Yahoo Tuning Groups Ultimate Backup tuning 5-limit maximal tetradic comma pumps

I think Mike will like this.

We can turn a 5-limit major triad into a 5-limit major tetrad by adding
either 15/8 or 5/3, depending on which of the two spans more generators in a
particular temperament.
This allows us to find 5-limit tetradic comma pumps with the maximum number
of tones possible.

For example, in meantone, 15/8 spans the highest number of generators and
5/4 comes right after that. Therefore, we'll go for 8:10:12:15 as our
tetradic model. A possible maximal tetradic comma pump in meantone may
consist the following tones:
C, E, F, A, Bb, D, F#, G, C
Which of the tones we finally use in our first chord or second chord or
whatever, is then upon our personal decision.
One possible rendition is:
C-E-F-A, F-A-Bb-D, D-F#-G-B, G-B-C-E, C-E-F-A

We can apply this concept to other temperaments like this:

! hans_tcp.scl
!
Hanson tetradic comma pump
17
!
497.56810
385.35992
882.92802
180.49612
68.28793
565.85603
1063.42413
951.21595
248.78405
746.35215
634.14397
1131.71207
1019.50388
317.07198
814.64008
702.43190
2/1

! ssixth_tcp.scl
!
Semisixth tetradic comma pump
25
!
386.84967
499.11692
885.96659
72.81627
185.08351
571.93319
684.20043
1071.05011
257.89978
370.16703
757.01670
1143.86638
56.13362
442.98330
829.83297
942.10022
128.94989
515.79957
628.06681
1014.91649
1127.18373
314.03341
700.88308
813.15033
2/1

Other temperaments use more generators for 5/3 than for 15/8 and therefore
we can find maximal tetradic pumps if we use 12:15:18:20 as our tetradic
model.
In the following examples, I've tempered the octaves as well to be able to
get even less mistuning in the target intervals.

! dias_tcp.scl
!
Diaschismatic tetradic comma pump
14
!
885.14082
704.30131
390.22423
209.38472
1094.52554
780.44846
599.60895
285.53187
104.69236
989.83318
808.99367
494.91659
180.83951
1199.21790

! orw_tcp.scl
!
5-limit orwell tetradic comma pump
27
!
884.14911
701.32619
385.26569
69.20520
1086.59188
770.53139
587.70847
271.64797
1155.79708
972.97416
656.91366
340.85317
158.03025
1042.17936
859.35644
543.29594
227.23545
44.41252
928.56163
612.50114
429.67822
113.61772
997.76683
814.94391
498.88342
316.06050
1200.20960

! mir_tcp.scl
!
5-limit miracle tetradic comma pump
32
!
883.80490
700.29356
383.54465
200.03332
1083.83822
767.08931
583.57797
266.82906
83.31772
967.12262
650.37371
466.86238
150.11347
1167.15594
850.40703
533.65812
350.14678
33.39787
1050.44035
733.69144
416.94252
233.43119
1117.23609
933.72475
616.97584
300.22693
116.71559
1000.52050
817.00916
500.26025
183.51134
1200.55381

--------------------

Petr

🔗genewardsmith <genewardsmith@...>

🔗Petr Parízek <petrparizek2000@...>

Gene wrote:

> Not to mention 225/32.

#1. 225/64 is 15/8 squared and therefore adding 225/64 to the list of allowed steps between consecutive pitches may result in stripping out some pitches in situations where there would otherwise be two 15/8s in a row.

#2. In order the procedure could work using no more than two different step sizes before octave reduction, the chords have to be symmetric -- i.e. 8:10:12:15 contains two fifths a major third apart or two major thirds a fifth apart. Likewise, 10:12:15:18 contains two fifths a minor third apart or two minor thirds a fifth apart.

Petr

🔗Petr Parízek <petrparizek2000@...>

I wrote:

> #2. In order the procedure could work using no more than two different > step
> sizes before octave reduction, the chords have to be symmetric -- i.e.
> 8:10:12:15 contains two fifths a major third apart or two major thirds a
> fifth apart. Likewise, 10:12:15:18 contains two fifths a minor third apart
> or two minor thirds a fifth apart.

They obviously don't have to be symmetric within the entire progression, I was inexact.
The fact is that the chords should only be constructed using two different step sizes before octave reduction -- i.e. we can, if we wish, choose the pitches in such a way that we get tetrads like 20:25:30:36. If our step sizes are 5/4 and 6/5, this is perfectly okay.

