Using small intervals in scales

Hi tuners,

this is interesting. I remember myself having said the other day that I didn't like using intervals less than about 60 cents as melodic steps as I found them rather "unnatural", for one thing, and, for another thing, not easy to sing or to play on an non-fixed-pitch instrument (particularly a sequence of more like these played right one after another sounds pretty "harsh" to my ear sometimes). Then, a few months later, I found yet another point to this topic. And today, this was "reconfirmed" to me in such a strong way that I just think it deserves being shared.
I didn't manage to follow your "con/dissonance" discussion very carefully but I think this might be one of many possible examples where larger intervals can serve better than smaller ones.
This afternoon, I was improvising some sort of variations on an old Baroque "ciacona" (hope I can call it like that) in a tuning close to meantone (only some fifths were slightly wider or narrower than others). None of the voices used any step between consecutive tones which would be smaller than about 115 cents or so. I ran it through a very strong reverb effect and the triads started blending nicely. Just for an experiment, I then took a part of my earlier semisixth piece, ran it through the same reverb, and what happened was exactly what I expected would happen. Whenever any of the voices went up or down by the ~56 cent step (which is the distance of 3 octaves minus 8 "semi-sixths"), the difference tones between the overlapping pitches got so slow that I could actually hear beating in some places.
For conclusion, here's what semisixth in reverb sounds like:
www.sendspace.com/file/o8hv0u

Petr

Well I for one love small intervals. They're great for
jazz -- scat singing for example. And great in other
applications too.

-Carl

One funny paradox....
...Is that very small intervals can be very consonant while small intervals are not at all consonant but then get more consonant as they get larger.  I believe anything within about 8-cents of a note (IE a very small interval) will sound fairly consonant when played with that note...and you can use this purely for emotional effect, as micro-tonal often does, without "destroying" the naturalness of the tone. 
 
I have read about this and found it works in practice: it's as if something very close to a note is read in the mind as "the same note" while something kind of close makes the brain stumble asking the question "is this the same note or not?" thus making that note sound un-natural.
 

--- On Sat, 11/8/08, Carl Lumma <carl@...> wrote:

From: Carl Lumma <carl@...>
Subject: [tuning] Re: Using small intervals in scales
To: tuning@yahoogroups.com
Date: Saturday, November 8, 2008, 10:17 PM

Well I for one love small intervals. They're great for
jazz -- scat singing for example. And great in other
applications too.

-Carl

This sounded extremely cool to me (in my tongue, cool=good)

If you have patience, can you give tuning?

caleb

On Nov 8, 2008, at 6:26 PM, Petr Pařízek wrote:

> Hi tuners,
>
> this is interesting. I remember myself having said the other day
> that I
> didn't like using intervals less than about 60 cents as melodic
> steps as I
> found them rather "unnatural", for one thing, and, for another
> thing, not
> easy to sing or to play on an non-fixed-pitch instrument
> (particularly a
> sequence of more like these played right one after another sounds
> pretty
> "harsh" to my ear sometimes). Then, a few months later, I found yet
> another
> point to this topic. And today, this was "reconfirmed" to me in such a
> strong way that I just think it deserves being shared.
> I didn't manage to follow your "con/dissonance" discussion very
> carefully
> but I think this might be one of many possible examples where larger
> intervals can serve better than smaller ones.
> This afternoon, I was improvising some sort of variations on an old
> Baroque
> "ciacona" (hope I can call it like that) in a tuning close to
> meantone (only
> some fifths were slightly wider or narrower than others). None of
> the voices
> used any step between consecutive tones which would be smaller than
> about
> 115 cents or so. I ran it through a very strong reverb effect and
> the triads
> started blending nicely. Just for an experiment, I then took a part
> of my
> earlier semisixth piece, ran it through the same reverb, and what
> happened
> was exactly what I expected would happen. Whenever any of the voices
> went up
> or down by the ~56 cent step (which is the distance of 3 octaves
> minus 8
> "semi-sixths"), the difference tones between the overlapping pitches > got so
> slow that I could actually hear beating in some places.
> For conclusion, here's what semisixth in reverb sounds like:
> www.sendspace.com/file/o8hv0u
>
> Petr
>
>
>
> ------------------------------------
>
> You can configure your subscription by sending an empty email to one
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>
>

