Reforms of tonality

Thinking about the diatonic class of scales as defined by being proper
and having a circle of thirds, I decided to look at the
diaschmic/pajara 10-note scales. As expected, the four 22-et strictly
proper scales make their appearance, but it turns out that these are
the only ones which are both strictly proper and have a circle of
thirds, which is a generic fact for 22, 34 or 46 somewhat like the
corresponding facts for diatonic scales.

We have the two from Paul's paper, which I'm expressing as a circle of
major (M) and minor (m) thirds:

Symmetrical major = Pajara[10] = Diaschismic[10]
MmMmMMmMmM

Asymmetrical major
MmMmMMmMMm

Then there are the two new ones I've mentioned in connection with
22-et, but which can be seen more generally as the other two proper
ten-note third-circle diaschismic scales:

Alternative proper decatonic scale
MmMmMMMmMm

Exotic symmetrical decatonic
MmmMMMmmM

Gene wrote...
>Thinking about the diatonic class of scales as defined by being
>proper and having a circle of thirds, I decided to look at the
>diaschmic/pajara 10-note scales. As expected, the four 22-et strictly
>proper scales make their appearance, but it turns out that these are
>the only ones which are both strictly proper and have a circle of
>thirds, which is a generic fact for 22, 34 or 46 somewhat like the
>corresponding facts for diatonic scales.
>
>We have the two from Paul's paper, which I'm expressing as a circle
>of major (M) and minor (m) thirds:
>
>Symmetrical major = Pajara[10] = Diaschismic[10]
>MmMmMMmMmM
>
>Asymmetrical major
>MmMmMMmMMm
>
>Then there are the two new ones I've mentioned in connection with
>22-et, but which can be seen more generally as the other two proper
>ten-note third-circle diaschismic scales:
>
>Alternative proper decatonic scale
>MmMmMMMmMm
>
>Exotic symmetrical decatonic
>MmmMMMmmM

Here's a case where using diatonic nomenclature for specific
intervals can hold us back. You're looking at fourths, not
thirds, which contain 5:4s and 6:5s. The fact that they form
a circle is I think not as important as the fact that there
are a lot of them.

I believe your procedure effectively finds scales that have
lots of 5-limit triads and a map from them to a pattern of
scale steps, like 1-4-7.

The other 'circle' (I'll assume it is one) you'll find in the
decatonic scale is 9ths, containing 7:4 and 12:7.

I believe the real procedure to use here is to find a comma
that bridges between two consonances (like in the above
examples; they need not be of the same odd limit, but they'll
tend to be since getting a lot of chords in JI favors using
otonal/utonal relationships) and find periodicity blocks of
5-10 notes based on that comma, and other commas which are
tempered out.

Both Paul and Kalle attempted to search for scales like this,
but a properly thorough search hasn't been done yet.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:

> Here's a case where using diatonic nomenclature for specific
> intervals can hold us back. You're looking at fourths, not
> thirds, which contain 5:4s and 6:5s.

I don't know why you say I am looking at fourths. I was going to do
the circle of major thirds and fourths related to Hanson[11], but
these are not proper so I need another way of cherry-picking the best.

The fact that they form
> a circle is I think not as important as the fact that there
> are a lot of them.

Agreed, but the circle does matter. For one thing, you get a cycle of
harmonically linked triads, including perhaps diminished and augmeted
triads.

> The other 'circle' (I'll assume it is one) you'll find in the
> decatonic scale is 9ths, containing 7:4 and 12:7.

Or looking at complements within 600 cents, 8/7 and 7/6.

> I believe the real procedure to use here is to find a comma
> that bridges between two consonances (like in the above
> examples; they need not be of the same odd limit, but they'll
> tend to be since getting a lot of chords in JI favors using
> otonal/utonal relationships) and find periodicity blocks of
> 5-10 notes based on that comma, and other commas which are
> tempered out.

Bridges between two consonances doesn't sound much like a comma which
comletes a cycle, which is where I started from. Cycles must have such
a comma or they can't close.

Of course if you allow yourself bigger scales you can find beasts like
the cycles of thirds in 17 notes of schismatic, or a kind of ultimate,
the comton/waage circle of 24 thirds.

