diff --git a/packages/core/fraction.mjs b/packages/core/fraction.mjs
index ce614246..a339567f 100644
--- a/packages/core/fraction.mjs
+++ b/packages/core/fraction.mjs
@@ -6,6 +6,7 @@ This program is free software: you can redistribute it and/or modify it under th
 
 import Fraction from 'fraction.js';
 import { TimeSpan } from './timespan.mjs';
+import { removeUndefineds } from './util.mjs';
 
 // Returns the start of the cycle.
 Fraction.prototype.sam = function () {
@@ -64,6 +65,22 @@ Fraction.prototype.min = function (other) {
   return this.lt(other) ? this : other;
 };
 
+Fraction.prototype.mulmaybe = function (other) {
+  return other !== undefined ? this.mul(other) : undefined;
+};
+
+Fraction.prototype.divmaybe = function (other) {
+  return other !== undefined ? this.div(other) : undefined;
+};
+
+Fraction.prototype.addmaybe = function (other) {
+  return other !== undefined ? this.add(other) : undefined;
+};
+
+Fraction.prototype.submaybe = function (other) {
+  return other !== undefined ? this.sub(other) : undefined;
+};
+
 Fraction.prototype.show = function (/* excludeWhole = false */) {
   // return this.toFraction(excludeWhole);
   return this.s * this.n + '/' + this.d;
@@ -89,11 +106,24 @@ const fraction = (n) => {
 };
 
 export const gcd = (...fractions) => {
+  fractions = removeUndefineds(fractions);
+  if (fractions.length === 0) {
+    return undefined;
+  }
+
   return fractions.reduce((gcd, fraction) => gcd.gcd(fraction), fraction(1));
 };
 
 export const lcm = (...fractions) => {
-  return fractions.reduce((lcm, fraction) => lcm.lcm(fraction), fraction(1));
+  fractions = removeUndefineds(fractions);
+  if (fractions.length === 0) {
+    return undefined;
+  }
+
+  return fractions.reduce(
+    (lcm, fraction) => (lcm === undefined || fraction === undefined ? undefined : lcm.lcm(fraction)),
+    fraction(1),
+  );
 };
 
 fraction._original = Fraction;
diff --git a/packages/core/pattern.mjs b/packages/core/pattern.mjs
index 91174857..7291c12f 100644
--- a/packages/core/pattern.mjs
+++ b/packages/core/pattern.mjs
@@ -42,16 +42,16 @@ export class Pattern {
    */
   constructor(query, tactus = undefined) {
     this.query = query;
-    this._Pattern = true; // this property is used to detect if a pattern that fails instanceof Pattern is an instance of another Pattern
-    this.__tactus = Fraction(tactus); // in terms of number of steps per cycle
+    this._Pattern = true; // this property is used to detectinstance of another Pattern
+    this.tactus = tactus; // in terms of number of steps per cycle
   }
 
   get tactus() {
-    return this.__tactus ?? Fraction(1);
+    return this.__tactus;
   }
 
   set tactus(tactus) {
-    this.__tactus = Fraction(tactus);
+    this.__tactus = tactus === undefined ? undefined : Fraction(tactus);
   }
 
   setTactus(tactus) {
@@ -63,6 +63,10 @@ export class Pattern {
     return new Pattern(this.query, this.tactus === undefined ? undefined : f(this.tactus));
   }
 
+  get hasTactus() {
+    return this.tactus !== undefined;
+  }
+
   //////////////////////////////////////////////////////////////////////
   // Haskell-style functor, applicative and monadic operations
 
@@ -1181,7 +1185,7 @@ export const pm = s_polymeter;
  * @example
  * gap(3) // "~@3"
  */
-export const gap = (tactus) => new Pattern(() => [], Fraction(tactus));
+export const gap = (tactus) => new Pattern(() => [], tactus);
 
 /**
  * Does absolutely nothing..
@@ -1762,7 +1766,7 @@ export const { focusSpan, focusspan } = register(['focusSpan', 'focusspan'], fun
  */
 export const ply = register('ply', function (factor, pat) {
   const result = pat.fmap((x) => pure(x)._fast(factor)).squeezeJoin();
-  result.tactus = pat.tactus.mul(factor);
+  result.tactus = Fraction(factor).mulmaybe(pat.tactus);
   return result;
 });
 
@@ -1784,7 +1788,7 @@ export const { fast, density } = register(['fast', 'density'], function (factor,
   factor = Fraction(factor);
   const fastQuery = pat.withQueryTime((t) => t.mul(factor));
   const result = fastQuery.withHapTime((t) => t.div(factor));
-  result.tactus = factor.mul(pat.tactus);
+  result.tactus = factor.mulmaybe(pat.tactus);
   return result;
 });
 