Ptolemy put the smallest melodic limit at the 45/44. In a blank test outside of context a 55/54 i have been fine with. D haven't tested much further.
But context can define things in uncanny ways. In a structure such as the eikosany one adds pitches to complete a constant structure, some can be as small as 4.5 cents. In that context in using the scale where each note has a unique role, i would be at a lost if it wasn't there. but possibly this still means that i might be hearing a tone that fluctuates between two values. like a double star, or something with a unique 'life'.
I think Carl is touching upon the same type of phenomenon, that sometimes we need notes that moves. Scelsi would have gone along with this.
--

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

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

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

Caleb wrote:

> If you have patience, can you give tuning?

It's a 16-tone chain of, let's say, "extremely large major thirds". More specifically, it's a 2D temperament where the octave is 2/1 and the generator is about 443 cents -- or, to be precise, 16th root of 60. The whole piece is the one I posted here back in May. But I can repost, if you like. The new link is here: www.sendspace.com/file/sfe1v8

Petr

1st, caleb picks jaw off floor, all covered in lint and dust-mites.

2cnd--that's an amazing piece--you hear these pure triads, and a lot
of I-VI, and then at around 1:03 the first strange shift is a
wonderful surprise

when the plucked instrument came in, it was welcome, and I wanted it
to continue or come back

but the rest of the piece was even more interesting than the first part

=====================================

3rd--allright, I know nothing, or very little about tuning, I guess.

16th root of 60 is 1.29161908

chain of 16 443's: I know this is approximate, but I just want to see:

0
443
886
1329-1200=129
1772-1200=572
2215
2658
3101
3544
3987
4430
4873
5316
5759
6202
6645
7088

questions: am I almost on the right track?

how do I get from 1.2916...to 443?

On Nov 9, 2008, at 5:36 AM, Petr Pařízek wrote:

> Caleb wrote:
>
>> If you have patience, can you give tuning?
>
> It's a 16-tone chain of, let's say, "extremely large major thirds".
> More
> specifically, it's a 2D temperament where the octave is 2/1 and the
> generator is about 443 cents -- or, to be precise, 16th root of 60.
> The
> whole piece is the one I posted here back in May. But I can repost,
> if you
> like. The new link is here: www.sendspace.com/file/sfe1v8
>
> Petr
>
>
>
>
>
>
> ------------------------------------
>
> You can configure your subscription by sending an empty email to one
> of these addresses (from the address at which you receive the list):
> tuning-subscribe@yahoogroups.com - join the tuning group.
> tuning-unsubscribe@yahoogroups.com - leave the group.
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> tuning-help@yahoogroups.com - receive general help information.
> Yahoo! Groups Links
>
>
>

This is my experience, too.
I love neutral or semitones chords, while I have some difficulties with
quartertones chords (outside of pure melodic purpose). And then I love comma
chords again and all totally beating things myself, as large as 56/55 or
even 49/48 (sometimes).

I wonder if it has not some relation with their difference tone range, or
some musical memory of that.
Most people love "normally slow" vibratos, but not when they become two
fast, like between normally "rythmical" tempos and the lowest bass sounds.
Same with beating chords ?

About "small" melodic pitch variations, such as indian music gamaks, and
many more examples, it does not necessarily means these are expressing one
"same" note, but simply their double nature, in a context where those commas
are necessary.
I suppose many of you already experienced the very indian-style glides you
can do between 19/16 and 6/5, both minor thirds but of two complementary
emotions.
(or even in the "triple star" version if you wish, along with 32/27...)

- - - - - - - - - - -
Jacques Dudon

le 9/11/08 9:36, Michael Sheiman à djtrancendance@... a écrit :

One funny paradox....
...Is that very small intervals can be very consonant while small intervals
are not at all consonant but then get more consonant as they get larger. I
believe anything within about 8-cents of a note (IE a very small interval)
will sound fairly consonant when played with that note...and you can use
this purely for emotional effect, as micro-tonal often does, without
"destroying" the naturalness of the tone.