>> Here's a case where using diatonic nomenclature for specific
>> intervals can hold us back. You're looking at fourths, not
>> thirds, which contain 5:4s and 6:5s.
>
>I don't know why you say I am looking at fourths.

Because 5:4 and 6:5 are fourths in the decatonic scale. Calling
the intervals 5:4 and 6:5 "thirds" is lame factor 3000 in my opinion.

>> The fact that they form
>> a circle is I think not as important as the fact that there
>> are a lot of them.
>
>Agreed, but the circle does matter. For one thing, you get a cycle of
>harmonically linked triads, including perhaps diminished and augmeted
>triads.

If every mode has a "third" on it, they'd have to be linked.
If we're talking about a scale with "thirds" in half of its
modes, I suppose it is better if they're linked. I'm not sure
how big a portion of modes can have unlinked "thirds", but it
seems like it might not be very high.

>> I believe the real procedure to use here is to find a comma
>> that bridges between two consonances (like in the above
>> examples; they need not be of the same odd limit, but they'll
>> tend to be since getting a lot of chords in JI favors using
>> otonal/utonal relationships) and find periodicity blocks of
>> 5-10 notes based on that comma, and other commas which are
>> tempered out.
>
>Bridges between two consonances doesn't sound much like a comma
>which comletes a cycle, which is where I started from. Cycles
>must have such a comma or they can't close.

The comma in question for "thirds" is 25:24. For 12:7 and 7:4,
it's 49:48.

I know you started from cycles, and they may interest you (more
power to you), but I wanted to point out that what makes these
kind of cycles different from the usual MOS kind is that they're
made up of more than one specific interval, but still one generic
interval. In my view this is the single most important, drop-down
drag-out feature of "diatonic" scales. Cycles optional.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:

> I believe the real procedure to use here is to find a comma
> that bridges between two consonances (like in the above
> examples; they need not be of the same odd limit, but they'll
> tend to be since getting a lot of chords in JI favors using
> otonal/utonal relationships) and find periodicity blocks of
> 5-10 notes based on that comma, and other commas which are
> tempered out.
>
> Both Paul and Kalle attempted to search for scales like this,
> but a properly thorough search hasn't been done yet.

When and where did they do this, by the way? I spent a lot of
gunpowder on MOS, looking at how various choices of chroma affected
not just circles of notds, but circles of chords, and I looked at this
circle of notes business before also. But I can't recall anyone else
doing the like.

>> I believe the real procedure to use here is to find a comma
>> that bridges between two consonances (like in the above
>> examples; they need not be of the same odd limit, but they'll
>> tend to be since getting a lot of chords in JI favors using
>> otonal/utonal relationships) and find periodicity blocks of
>> 5-10 notes based on that comma, and other commas which are
>> tempered out.
>>
>> Both Paul and Kalle attempted to search for scales like this,
>> but a properly thorough search hasn't been done yet.
>
>When and where did they do this, by the way? I spent a lot of
>gunpowder on MOS, looking at how various choices of chroma affected
>not just circles of notds, but circles of chords, and I looked at this
>circle of notes business before also. But I can't recall anyone else
>doing the like.

Yes, it's equivalent to looking for MOS where L-s = a comma
that bridges two consonances.

I don't recall the gunpowder you mention. Can you find it
(or tell me what to look for)?

Kalle and Paul did what I mentioned on this list, maybe in
2003 (?), but I ...

...can't find it.

-Carl

I still can't find the original thread with Kalle, but in a
thread called "this T[n] business" you seemed to be looking
for MOS and NMOS with square- and triangular-number chroma.