For example, if we suppose that amity uses 13 generators for 5/1 and 18 generators for 15/1, we can make a maximal tetradic pump by alternating steps of +13 and -18 generators like this:

0
13
-5
8
-10
3
-15
-2
11
-7
6
-12
1
14
-4
9
-9
4
-14
-1
12
-6
7
-11
2
15
-3
10
-8
5
-13
0

Petr

🔗Carl Lumma <carl@...>

🔗Mike Battaglia <battaglia01@...>

This looks like the greatest thing I have ever seen. Will delve into these
soon.

Sent from my iPhone

On Jul 15, 2011, at 2:25 PM, "Petr Pařízek" <petrparizek2000@...>
wrote:

I think Mike will like this.

We can apply this concept to other temperaments like this:

! dias_tcp.scl
!
Diaschismatic tetradic comma pump
14
!
885.14082
704.30131
390.22423
209.38472
1094.52554
780.44846
599.60895
285.53187
104.69236
989.83318
808.99367
494.91659
180.83951
1199.21790

--------------------

Petr

🔗genewardsmith <genewardsmith@...>

🔗Petr Parízek <petrparizek2000@...>

Carl wrote:

> I don't understand the connection between 8:10:12:15
> and 12:15:18:20. How are these related? From the point
> of view of usual chord theory, the former is a 15-limit
> chord and the latter a 9-limit chord. Maybe you are saying
> they are the two most concordant tetrads available in
> 5-limit temperaments?

Basically yes.
If you have an ordinary major triad like 4:5:6, the most straight-forward way to turn it into a tetrad is to add a fifth or fourth from the original third -- i.e. 15/8 is a fifth from 5/4, 5/3 is a fourth from 5/4.

Petr

🔗Petr Parízek <petrparizek2000@...>

Gene wrote:

> Can you explain it? It's not making sense to me.

Instead of expressing the vanishing ratio using exponents of three prime numbers, it's like doing the same thing using exponents of two numbers which don't have to be integers.

For example, in the context of 5-limit triads in amity (now that I've started discussing amity), major thirds use the highest number of generators, followed by minor thirds, and fifths come last. So to get an amity pump containing the maximum number of triads within a contiguous generator range, we can assign, for example, "a=24/5, b=5/64", which makes "a^13 * b^8" equal to 1600000/1594323.

This means that an amity pump using these step sizes would contain 13+8 = 21 pitches. If you divide both "a" and "b" by the 21st root of 1600000/1594323, the last pitch turns out to be 1/1. Reducing the resulting pitch sequence into one octave range then gives the tones suggested for the progression.

That's how most of my comma pump examples were made.

Petr

🔗petrparizek2000 <petrparizek2000@...>

🔗Mike Battaglia <battaglia01@...>

On Fri, Jul 15, 2011 at 9:21 PM, Carl Lumma <carl@...> wrote:
>
> Hi Petr,
>
> I don't understand the connection between 8:10:12:15
> and 12:15:18:20. How are these related? From the point
> of view of usual chord theory, the former is a 15-limit
> chord and the latter a 9-limit chord. Maybe you are saying
> they are the two most concordant tetrads available in
> 5-limit temperaments?

8:10:12:15 is a major 7 chord, and 10:12:15:18 is a minor 7 chord.
12:15:18:20 is a major 6 chord. I'm not sure exactly what you're
saying, but please don't tell me that you mean that major and minor 7
and major 6 chords are dissonant because of odd-limit. That will break
my heart, and leave me a broken shell of a man.

-Mike

🔗Carl Lumma <carl@...>

🔗Mike Battaglia <battaglia01@...>

Petr wants to make comma pumps based on major 7 and minor 7 chords.
Major 7 chords are amazing. My theory for the sound of this chord is
that rather than the notes than fusing into a single virtual pitch,
the chord sort of semi-fuses into a series of virtual pitches that are
all harmonically related to one another in a 4:5:6 ratio, giving you a
kaleidoscope effect. Or, more precisely, the 4:5:6 in the chord fuses,
and the other pitches are sort of there in a background sense sort of
popping in and out of existence slightly and subconsciously
harmonizing with the fundamental, thus mysteriously enabling direct
transmission of the feeling of "bliss" from one human being to another
when the chord is played. That's a nice thing to be able to do
whenever you want. 8:10:12:15:18 is even better.