I have read about this and found it works in practice: it's as if something
very close to a note is read in the mind as "the same note" while something
kind of close makes the brain stumble asking the question "is this the same
note or not?" thus making that note sound un-natural.

--- On Sat, 11/8/08, Carl Lumma <carl@...> wrote:
From: Carl Lumma <carl@...>
Subject: [tuning] Re: Using small intervals in scales
To: tuning@yahoogroups.com
Date: Saturday, November 8, 2008, 10:17 PM

Well I for one love small intervals. They're great for
jazz -- scat singing for example. And great in other
applications too.

-Carl

second listen:

the plucked instrument doesn't go away, it just blends

make that first startling change 1:00, not 1:03

nice how the harmonic rhythm really picks up toward the end

On Nov 9, 2008, at 8:01 AM, caleb morgan wrote:

>
> 1st, caleb picks jaw off floor, all covered in lint and dust-mites.
>
> 2cnd--that's an amazing piece--you hear these pure triads, and a lot
> of I-VI, and then at around 1:03 the first strange shift is a
> wonderful surprise
>
> when the plucked instrument came in, it was welcome, and I wanted it
> to continue or come back
>
> but the rest of the piece was even more interesting than the first
> part
>
> =====================================
>
> 3rd--allright, I know nothing, or very little about tuning, I guess.
>
> 16th root of 60 is 1.29161908
>
> chain of 16 443's: I know this is approximate, but I just want to
> see:
>
> 0
> 443
> 886
> 1329-1200=129
> 1772-1200=572
> 2215
> 2658
> 3101
> 3544
> 3987
> 4430
> 4873
> 5316
> 5759
> 6202
> 6645
> 7088
>
> questions: am I almost on the right track?
>
> how do I get from 1.2916...to 443?
>
>
>
>
>
>
>
> On Nov 9, 2008, at 5:36 AM, Petr Pařízek wrote:
>
>> Caleb wrote:
>>
>>> If you have patience, can you give tuning?
>>
>> It's a 16-tone chain of, let's say, "extremely large major thirds".
>> More
>> specifically, it's a 2D temperament where the octave is 2/1 and the
>> generator is about 443 cents -- or, to be precise, 16th root of 60.
>> The
>> whole piece is the one I posted here back in May. But I can repost,
>> if you
>> like. The new link is here: www.sendspace.com/file/sfe1v8
>>
>> Petr
>>
>>
>>
>>
>>
>>
>> ------------------------------------
>>
>> You can configure your subscription by sending an empty email to one
>> of these addresses (from the address at which you receive the list):
>> tuning-subscribe@yahoogroups.com - join the tuning group.
>> tuning-unsubscribe@yahoogroups.com - leave the group.
>> tuning-nomail@yahoogroups.com - turn off mail from the group.
>> tuning-digest@yahoogroups.com - set group to send daily digests.
>> tuning-normal@yahoogroups.com - set group to send individual emails.
>> tuning-help@yahoogroups.com - receive general help information.
>> Yahoo! Groups Links
>>
>>
>>
>
>
>
> ------------------------------------
>
> You can configure your subscription by sending an empty email to one
> of these addresses (from the address at which you receive the list):
> tuning-subscribe@yahoogroups.com - join the tuning group.
> tuning-unsubscribe@yahoogroups.com - leave the group.
> tuning-nomail@yahoogroups.com - turn off mail from the group.
> tuning-digest@yahoogroups.com - set group to send daily digests.
> tuning-normal@yahoogroups.com - set group to send individual emails.
> tuning-help@yahoogroups.com - receive general help information.
> Yahoo! Groups Links
>
>
>

Hi Caleb

Thanks for your great appreciation.

You wrote:

> 16th root of 60 is 1.29161908

Agreed.

> chain of 16 443's: I know this is approximate, but I just want to see:
>
> 0
> 443
> 886
> 1329-1200=129
> 1772-1200=572
> 2215
> 2658
> 3101
> 3544
> 3987
> 4430
> 4873
> 5316
> 5759
> 6202
> 6645
> 7088
>
> questions: am I almost on the right track?