@@ -1954,7 +1958,7 @@ export const zoom = register('zoom', function (s, e, pat) {
     return nothing;
   }
   const d = e.sub(s);
-  const tactus = pat.tactus.mul(d);
+  const tactus = pat.tactus.mulmaybe(d);
   return pat
     .withQuerySpan((span) => span.withCycle((t) => t.mul(d).add(s)))
     .withHapSpan((span) => span.withCycle((t) => t.sub(s).div(d)))
@@ -2408,15 +2412,15 @@ Pattern.prototype.stepJoin = function () {
 };
 
 export function _retime(timedHaps) {
-  const occupied_perc = timedHaps.filter((t, pat) => pat.tactus != undefined).reduce((a, b) => a.add(b), Fraction(0));
+  const occupied_perc = timedHaps.filter((t, pat) => pat.hasTactus).reduce((a, b) => a.add(b), Fraction(0));
   const occupied_tactus = removeUndefineds(timedHaps.map((t, pat) => pat.tactus)).reduce(
     (a, b) => a.add(b),
     Fraction(0),
   );
-  const total_tactus = occupied_perc.eq(0) ? 0 : occupied_tactus.div(occupied_perc);
+  const total_tactus = occupied_perc.eq(0) ? undefined : occupied_tactus.div(occupied_perc);
   function adjust(dur, pat) {
     if (pat.tactus === undefined) {
-      return [dur.mul(total_tactus), pat];
+      return [dur.mulmaybe(total_tactus), pat];
     }
     return [pat.tactus, pat];
   }
@@ -2454,7 +2458,10 @@ export function _match(span, hap_p) {
  * // The same as s("{bd sd cp}%4")
  */
 export const steps = register('steps', function (targetTactus, pat) {
-  if (pat.tactus.eq(0)) {
+  if (pat.tactus === undefined) {
+    return pat;
+  }
+  if (pat.tactus.eq(Fraction(0))) {
     // avoid divide by zero..
     return nothing;
   }
@@ -2520,10 +2527,16 @@ export function s_polymeter(...args) {
     return _polymeterListSteps(0, ...args);
   }
 
+  // TODO currently ignoring arguments without tactus...
+  args = args.filter((arg) => arg.hasTactus);
+
   if (args.length == 0) {
     return silence;
   }
   const tactus = args[0].tactus;
+  if (tactus.eq(Fraction(0))) {
+    return nothing;
+  }
   const [head, ...tail] = args;
 
   const result = stack(head, ...tail.map((pat) => pat._slow(pat.tactus.div(tactus))));
@@ -2543,8 +2556,26 @@ export function s_polymeter(...args) {
  * // the same as "bd sd cp hh hh".sound()
  */
 export function s_cat(...timepats) {
+  if (timepats.length === 0) {
+    return nothing;
+  }
   const findtactus = (x) => (Array.isArray(x) ? x : [x.tactus, x]);
   timepats = timepats.map(findtactus);
+  if (timepats.find((x) => x[0] === undefined)) {
+    const times = timepats.map((a) => a[0]).filter((x) => x !== undefined);
+    if (times.length === 0) {
+      return fastcat(...timepats.map((x) => x[1]));
+    }
+    if (times.length === timepats.length) {
+      return nothing;
+    }
+    const avg = times.reduce((a, b) => a.add(b), Fraction(0)).div(times.length);
+    for (let timepat of timepats) {
+      if (timepat[0] === undefined) {
+        timepat[0] = avg;
+      }
+    }
+  }
   if (timepats.length == 1) {
     const result = reify(timepats[0][1]);
     result.tactus = timepats[0][0];
@@ -2588,7 +2619,7 @@ export function s_alt(...groups) {
   for (let cycle = 0; cycle < cycles; ++cycle) {
     result.push(...groups.map((x) => (x.length == 0 ? silence : x[cycle % x.length])));
   }
-  result = result.filter((x) => x.tactus > 0);
+  result = result.filter((x) => x.hasTactus && x.tactus > 0);
   const tactus = result.reduce((a, b) => a.add(b.tactus), Fraction(0));
   result = s_cat(...result);
   result.tactus = tactus;
@@ -2599,6 +2630,9 @@ export function s_alt(...groups) {
  * *EXPERIMENTAL* - Retains the given number of steps in a pattern (and dropping the rest), according to its 'tactus'.
  */
 export const s_add = stepRegister('s_add', function (i, pat) {
+  if (!pat.hasTactus) {
+    return nothing;
+  }
   if (pat.tactus.lte(0)) {
     return nothing;
   }
@@ -2627,6 +2661,10 @@ export const s_add = stepRegister('s_add', function (i, pat) {
  * *EXPERIMENTAL* - Removes the given number of steps from a pattern, according to its 'tactus'.
  */
 export const s_sub = stepRegister('s_sub', function (i, pat) {
+  if (!pat.hasTactus) {
+    return nothing;
+  }
+
   i = Fraction(i);
   if (i.lt(0)) {
     return pat.s_add(Fraction(0).sub(pat.tactus.add(i)));
@@ -2647,10 +2685,15 @@ export const s_contract = stepRegister('s_contract', function (factor, pat) {
  */
 Pattern.prototype.s_taperlist = function (amount, times) {
   const pat = this;
+
+  if (!pat.hasTactus) {
+    return [pat];
+  }
+
   times = times - 1;
 