-Carl

At 09:58 AM 7/22/2006, you wrote:
>>> I believe the real procedure to use here is to find a comma
>>> that bridges between two consonances (like in the above
>>> examples; they need not be of the same odd limit, but they'll
>>> tend to be since getting a lot of chords in JI favors using
>>> otonal/utonal relationships) and find periodicity blocks of
>>> 5-10 notes based on that comma, and other commas which are
>>> tempered out.
>>>
>>> Both Paul and Kalle attempted to search for scales like this,
>>> but a properly thorough search hasn't been done yet.
>>
>>When and where did they do this, by the way? I spent a lot of
>>gunpowder on MOS, looking at how various choices of chroma affected
>>not just circles of notds, but circles of chords, and I looked at this
>>circle of notes business before also. But I can't recall anyone else
>>doing the like.
>
>Yes, it's equivalent to looking for MOS where L-s = a comma
>that bridges two consonances.
>
>I don't recall the gunpowder you mention. Can you find it
>(or tell me what to look for)?
>
>Kalle and Paul did what I mentioned on this list, maybe in
>2003 (?), but I ...
>
>...can't find it.
>
>-Carl
>
>
>
>
>
>Yahoo! Groups Links
>
>
>
>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> I still can't find the original thread with Kalle, but in a
> thread called "this T[n] business" you seemed to be looking
> for MOS and NMOS with square- and triangular-number chroma.
>
> -Carl

Hi Carl,

The thread was in the tuning list, it starts about here

/tuning/topicId_41347.html#41360

We were discussing scales where the same scale pattern produces major
and minor tetrads in root position and the fact that if the scales are
tuned in rank 2 temperaments it means that (25/24)/(49/48)=50/49
should be tempered out.

Lately I've been thinking about this more generally and will post
about it soon.

Kalle

>> I still can't find the original thread with Kalle, but in a
>> thread called "this T[n] business" you seemed to be looking
>> for MOS and NMOS with square- and triangular-number chroma.
>>
>> -Carl
>
>Hi Carl,
>
>The thread was in the tuning list, it starts about here
>
>/tuning/topicId_41347.html#41360
>
>We were discussing scales where the same scale pattern produces major
>and minor tetrads in root position and the fact that if the scales are
>tuned in rank 2 temperaments it means that (25/24)/(49/48)=50/49
>should be tempered out.
>
>Lately I've been thinking about this more generally and will post
>about it soon.

Great. I think there's a potential universe of scales that
transform things like 7:9:11 or 9:11:13 to 4:5:6, etc. etc.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:

> Great. I think there's a potential universe of scales that
> transform things like 7:9:11 or 9:11:13 to 4:5:6, etc. etc.

This is true but I'm afraid it's more modest than that. That is, I'm
still just thinking about otonal/utonal chord pairs.

Here it comes:

in diatonic scale the most common interval is the perfect fifth (or
fourth). That is the interval that is shared by major and minor triads
in root position. Given that the diatonic scale is a constant
structure the number of complete triads can't be larger than the
number of fifths.

Let us choose some other pair of chords i.e.

1/1 5/4 3/2 and
1/1 4/3 8/5

their 1st inversions are

1/1 6/5 8/5
1/1 6/5 3/2

which shows that (an approximate) 6/5 should be the most common
interval in a constant structure with as many 1/1 5/4 3/2 or 1/1 4/3
8/5 chords as possible.

Now let's start with a "scale"

1/1
5/4
3/2
2/1

and add pitches a 6/5 away from those in the scale

1/1
6/5
5/4
3/2
9/5
2/1

This is a constant structure and already contains two 1/1 5/4 3/2
chords and one 1/1 4/3 8/5 chord which are produced by the same scale
pattern! Tempering out 27/25 gives one more 1/1 4/3 8/5 chord. The
resulting scale is Bug[5].

We can add more pitches a 6/5 apart (this is easy to do in Scala:
generate a polychordal scale with intervals 3:4:5 and a multiplicity of 3)

1/1
10/9
125/108
4/3
25/18
5/3
50/27
2/1

The resulting scale is not a constant structure but one can easily
fill the gaps. To get a constant structure the interval matrix

1/1 : 10/9 125/108 4/3 25/18 5/3 50/27 2/1
10/9 : 25/24 6/5 5/4 3/2 5/3 9/5 2/1
125/108: 144/125 6/5 36/25 8/5 216/125 48/25 2/1
4/3 : 25/24 5/4 25/18 3/2 5/3 125/72 2/1
25/18 : 6/5 4/3 36/25 8/5 5/3 48/25 2/1
5/3 : 10/9 6/5 4/3 25/18 8/5 5/3 2/1
50/27 : 27/25 6/5 5/4 36/25 3/2 9/5 2/1
2/1

tells that one should multiply 25/18 with 25/24 or 144/125 or 10/9 or
27/25. I choose 25/18 * 27/25 = 3/2.