You are, only the last interval (7088 cents) is not included because that would result in a 17-tone scale, not 16, after you reduce it to a single octave range.

> how do I get from 1.2916...to 443?

When a pitch shifts an octave higher, it can be expressed either as a linear factor (i.e. frequency ratio) of 2/1 or as a size of +1200 cents. A falling octave, on the other hand, can be written either as a linear factor of 1/2 or as a size of -1200 cents. Similarly, the linear factor for my version of the "semisixth" interval is the 16th root of 60, which you can convert to cents either by converting this value in exactly the same way as I've described, or by converting 60 to cents and then dividing the result by 16. Whatever way you do it, you'll end up with ~443.01679 cents. -- If you are also interested in why I chose this particular value, you can ask me as well -- it has a lot to do with approximating 5-limit intervals.

Petr

Michael Sheiman wrote:

> I have read about this and found it works in practice:
> it's as if something very close to a note is read in the mind as "the same note"
> while something kind of close makes the brain stumble asking the question
> "is this the same note or not?" thus making that note sound un-natural.

Couldn't say it better.

Petr

Thanks Petr, this is a delight to hear !
It reminds me in the beginning of a harmonic flute (10-11-12 harmonics, with
that similar "pushed 11th harmonic" effect).
But such a simple and wonderful system -
That apparently lets you have very close to JI fifths and major thirds.
And a very present, but interesting I think, 31/24 approximation.
My calculator is out of service but I presume it's close.

May I add that this generator is giving you almost perfectly coherent
difference tones, as it shows in this JI-approximated series :
56 72 93 120...
where 93 - 72 = 21 (8/3 of 56) and 120 - 93 = 27 (8/3 of 72)

I would think it adds to the musicality of the shifts where you use it.

- - - -
Jacques

le 9/11/08 11:36, Petr Parízek à p.parizek@... a écrit :

> It's a 16-tone chain of, let's say, "extremely large major thirds". More
> specifically, it's a 2D temperament where the octave is 2/1 and the
> generator is about 443 cents -- or, to be precise, 16th root of 60. The
> whole piece is the one I posted here back in May. But I can repost, if you
> like. The new link is here: www.sendspace.com/file/sfe1v8
>
> Petr

To Jacques:

Thank you for your interesting suggestions.
My aim was to find a system specifically for tempering out the comma of 78732/78125 (i.e. prime exponents of "2 9 -7"). This is especially noticeable there at around 2:33 where the final chord of the progression is the same as the starting chord. If you try, for example, to do this in 12-EDO, then you'll end up one step higher than where you started because this interval doesn't turn into unison in 12-EDO.

Petr

You wrote:

> 16th root of 60 is 1.29161908

this scale is close to a recurrent sequence ( i found awhile ago) which converges on 441.716.
Hn=Hn-5+2Hn-4
which means that
O
443 an octave higher ( that is x2)
will almost form a proportional triad with 2215

A Quasi equal beating triad
if you lower the top one an octave but by then the errors might have accumulated too much.
worth trying as another chord source with this scale.

By coincidence i was looking at this scale yesterday as possible pelog analog, up to nine places
--

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

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

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

On Sun, 2008-11-09 at 21:30 +1100, Kraig Grady wrote:

> Ptolemy put the smallest melodic limit at the 45/44. In a blank test
> outside of context a 55/54 i have been fine with. D haven't tested much
> further.

45/44 is just a couple mills off of 1/31 of an octave, by the way.

In 72-tone, I've been using either 33 1/3 or 50 cents as a limit,
depending on a given situation, since the pitches of the 31-tone Miracle
subset are all two or three commas apart. But 33 cents can also function
as a comma, when treated as an equivalent to 64/63.

~D.

Too bad, the sendspace site won't let me hear your piece anymore today -
May be I should change of computer.
Thanks for the explanation anyway !
- - - - - - -
Jacques

le 9/11/08 19:57, Petr Parízek à p.parizek@... a écrit :

To Jacques:

Thank you for your interesting suggestions.
My aim was to find a system specifically for tempering out the comma of
78732/78125 (i.e. prime exponents of "2 9 -7"). This is especially
noticeable there at around 2:33 where the final chord of the progression is
the same as the starting chord. If you try, for example, to do this in
12-EDO, then you'll end up one step higher than where you started because
this interval doesn't turn into unison in 12-EDO.