   if (times === 0) {
-    return pat;
+    return [pat];
   }
 
   const list = [];
@@ -2675,6 +2718,10 @@ export const s_taperlist = (amount, times, pat) => pat.s_taperlist(amount, times
 export const s_taper = register(
   's_taper',
   function (amount, times, pat) {
+    if (!pat.hasTactus) {
+      return nothing;
+    }
+
     const list = pat.s_taperlist(amount, times);
     const result = s_cat(...list);
     result.tactus = list.reduce((a, b) => a.add(b.tactus), Fraction(0));
@@ -2726,7 +2773,7 @@ export const chop = register('chop', function (n, pat) {
   const func = function (o) {
     return sequence(slice_objects.map((slice_o) => Object.assign({}, o, slice_o)));
   };
-  return pat.squeezeBind(func).setTactus(pat.tactus.mul(n));
+  return pat.squeezeBind(func).setTactus(Fraction(n).mulmaybe(pat.tactus));
 });
 
 /**
diff --git a/packages/core/test/pattern.test.mjs b/packages/core/test/pattern.test.mjs
index 9c17d8be..4bed34cb 100644
--- a/packages/core/test/pattern.test.mjs
+++ b/packages/core/test/pattern.test.mjs
@@ -47,6 +47,10 @@ import {
   time,
   run,
   pick,
+  stackLeft,
+  stackRight,
+  stackCentre,
+  s_cat,
 } from '../index.mjs';
 
 import { steady } from '../signal.mjs';
@@ -1136,6 +1140,41 @@ describe('Pattern', () => {
       expect(sequence({ s: 'bev' }, { s: 'amenbreak' }).splice(4, sequence(0, 1, 2, 3)).tactus).toStrictEqual(
         Fraction(4),
       );
+      expect(sequence({ n: 0 }, { n: 1 }, { n: 2 }).chop(4).tactus).toStrictEqual(Fraction(12));
+      expect(
+        pure((x) => x + 1)
+          .setTactus(3)
+          .appBoth(pure(1).setTactus(2)).tactus,
+      ).toStrictEqual(Fraction(6));
+      expect(
+        pure((x) => x + 1)
+          .setTactus(undefined)
+          .appBoth(pure(1).setTactus(2)).tactus,
+      ).toStrictEqual(Fraction(2));
+      expect(
+        pure((x) => x + 1)
+          .setTactus(3)
+          .appBoth(pure(1).setTactus(undefined)).tactus,
+      ).toStrictEqual(Fraction(3));
+      expect(stack(fastcat(0, 1, 2), fastcat(3, 4)).tactus).toStrictEqual(Fraction(6));
+      expect(stack(fastcat(0, 1, 2), fastcat(3, 4).setTactus(undefined)).tactus).toStrictEqual(Fraction(3));
+      expect(stackLeft(fastcat(0, 1, 2, 3), fastcat(3, 4)).tactus).toStrictEqual(Fraction(4));
+      expect(stackRight(fastcat(0, 1, 2), fastcat(3, 4)).tactus).toStrictEqual(Fraction(3));
+      // maybe this should double when they are either all even or all odd
+      expect(stackCentre(fastcat(0, 1, 2), fastcat(3, 4)).tactus).toStrictEqual(Fraction(3));
+      expect(fastcat(0, 1).ply(3).tactus).toStrictEqual(Fraction(6));
+      expect(fastcat(0, 1).setTactus(undefined).ply(3).tactus).toStrictEqual(undefined);
+      expect(fastcat(0, 1).fast(3).tactus).toStrictEqual(Fraction(6));
+      expect(fastcat(0, 1).setTactus(undefined).fast(3).tactus).toStrictEqual(undefined);
+    });
+  });
+  describe('s_cat', () => {
+    it('can cat', () => {
+      expect(sameFirst(s_cat(fastcat(0, 1, 2, 3), fastcat(4, 5)), fastcat(0, 1, 2, 3, 4, 5)));
+      expect(sameFirst(s_cat(pure(1), pure(2), pure(3)), fastcat(1, 2, 3)));
+    });
+    it('calculates undefined tactuses as the average', () => {
+      expect(sameFirst(s_cat(pure(1), pure(2), pure(3).setTactus(undefined)), fastcat(1, 2, 3)));
     });
   });
   describe('s_taper', () => {
diff --git a/packages/tonal/tonal.mjs b/packages/tonal/tonal.mjs
index a2e01092..4c25d23e 100644
--- a/packages/tonal/tonal.mjs
+++ b/packages/tonal/tonal.mjs
@@ -188,61 +188,66 @@ export const scaleTranspose = register('scaleTranspose', function (offset /* : n
  * .s("piano")
  */
 