We get a scale with 3 otonal and 2 utonal triads with the same scale
pattern. To get the extra utonal triad one could temper out 648:625.
This results in Dimipent[8], the octatonic scale.

Now let's start with a polychordal scale with intervals 3:4:5 and a
multiplicity of 4:

1/1
10/9
125/108
4/3
25/18
125/81
5/3
50/27
625/324
2/1

This again is not a constant structure but studying the interval matrix

1/1 : 10/9 125/108 4/3 25/18 125/81 5/3 50/27 625/324 2/1
10/9 : 25/24 6/5 5/4 25/18 3/2 5/3 125/72 9/5 2/1
125/108: 144/125 6/5 4/3 36/25 8/5 5/3 216/125 48/25 2/1
4/3 : 25/24 125/108 5/4 25/18 625/432 3/2 5/3 125/72 2/1
25/18 : 10/9 6/5 4/3 25/18 36/25 8/5 5/3 48/25 2/1
125/81 : 27/25 6/5 5/4 162/125 36/25 3/2 216/125 9/5 2/1
5/3 : 10/9 125/108 6/5 4/3 25/18 8/5 5/3 50/27 2/1
50/27 : 25/24 27/25 6/5 5/4 36/25 3/2 5/3 9/5 2/1
625/324: 648/625 144/125 6/5 864/625 36/25 8/5 216/125 48/25 2/1
2/1

suggest adding 125/81 * 25/24 = 3125/1944 and 125/108 * 648/625 = 6/5.

The resulting 11-tone constant structure contains 4 otonal and 3
utonal chords. Tempering out 15625:15552 gives you an extra utonal triad.

A chain of five 3:4:5s results in a 11-tone scale with five 1/1 5/4
3/2 chords and four 1/1 4/3 8/5 chords. Tempering out 15625:15552
gives an extra utonal triad and the resulting scale is Hanson[11].

The same thing seems to work with tetrads too.

Kalle

>> Great. I think there's a potential universe of scales that
>> transform things like 7:9:11 or 9:11:13 to 4:5:6, etc. etc.
>
>
>This is true but I'm afraid it's more modest than that. That is, I'm
>still just thinking about otonal/utonal chord pairs.

A good place to start, since you get an extra common tone before
you start to temper.