Petr

Hi Kraig,
I don't know if you noticed but in my yesterdays's mail the recurrent
sequence I suggested is totally close to yours -
In fact if you complete my sequence that way :
56 72 93 120 155 200 (...258 333 430 555 and so on),
you will find some of the best possible starting whole numbers for the
sequence you're talking about, where 200 = 144 (two times 72) + 56, then
258 = 186 + 72 and so on.

My sequence points to Petr's generator's particular differential coherence
in his own octavial system, of which Hn = Hn-1 + (3/8) Hn-2 is the simplest
expression.
Since Petr's sequence aproximates so well octave transpositions of harmonics
3 and 5 and their combinations, you can find many other recurrent sequences
that express some of those 5-limit relationships :

1,290 683 644 73 resolves Hn = Hn-1 + (5/8) Hn-4 another one of those
1,290 648 801 35 resolves Hn = Hn-5 + 2 Hn-4 as you mentionned
1,290 569 415 04 resolves Hn = Hn-1 + (3/8) Hn-2 for mine

all these series of course have a strong attractor with 5/3 every two terms,
since
1,290 994 448 736 = square root of 5/3
and also other affinities :
1,290 909 090 9 = 71/55
1,290 322 580 6 = 40/31
1,291 666 666 67 = 31/24

(reminder :)
1,291 619 083 84 (Petr's) = 16th root of 60

Besides that I am not familiar with the terminology you use to explain
Petr's sequence
in the end of your message - I can't understand what you mean by :
"A Quasi equal beating triad
if you lower the top one an octave but by then the errors might have
accumulated too much."

What is supposed to beat the same here ?
Too bad we don't use the same terms - excuse my ignorance.
If you would use ratios, or frequencies exemples, I would probably
understand better.
(If everyone else does, don't bother !)
- - - - - -
Jacques

le 10/11/08 9:53, Kraig Grady à kraiggrady@anaphoria.com a écrit :

You wrote:

> 16th root of 60 is 1.29161908

this scale is close to a recurrent sequence ( i found awhile ago) which
converges on 441.716.
Hn=Hn-5+2Hn-4
which means that
O
443 an octave higher ( that is x2)
will almost form a proportional triad with 2215

A Quasi equal beating triad
if you lower the top one an octave but by then the errors might have
accumulated too much.
worth trying as another chord source with this scale.

Jacques wrote:

> Too bad, the sendspace site won't let me hear your piece anymore today ...

Don't know why the link is not working for you, it's still active.

Petr

HI Jacques ~
BTW i was enjoying your CD recently!

By equal beating i meant that the bottom and middle note will beat at the same rate as the top and middle.
An example might be 32:37:42 where the beating will be 5 (and of course 10 too)
I said quasi because Petr series is not quite this. That you have another series close to his would also give his added resources to play with.
--

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

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

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

yes, I know what equal beating means - my question was :
what are the equal or quasi-equal beating intervals here (either in Petr's
or in your recurrent sequence) ?
(that would work in mine I guess !)

56 72 93 120 155 200...

is it 200 - 144 = 56 or something else ?

- - - - - -
Jacques

le 11/11/08 10:23, Kraig Grady à kraiggrady@... a écrit :

HI Jacques ~
BTW i was enjoying your CD recently!

Glad you enjoyed - I know you're fond of fibonacci patterns, and you
probably noticed many golden rythmical polyrythms and sequences in
"Tournesols" and "Fleurs de lumiere" other pieces !

By equal beating i meant that the bottom and middle note will beat at
the same rate as the top and middle.
An example might be 32:37:42 where the beating will be 5 (and of course
10 too)
I said quasi because Petr series is not quite this. That you have
another series close to his would also give his added resources to play
with.

56 72 93 120 155 200...

is it 200 - 144 = 56 or something else ?

That is it as far as our recurrent sequence
and if you lower the top number we get the proportional triad
56-100-144

--

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

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

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