dissonant 25/21 is 'hidden' in the middle.

Wow, the difference in sound is substantial. Which is why one
should always compare all inversions. Shame on me.

> Now how about 35-42-49-60 ... that will have the dissonant 3rd in a
> different place and the oddity of a wide minor 3rd. I'm not overly
> confident about it.

I'll add it to the list of chords to make MIDI files for.

And now that I think about it, Gene, why are we tempering out
126/125 instead of 648/625? The former comma gives 4-tET I
assume, while the later equates two 6:5s and a 7:5 with the
octave...

-Carl

I wrote:
> Wow, the difference in sound is substantial. Which is why one
> should always compare all inversions. Shame on me.

But actually this is only appropriate if one assumes one is only
allowed a single dim7 chord, and that one wants to orchestrate
freely with it.

Otherwise, individual inversions must be considered as separate
chords in a way. This is one of the truths of extended JI.
Already in 12-equal, "voicings" are an indispensible part of
music. It often makes sense to talk about general chord
types (ignoring inversions), but slash notation in jazz already
says we need to sometimes be more specific. By the time you
get up to the 17-limit and beyond, inversions are no longer
alterations to a chord, they can generate entirely new sounds.
All one can say is, "you asked for it".

-Carl

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

> And now that I think about it, Gene, why are we tempering out
> 126/125 instead of 648/625?

Because we all know what 648/625 tempering sounds like in this
context. The whole thread, I thought, was asking how to move from
that to something similar but more consonant.

The former comma gives 4-tET I
> assume, while the later equates two 6:5s and a 7:5 with the
> octave...

Strictly speaking, as a 5-limit tempering, it doesn't deal with the 7-
limit at all. But if you add 50/49 to the mix, you get diminished
temperament, the 12&28 system. As you can see by this, tunings can
vary a lot, but a judicous mixture--for example, 26 parts <12 19 28
34| and 27 parts <28 44 65 79|, giving <1068 1682 2483 3017| is
probably better than either extreme. Of course a reasonable person
might ask why the heck I don't just suggest 40, which gives
essentially the same result, but 40 isn't what came out when I turned
the crank on my math machine. I suggest using it anyway if you want a
tuning in this vicinity.

> >... equates two 6:5s and a 7:5 with the
> > octave...
>
> Strictly speaking, as a 5-limit tempering, it doesn't deal with
> the 7-limit at all.

126/125 has a factor of 7 in it...

> But if you add 50/49 to the mix, you get diminished
> temperament, the 12&28 system. As you can see by this, tunings can
> vary a lot, but a judicous mixture--for example, 26 parts <12 19 28
> 34| and 27 parts <28 44 65 79|, giving <1068 1682 2483 3017| is
> probably better than either extreme. Of course a reasonable person
> might ask why the heck I don't just suggest 40, which gives
> essentially the same result, but 40 isn't what came out when I
> turned the crank on my math machine.

:)

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > >... equates two 6:5s and a 7:5 with the
> > > octave...
> >
> > Strictly speaking, as a 5-limit tempering, it doesn't deal with
> > the 7-limit at all.
>
> 126/125 has a factor of 7 in it...

Yes, but 648/625, which is what yo were talking about, does not. Put
them together, though, and you get the diminished temperament.

Hi Cameron.

Can you do some example of it?

----- Original Message -----
From: Cameron Bobro
To: tuning@yahoogroups.com
Sent: Saturday, December 30, 2006 10:25 PM
Subject: [tuning] Re: Diminished chord

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:

> Consonant interval-ratios are classified according to *odd* rather
> than *prime* limit, according to the largest *odd number* in the
> ratio. 81/64 would therefore not be considered a consonance below
> the 81-limit, so it would be a 3-limit dissonance.
> Likewise, 9/7 is a 9-limit (rather than a 7-limit) consonance.
>
> This relationship between consonance and harmonic limit, first
> proposed by Harry Partch, is backward-compatible with common-
practice
> (5-limit) usage and is widely accepted in this group.
>
> --George

While I agree that this is vastly superior than judging by primes
alone (as Carl suggested above, you can make
national-debt/centimeters-from-Pluto kinds of ratios with low primes),
I don't think, or hear, that it is telling the whole story.

For one thing, the n/2^x family must have some special status, at
least out to some distance, for obvious reasons, seems to me. I
promised Carl a recording, there's curious demonstration of this
in the recording.

Another thing, taking that idea further, is... outside of what
I can articulate with words. But you can try it yourself
and see if you agree that there's something in it, or not.

Simply take the denominator of a complex ratio and octave-reduce it,
check out what you get in relation to the 1/1. These are also
"families", I believe, and consonance/dissonance, especially
in terms of the tuning as a whole, depends on family affiliation as
well as odd number. Obviously this ties the idea of consonance with
function, but I think it's a real-life and audible thing.

-Cameron Bobro

Hi Tom,

SNIP

>
> Now how about 35-42-49-60 ... that will have the dissonant 3rd in a
> different place and the oddity of a wide minor 3rd. I'm not overly
> confident about it.
>

Compared to 35-42-50-60, this sounds like C-Eb-F-A.

> Happy New Year!
> ~~~T~~~
>
>

Hope we have a great one!
Oz.

--- In tuning@yahoogroups.com, "Lorenzo Frizzera"
<lorenzo.frizzera@...> wrote:
>
> Hi Cameron.
>
> Can you do some example of it?
>

I'm trying to get one of those free sites up so I can post some files.