>Here it comes:
>
>in diatonic scale the most common interval is the perfect fifth (or
>fourth). That is the interval that is shared by major and minor triads
>in root position. Given that the diatonic scale is a constant
>structure the number of complete triads can't be larger than the
>number of fifths.
>
>Let us choose some other pair of chords i.e.
>
>1/1 5/4 3/2 and
>1/1 4/3 8/5
>
>their 1st inversions are
>
>1/1 6/5 8/5
>1/1 6/5 3/2
>
>which shows that (an approximate) 6/5 should be the most common
>interval in a constant structure with as many 1/1 5/4 3/2 or 1/1 4/3
>8/5 chords as possible.
>
>Now let's start with a "scale"
>
>1/1
>5/4
>3/2
>2/1
>
>and add pitches a 6/5 away from those in the scale
>
>1/1
>6/5
>5/4
>3/2
>9/5
>2/1
>
>This is a constant structure and already contains two 1/1 5/4 3/2
>chords and one 1/1 4/3 8/5 chord which are produced by the same scale
>pattern! Tempering out 27/25 gives one more 1/1 4/3 8/5 chord. The
>resulting scale is Bug[5].
>
>We can add more pitches a 6/5 apart (this is easy to do in Scala:
>generate a polychordal scale with intervals 3:4:5 and a multiplicity of 3)
>
>1/1
>10/9
>125/108
>4/3
>25/18
>5/3
>50/27
>2/1
>
>The resulting scale is not a constant structure but one can easily
>fill the gaps. To get a constant structure the interval matrix
>
>1/1 : 10/9 125/108 4/3 25/18 5/3 50/27 2/1
>10/9 : 25/24 6/5 5/4 3/2 5/3 9/5 2/1
>125/108: 144/125 6/5 36/25 8/5 216/125 48/25 2/1
>4/3 : 25/24 5/4 25/18 3/2 5/3 125/72 2/1
>25/18 : 6/5 4/3 36/25 8/5 5/3 48/25 2/1
>5/3 : 10/9 6/5 4/3 25/18 8/5 5/3 2/1
>50/27 : 27/25 6/5 5/4 36/25 3/2 9/5 2/1
>2/1
>
>tells that one should multiply 25/18 with 25/24 or 144/125 or 10/9 or
>27/25. I choose 25/18 * 27/25 = 3/2.
>
>We get a scale with 3 otonal and 2 utonal triads with the same scale
>pattern. To get the extra utonal triad one could temper out 648:625.
>This results in Dimipent[8], the octatonic scale.
>
>Now let's start with a polychordal scale with intervals 3:4:5 and a
>multiplicity of 4:
>
>1/1
>10/9
>125/108
>4/3
>25/18
>125/81
>5/3
>50/27
>625/324
>2/1
>
>This again is not a constant structure but studying the interval matrix
>
>1/1 : 10/9 125/108 4/3 25/18 125/81 5/3 50/27 625/324 2/1
>10/9 : 25/24 6/5 5/4 25/18 3/2 5/3 125/72 9/5 2/1
>125/108: 144/125 6/5 4/3 36/25 8/5 5/3 216/125 48/25 2/1
>4/3 : 25/24 125/108 5/4 25/18 625/432 3/2 5/3 125/72 2/1
>25/18 : 10/9 6/5 4/3 25/18 36/25 8/5 5/3 48/25 2/1
>125/81 : 27/25 6/5 5/4 162/125 36/25 3/2 216/125 9/5 2/1
>5/3 : 10/9 125/108 6/5 4/3 25/18 8/5 5/3 50/27 2/1
>50/27 : 25/24 27/25 6/5 5/4 36/25 3/2 5/3 9/5 2/1
>625/324: 648/625 144/125 6/5 864/625 36/25 8/5 216/125 48/25 2/1
>2/1
>
>suggest adding 125/81 * 25/24 = 3125/1944 and 125/108 * 648/625 = 6/5.
>
>The resulting 11-tone constant structure contains 4 otonal and 3
>utonal chords. Tempering out 15625:15552 gives you an extra utonal triad.
>
>A chain of five 3:4:5s results in a 11-tone scale with five 1/1 5/4
>3/2 chords and four 1/1 4/3 8/5 chords. Tempering out 15625:15552
>gives an extra utonal triad and the resulting scale is Hanson[11].
>
>
>The same thing seems to work with tetrads too.

Sweet! Great work!

It'd be nice to do a search for these. For examples like
9:11:13 -> 4:5:6, NMOS are probably needed (I think it's possible
to get away with a plain MOS in such cases where L-s can
serve as two separate chroma via transformation by a comma
that *is* tempered out).

-Carl

Hi,

I realized that all that hassle about filling the gaps can be bypassed
by generating polychordal scales by chaining 3:4:5s, 4:5:6s, 5:6:8s,
3:5:7s or 4:5:7:12s etc. and keeping record of all the resulting one
step intervals. These are potential commas for periodicity blocks. One
should also divide these with the chord pair chroma to get more commas.