Some variants -
- 8:12:14:21 or 8:12:14:18:21 is sort of the 7-limit version of this,
with perhaps 8:12:14:18:21 strengthening the effect a bit, but that
21/8 on the end sort of sounds like a "fourth" and messes with my
concept of the root a bit.
- 20:26:30:39 or 20:26:30:39:45 is the obvious 10:13:15 version that
sounds "sadder" than a major 7 chord to my ears.
- 18:33:27:22 is the delightfully cloudy neutral 7 version.
- 16:18:22:24:27:33:36 is the version of this chord applied to
8:9:11:12, which is a chord everyone should play as much of as
possible. It's kaleidoscopic and intense at the same time.
- 16:20:24:22:30:36:33 is a bit more bittersweet and consists of two
similarly interlocking 8:10:11:12 tetrads, and 16:20:24:26:30:36:39 is
pretty trippy and consists of two similarly interlocking 8:10:12:13
tetrads.

So in light of that, I hope you aren't going to tell me that you think
that major 7 chords are dissonant because the 7th needs to resolve
upward, because the strain may simply cause my feeble psyche to just
snap. There's only so much one man can take, after all.

-Mike

On Sat, Jul 16, 2011 at 4:24 AM, genewardsmith
<genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > This looks like the greatest thing I have ever seen. Will delve into these
> > soon.
>
> Can you explain it? It's not making sense to me.
>
>

🔗Mike Battaglia <battaglia01@...>

2011/7/15 Petr Pařízek <petrparizek2000@...>
>
> I think Mike will like this.
>
> We can turn a 5-limit major triad into a 5-limit major tetrad by adding
> either 15/8 or 5/3, depending on which of the two spans more generators in a
> particular temperament.
> This allows us to find 5-limit tetradic comma pumps with the maximum number
> of tones possible.
>
> For example, in meantone, 15/8 spans the highest number of generators and
> 5/4 comes right after that. Therefore, we'll go for 8:10:12:15 as our
> tetradic model. A possible maximal tetradic comma pump in meantone may
> consist the following tones:
> C, E, F, A, Bb, D, F#, G, C
> Which of the tones we finally use in our first chord or second chord or
> whatever, is then upon our personal decision.
> One possible rendition is:
> C-E-F-A, F-A-Bb-D, D-F#-G-B, G-B-C-E, C-E-F-A
>
> We can apply this concept to other temperaments like this:

Petr, these sound great, but I'm not sure I'm managing to play them
all right. Since I don't know what tones to hold constant in each
case, I end up landing on a lot of maj7#5 and min/maj7 and m7b5
chords. Is there a consistent pattern for some of these? I managed to
get Hanson's to work though and it sounded amazing.

Maybe I should just make a list of 5-limit renderings of all of the
hexads we use in 12-tet or something.

-Mike

🔗Kalle Aho <kalleaho@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> So in light of that, I hope you aren't going to tell me that you think
> that major 7 chords are dissonant because the 7th needs to resolve
> upward, because the strain may simply cause my feeble psyche to just
> snap. There's only so much one man can take, after all.

To make your inner life less straining you could adopt a distinction
between discordance and dissonance.

The very mild discordance of maj7 chord doesn't prevent it from being
dissonant in common practice harmony where 5-limit triads are the basic
harmonies. Dissonant isn't the same as painful, it just wants to
resolve.

That said, sevenths resolve even in the archetypal circle of fifths
jazz progressions.

Kalle

🔗Petr Pařízek <petrparizek2000@...>

Mike wrote:

> Petr, these sound great, but I'm not sure I'm managing to play them
> all right. Since I don't know what tones to hold constant in each
> case, I end up landing on a lot of maj7#5 and min/maj7 and m7b5
> chords. Is there a consistent pattern for some of these?

For the time being, if I want to prevent this from happening, the only way I've found so far is that in situations where the same step size occurs twice in a row in the interval sequence, I use one of them for one chord and the other for the next chord. For example, in the aforementioned meantone example, since there's a major third between Bb-D and also between D-F#, I use the D as the common tone of the two chords and the three other pitches are changed. This gives me a meantone pump where there's only one case in which one tone between adjacent chords is preserved, while two tones are preserved in the rest. This has obviously some limitations but it's better than nothing.

> I managed to
> get Hanson's to work though and it sounded amazing.

Yeah, I'm not sure if I'm going to finally make a sequence file for that one as well but I agree that the minor third progressions combined with this 8:10:12:15 stuff may sound even more "tonally distant" than my example in amity.

> Maybe I should just make a list of 5-limit renderings of all of the
> hexads we use in 12-tet or something.

Not sure what you're talking about but the idea sounds interesting.