-export const scale = register('scale', function (scale, pat) {
-  // Supports ':' list syntax in mininotation
-  if (Array.isArray(scale)) {
-    scale = scale.flat().join(' ');
-  }
-  return (
-    pat
-      .fmap((value) => {
-        const isObject = typeof value === 'object';
-        let step = isObject ? value.n : value;
-        if (isObject) {
-          delete value.n; // remove n so it won't cause trouble
-        }
-        if (isNote(step)) {
-          // legacy..
-          return pure(step);
-        }
-        let asNumber = Number(step);
-        let semitones = 0;
-        if (isNaN(asNumber)) {
-          step = String(step);
-          if (!/^[-+]?\d+(#*|b*){1}$/.test(step)) {
-            logger(
-              `[tonal] invalid scale step "${step}", expected number or integer with optional # b suffixes`,
-              'error',
-            );
-            return silence;
+export const scale = register(
+  'scale',
+  function (scale, pat) {
+    // Supports ':' list syntax in mininotation
+    if (Array.isArray(scale)) {
+      scale = scale.flat().join(' ');
+    }
+    return (
+      pat
+        .fmap((value) => {
+          const isObject = typeof value === 'object';
+          let step = isObject ? value.n : value;
+          if (isObject) {
+            delete value.n; // remove n so it won't cause trouble
           }
-          const isharp = step.indexOf('#');
-          if (isharp >= 0) {
-            asNumber = Number(step.substring(0, isharp));
-            semitones = step.length - isharp;
-          } else {
-            const iflat = step.indexOf('b');
-            asNumber = Number(step.substring(0, iflat));
-            semitones = iflat - step.length;
+          if (isNote(step)) {
+            // legacy..
+            return pure(step);
           }
-        }
-        try {
-          let note;
-          if (isObject && value.anchor) {
-            note = stepInNamedScale(asNumber, scale, value.anchor);
-          } else {
-            note = scaleStep(asNumber, scale);
+          let asNumber = Number(step);
+          let semitones = 0;
+          if (isNaN(asNumber)) {
+            step = String(step);
+            if (!/^[-+]?\d+(#*|b*){1}$/.test(step)) {
+              logger(
+                `[tonal] invalid scale step "${step}", expected number or integer with optional # b suffixes`,
+                'error',
+              );
+              return silence;
+            }
+            const isharp = step.indexOf('#');
+            if (isharp >= 0) {
+              asNumber = Number(step.substring(0, isharp));
+              semitones = step.length - isharp;
+            } else {
+              const iflat = step.indexOf('b');
+              asNumber = Number(step.substring(0, iflat));
+              semitones = iflat - step.length;
+            }
           }
-          if (semitones != 0) note = Note.transpose(note, Interval.fromSemitones(semitones));
-          value = pure(isObject ? { ...value, note } : note);
-        } catch (err) {
-          logger(`[tonal] ${err.message}`, 'error');
-          value = silence;
-        }
-        return value;
-      })
-      .outerJoin()
-      // legacy:
-      .withHap((hap) => hap.setContext({ ...hap.context, scale }))
-  );
-});
+          try {
+            let note;
+            if (isObject && value.anchor) {
+              note = stepInNamedScale(asNumber, scale, value.anchor);
+            } else {
+              note = scaleStep(asNumber, scale);
+            }
+            if (semitones != 0) note = Note.transpose(note, Interval.fromSemitones(semitones));
+            value = pure(isObject ? { ...value, note } : note);
+          } catch (err) {
+            logger(`[tonal] ${err.message}`, 'error');
+            value = silence;
+          }
+          return value;
+        })
+        .outerJoin()
+        // legacy:
+        .withHap((hap) => hap.setContext({ ...hap.context, scale }))
+    );
+  },
+  true,
+  true, // preserve tactus
+);