Kalle

--- In tuning-math@yahoogroups.com, "Kalle Aho" <kalleaho@...> wrote:
>
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@> wrote:
>
> > Great. I think there's a potential universe of scales that
> > transform things like 7:9:11 or 9:11:13 to 4:5:6, etc. etc.
>
>
> This is true but I'm afraid it's more modest than that. That is, I'm
> still just thinking about otonal/utonal chord pairs.
>
> Here it comes:
>
> in diatonic scale the most common interval is the perfect fifth (or
> fourth). That is the interval that is shared by major and minor triads
> in root position. Given that the diatonic scale is a constant
> structure the number of complete triads can't be larger than the
> number of fifths.
>
> Let us choose some other pair of chords i.e.
>
> 1/1 5/4 3/2 and
> 1/1 4/3 8/5
>
> their 1st inversions are
>
> 1/1 6/5 8/5
> 1/1 6/5 3/2
>
> which shows that (an approximate) 6/5 should be the most common
> interval in a constant structure with as many 1/1 5/4 3/2 or 1/1 4/3
> 8/5 chords as possible.
>
> Now let's start with a "scale"
>
> 1/1
> 5/4
> 3/2
> 2/1
>
> and add pitches a 6/5 away from those in the scale
>
> 1/1
> 6/5
> 5/4
> 3/2
> 9/5
> 2/1
>
> This is a constant structure and already contains two 1/1 5/4 3/2
> chords and one 1/1 4/3 8/5 chord which are produced by the same scale
> pattern! Tempering out 27/25 gives one more 1/1 4/3 8/5 chord. The
> resulting scale is Bug[5].
>
> We can add more pitches a 6/5 apart (this is easy to do in Scala:
> generate a polychordal scale with intervals 3:4:5 and a multiplicity
of 3)
>
> 1/1
> 10/9
> 125/108
> 4/3
> 25/18
> 5/3
> 50/27
> 2/1
>
> The resulting scale is not a constant structure but one can easily
> fill the gaps. To get a constant structure the interval matrix
>
> 1/1 : 10/9 125/108 4/3 25/18 5/3 50/27 2/1
> 10/9 : 25/24 6/5 5/4 3/2 5/3 9/5 2/1
> 125/108: 144/125 6/5 36/25 8/5 216/125 48/25 2/1
> 4/3 : 25/24 5/4 25/18 3/2 5/3 125/72 2/1
> 25/18 : 6/5 4/3 36/25 8/5 5/3 48/25 2/1
> 5/3 : 10/9 6/5 4/3 25/18 8/5 5/3 2/1
> 50/27 : 27/25 6/5 5/4 36/25 3/2 9/5 2/1
> 2/1
>
> tells that one should multiply 25/18 with 25/24 or 144/125 or 10/9 or
> 27/25. I choose 25/18 * 27/25 = 3/2.
>
> We get a scale with 3 otonal and 2 utonal triads with the same scale
> pattern. To get the extra utonal triad one could temper out 648:625.
> This results in Dimipent[8], the octatonic scale.
>
> Now let's start with a polychordal scale with intervals 3:4:5 and a
> multiplicity of 4:
>
> 1/1
> 10/9
> 125/108
> 4/3
> 25/18
> 125/81
> 5/3
> 50/27
> 625/324
> 2/1
>
> This again is not a constant structure but studying the interval matrix
>
> 1/1 : 10/9 125/108 4/3 25/18 125/81 5/3 50/27 625/324 2/1
> 10/9 : 25/24 6/5 5/4 25/18 3/2 5/3 125/72 9/5 2/1
> 125/108: 144/125 6/5 4/3 36/25 8/5 5/3 216/125 48/25 2/1
> 4/3 : 25/24 125/108 5/4 25/18 625/432 3/2 5/3 125/72 2/1
> 25/18 : 10/9 6/5 4/3 25/18 36/25 8/5 5/3 48/25 2/1
> 125/81 : 27/25 6/5 5/4 162/125 36/25 3/2 216/125 9/5 2/1
> 5/3 : 10/9 125/108 6/5 4/3 25/18 8/5 5/3 50/27 2/1
> 50/27 : 25/24 27/25 6/5 5/4 36/25 3/2 5/3 9/5 2/1
> 625/324: 648/625 144/125 6/5 864/625 36/25 8/5 216/125 48/25 2/1
> 2/1
>
> suggest adding 125/81 * 25/24 = 3125/1944 and 125/108 * 648/625 = 6/5.
>
> The resulting 11-tone constant structure contains 4 otonal and 3
> utonal chords. Tempering out 15625:15552 gives you an extra utonal
triad.
>
> A chain of five 3:4:5s results in a 11-tone scale with five 1/1 5/4
> 3/2 chords and four 1/1 4/3 8/5 chords. Tempering out 15625:15552
> gives an extra utonal triad and the resulting scale is Hanson[11].
>
>
> The same thing seems to work with tetrads too.