Petr

🔗Mike Battaglia <battaglia01@...>

On Sun, Jul 17, 2011 at 6:16 AM, Kalle Aho <kalleaho@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > So in light of that, I hope you aren't going to tell me that you think
> > that major 7 chords are dissonant because the 7th needs to resolve
> > upward, because the strain may simply cause my feeble psyche to just
> > snap. There's only so much one man can take, after all.
>
> To make your inner life less straining you could adopt a distinction
> between discordance and dissonance.
>
> The very mild discordance of maj7 chord doesn't prevent it from being
> dissonant in common practice harmony where 5-limit triads are the basic
> harmonies. Dissonant isn't the same as painful, it just wants to
> resolve.

I think there's more to the issue of discordance vs dissonance than
that. For example, 5:6:7 sounds dissonant if you imagine the root as
5, and consonant if you imagine the root as 4.

> That said, sevenths resolve even in the archetypal circle of fifths
> jazz progressions.

Major 7ths don't resolve upward in jazz. They're used over the I chord
all the time. It's more common to hear major 7 chords than major
triads in jazz. But this is a more general facet of extended diatonic
harmony than something that has specifically to do with "jazz."

-Mike

🔗Mike Battaglia <battaglia01@...>

2011/7/17 Petr Pařízek <petrparizek2000@...>
>
> > I managed to
> > get Hanson's to work though and it sounded amazing.
>
> Yeah, I'm not sure if I'm going to finally make a sequence file for that one
> as well but I agree that the minor third progressions combined with this
> 8:10:12:15 stuff may sound even more "tonally distant" than my example in
> amity.

I hear amity as not being too tonally distant at all, really. It
sounds like a chord progression in 12-tet in which the pitch slowly
drifts until you end back from where you started. Which is what comma
pumps often sound like (although I'm really getting the hang of
porcupine as its own "thing" now), but amity's is so gradual that it
almost feels "artificial" to me. I feel like the non-major 7 one was
easier to hear as a "different" thing. Not sure why that is.

-Mike

🔗Petr Pařízek <petrparizek2000@...>

Mike wrote:

> I hear amity as not being too tonally distant at all, really. It
> sounds like a chord progression in 12-tet in which the pitch slowly
> drifts until you end back from where you started. Which is what comma
> pumps often sound like

Maybe that's because the first and last chord of the smaller part of the progression are essentially a syntonic comma apart (like F, Bb, Eb, C\, F\).

> (although I'm really getting the hang of
> porcupine as its own "thing" now), but amity's is so gradual that it
> almost feels "artificial" to me. I feel like the non-major 7 one was
> easier to hear as a "different" thing. Not sure why that is.

I thought about your porcupine suggestion yesterday and tried to make a list of step sizes which would make a maximal tetradic pump in some temperaments which I've selected for the time being.
Here I'm disregarding the fact if the pump goes in one direction or the other, I've just chosen the step sizes in such a way that all of them could be greater than 1/1.
If you find mistakes there, don't be surprised - I was doing it all in my head.

So, this is what I've got:
Meantone, porcupine, tetracot, superpyth, semisixth, unicorn, amity
5/4, 16/15

Hanson
3/2, 16/15

Magic, wuerschmidt
3/2, 10/9

Diaschismatic, negri, schismatic, passion, orwell, miracle
6/5, 10/9

Petr

🔗Kalle Aho <kalleaho@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sun, Jul 17, 2011 at 6:16 AM, Kalle Aho <kalleaho@...> wrote:
> >
> > --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> > >
> > > So in light of that, I hope you aren't going to tell me that you think
> > > that major 7 chords are dissonant because the 7th needs to resolve
> > > upward, because the strain may simply cause my feeble psyche to just
> > > snap. There's only so much one man can take, after all.
> >
> > To make your inner life less straining you could adopt a distinction
> > between discordance and dissonance.
> >
> > The very mild discordance of maj7 chord doesn't prevent it from being
> > dissonant in common practice harmony where 5-limit triads are the basic
> > harmonies. Dissonant isn't the same as painful, it just wants to
> > resolve.
>
> I think there's more to the issue of discordance vs dissonance than
> that. For example, 5:6:7 sounds dissonant if you imagine the root as
> 5, and consonant if you imagine the root as 4.

Always? Are you sure you understand the distinction? According to
Blackwood, 4:5:6:7 is concordant but dissonant in common practice
context.

> > That said, sevenths resolve even in the archetypal circle of fifths
> > jazz progressions.
>
> Major 7ths don't resolve upward in jazz. They're used over the I chord
> all the time. It's more common to hear major 7 chords than major
> triads in jazz. But this is a more general facet of extended diatonic
> harmony than something that has specifically to do with "jazz."

What's the significance of resolving/not resolving upwards? It still
resolves or at least appears to resolve. And I wasn't talking about
some "jazz harmony", just about that specific progression, whatever
it's called.

Kalle